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Little Giant (1946)
Approved | 1h 31min | Comedy | 22 February 1946 (USA)
Lou Costello plays a country bumpkin vacuum-cleaner salesman, working for the company run by the crooked Bud Abbott. To try to keep him under his thumb, Abbott convinces Costello that he's ... See full summary »
Walter DeLeon (screenplay) (as Walter De Leon), Paul Jarrico (story) | 1 more credit »
Bud Abbott, Lou Costello, Brenda Joyce | See full cast & crew »
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Watched (1945-1949)
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Actor - Bud Abbott and Lou Costello
Search for "Little Giant" on Amazon.com
Title: Little Giant (1946)
Bud Abbott ... Eddie L. Morrison / T.S. Chandler
Lou Costello ... Benny Miller
Brenda Joyce ... Miss Ruby Burke
Jacqueline deWit ... Hazel Temple Morrison (as Jacqueline de Wit)
George Cleveland ... Clarence Goodring
Elena Verdugo ... Martha Hill
Mary Gordon ... Ma Miller
Pierre Watkin ... P.S. Van Loon
Donald MacBride ... Bus Conductor
Victor Kilian ... Gus Anderson - Salesman
George Chandler ... O'Brien - Salesman
Joe Kirk ... Salesman
Harry Brown ... Larry, Salesman
Beatrice Gray ... Miss King
Margaret Dumont ... Mrs. Hendrickson
Lou Costello plays a country bumpkin vacuum-cleaner salesman, working for the company run by the crooked Bud Abbott. To try to keep him under his thumb, Abbott convinces Costello that he's a crackerjack salesman. This comedy is somewhat like The Time of Their Lives (1946), in that Abbott and Costello don't have much screen time together and there are very few vaudeville bits woven into the plot. Written by Dan Weckerly <Daniel_Weckerly@providentmutual.com>
farce | cigarette smoking | 1940s | treachery | vacuum cleaner salesman | See All (23) »
Also - See Alvino Rey and Orchestra in "Swinging Down Scale" (print ad - Lubbock Morning Avalanche - Tower Theatre - Lubbock, Texas - August 16, 1946) See more »
William A. Seiter had previously directed The Marx Brothers and was able to get Margaret Dumont to do a cameo as one of Benny's intended customers. See more »
Abbott's toupee shifts noticeably during the "7 times 13 = 28" scene. (The "shift" is due to the fact that the scene was filmed after principal photography was completed. It was felt that at least one classic "routine" had to be inserted into the picture. You will notice that Lou is also heavier during this footage. Also filmed at this time was the routine with Sidney Fields, replacing a less confrontational sequence filmed with Eddy Waller.) See more »
Benny Miller: I feel like a tin can with a dog's tail tied to it.
Featured in The World of Abbott and Costello (1965) See more »
A truly unique Abbott and Costello film that I really liked...really!
19 August 2009 | by MartinHafer – See all my reviews
This is one of Abbott and Costello's most unusual films, as it's the first of two that made where the characters were NOT friends. In addition, Bud Abbott plays dual roles--one a nice enough guy and the other a total scum-bag! Plus, and here's the oddest part, the film is a traditional story in many ways-as both play honest-to-goodness people! As a result, it really isn't a comedy per se--as the film is not built around gags but people. Sure, there are a few laughs here and there, but that is all.
While I know that the film was a bit of a flop and many people disliked its style, I frankly liked it because it was such a departure. You see, by 1946, the team had already made 16 films in only 6 years--and all but one of them (WHO DONE IT!) followed roughly the same formula. With this formula, there was a love story, some Abbott and Costello comedy and lots of singing. As far as the love story goes, this time it actually involves Lou and a girl back home. However, there is no singing and little what anyone would consider comedy.
I think one thing that bothered people is the pathos in the film. Lou plays a nice guy who gets hurt pretty badly at times in the film. You want him to win but time and again, jerks take advantage of him. Near the very end, this hit practically epic proportions, though smartly, the film didn't stay mired too long in pathos--coming to a nice and quick resolution.
The film begins with Lou living on the farm with his Mom. He wants to make good, so he's been taking a correspondence course in salesmanship. Unfortunately, he isn't very good at it and when he goes to the big city to make his mark, he makes a mess of it. He loses his job and gets another job with the same vacuum cleaner company in another town. However, in an odd twist, his co-workers play a joke on him--convincing him he's psychic. The gag works too well, as Lou is convinced it's real. The jokes on them when he turns into an amazing salesman--setting a sales record the very next day.
As for Bud, in Los Angeles, he plays a crooked and thoroughly nasty jerk. He takes pleasure in firing Lou and it's interesting to see them working against each other instead of with each other. In his next job, Lou goes to work for Bud's cousin--played once again by Bud (with a slightly different hairdo). This time, he's more of a normal guy and confides in Lou that he can't stand his stupid cousin in L.A.! It was an interesting acting challenge for Bud--as rarely did any of his characters in other film have any depth. Here, he plays two parts and quite well. In fact, it worked out well enough that they had him do the same in the next film, THE TIME OF THEIR LIVES. Another, more practical reason they did this was because Bud and Lou were having a horrible spat at the time--and they would only play in films where they acted separately!! Fortunately for the team, the managed to patch things up for other films.
One of the only comedy routines in the film was also used in one of the team's earlier films, IN THE NAVY. This is the funny math routine where Lou explains (rather convincingly) that 7x13=21! While it is a retread, it's redone well.
Overall, while this is hated by most people, I liked the film a lot--nearly giving it a 9. Why? Originality and both Bud and Lou stretched themselves--trying new things even if the public wasn't 100% ready for THIS big a change. Maybe much of the reason I respect this film so much is that I have re-seen all the Abbott and Costello films leading up to LITTLE GIANT and it just felt like a breath of fresh air seeing such a completely original film.
Q: What is the Hercules Vacuum Cleaner Company's slogan?
Q: Midwest Premiere Happened When & Where?
The Boy Wonder See more »
Universal Studios - 100 Universal City Plaza, Universal City, California, USA
C.S. Productions, Universal Pictures See more »
Mono (Western Electric Recording)


Going along with the health-themed posts this week, I've invited food blogger Hilary from For the Love of Basil to join in on the fun!
Where did your passion for health come from?
I have always been passionate about health, nutrition, and cooking. I pursued my interest in health by receiving my Master's of Public Health and became certified as a nutrition consultant. During my work with the World Health Organization in Fiji and interning with the American Diabetes Association, I witnessed how crucial it is to have a healthy relationship with food and eat food that is good for your body. The main reasons people don't eat healthy is (1) they don't know a lot about nutrition or eating healthy or (2) they know what they should be eating, but wither a crazy schedule or lack of desire (which usually changes once people understand the impacts of a healthy diet) prevent them from actually doing it! I started my food blog both as a creative outlet and a way to share nutritional information and easy healthy recipes. Good food doesn't have to be hard!
I have always love smoothies, but have fallen head-over-heels in love again ever since adding avocados to my smoothies! My Avocado & Mango Protein Smoothie recipe is definitely my favorite way to start my day.
The avocado provides a creamy texture that is hard to find in dairy-free smoothies, while the mango adds a natural subtle sweetness, coconut water to keep you hydrated, protein powder to keep you full, but this Avocado & Mango Protein Smoothie truly becomes superb by adding lime and mint.
Why all the hype about avocados?
There are so many reasons that avocados deserve the credit and craze they have been given!
2. An avocado has more potassium than a banana.
3. Avocados are packed with fiber, folate, and vitamin K.
4. The healthy fats in avocados make it easier for the body to absorb all the other good stuff in smoothies like vitamins A, C, and K.
5. They make a great replacement for butter in healthy baking.
6. Finally, let's face it- there are a million delicious ways to add avocado into your diet. My husband's favorite way is my Avocado Hummus!
THANK YOU HILARY!! I can't wait to try the smoothie! Also, Isaac recently discovered a love for hummus, so I might have to make the Avocado Hummus for him too! Have a great day everyone!
YUM I love avocado hummus, and just avocados in general. Thanks for sharing!


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Enjoy the flavors of a delicious BLT with the added bonus of Johnsonville Summer Sausage. This topping is a snap to make, using mayonnaise, whipped cream cheese and, of course, the BLT. Throw in Monterey Jack for some added goodness, and top it on a delicious slice of sausage. For the flavor fanatics, stack the slices for a double-decker or triple-decker taste explosion!
In a bowl, combine the mayonnaise and cream cheese until smooth. Fold in the bacon, lettuce, tomato and cheese.
Spoon BLT mixture on top of sliced summer sausage.


Small animals spend the bulk of their time inside their cage or hutch, so it's important that you provide them with opportunities to exercise and keep occupied when you are unavailable to play with them.
Our range of exercise equipment, gnaws and toys for small animals includes a huge range of items for pets of all sizes and needs from tiny dwarf hamsters to ferrets and rabbits. An exercise wheel is an indispensable addition to a cage which will make the most out of your pets' confined living space. Providing a wheel for them to run on is a great way to prevent common ailments such as obesity.
Gnaws are a great way to help keep teeth from getting too long, and providing your pet with a gnaw can distract them from chewing on their cage!


Three little words that every parent covets. Three little words that mean more than anything when you're expecting a baby. These three little words are the deepest prayer of my heart right now. When you're waiting for a baby, nothing else matters. Sure, we look forward to the first crawl. We look forward to first words, and first steps, and the first day of school. That's what being a parent is about, but none of it matters more than happy and healthy.
Today, Beth and I are headed to the perinatologist for an amniocentesis. It's hard to really wrap my brain around something so simple, and yet so complex. The reality is that an amniocentesis takes about 45 seconds. It's routine. It's a simple procedure with a needle and some fluid. And yet, it's a miracle of the complex - that a simple needle can reach down into our little baby's world and tell us so much.
We won't know anything today. Today is the simple part. The complicated part comes later. The complicated part is waiting for the little cells to grow, and give us a picture of our little baby. The complicated part is waiting the 10 days for answers. The complicated part is uncovering what might be going on inside. The complicated part is understanding that the God who loves us more than anything, created this little baby for us. And he created in us this love and desire as parents, for everything to be ok.
So we keep praying. We keep hoping and praying for this little person inside Beth. We keep imagining and dreaming of first steps, and first birthdays, and first cuddles and first cries. We hope and pray for all of those things, but most of all... right now, we pray for happy and healthy.


Filters allow you to show only a certain subset of your Story's characters. This is particularly useful as your Story grows to have so many characters that you begin to get overwhelmed and find it hard to navigate. With filters, you can define certain criteria based on any of your characters' information and only show those characters who fit your criteria.
Navigate to Characters -> Filters. Then, you can add a Filter by clicking on the 'Add Filter' button. For each filter you add, you'll want to choose a field (like 'First Name', 'Relationships', or 'Events'), and then specify your desired value. For standard text fields, your options are Exactly Equals or Contains. Exactly Equals will only show those characters whose value for the chosen field is identical to the value you provide. Contains will show any characters whose value for the chosen field contains the value you provide. For example, if you add the filter First Name Exactly Equals Hanna, it will show characters named Hanna but not Hannah. However, if you changed the filter to be First Name Contains Hanna, then both Hanna and Hannah would be displayed.
To delete a filter, just click the Delete button next to it. Finally, once you've added your filters, click Apply Filters to apply them.
Next, read about Importing Characters From CSV.


A lightning-caused fire that started in Yellowstone National Park on Aug. 4 swelled to more than 900 acres on Sunday, and is currently the only active fire burning in the park, according to a statement released by the National Park Service Sunday afternoon.
The Fawn Fire is burning near Fawn Pass in the northwest section of Yellowstone National Park, approximately 35 miles southeast of Big Sky, Montana. The blaze, which was reported by an outfitter the evening of Aug. 4, grew significantly from 500 acres Saturday to 915 acres Sunday.
– Bighorn Pass trail eastbound at the junction of the cut-off trail to the Fawn Pass Trail.
– Bighorn Pass trail westbound at Bighorn Pass.
– Fawn Pass trail eastbound at the junction of the cut-off trail to the Big Horn Pass Trail.
– Fawn Pass trail westbound at campsite 1F1.
The Fawn Fire from the air on Aug. 5.
Rain and thunderstorm activity Saturday and Sunday deposited 0.29 inches of rain and kept temperatures on the cooler side.
"Fire activity is expected to increase in the coming days [particularly] in the afternoons as temperatures rise, relative humidity levels drop, and gusty winds increase," the press release said.
Fire staff are putting structure protection around the Fawn Pass cabin today by setting up sprinkler systems and wrapping the structure in protective material. Yellowstone Superintendent Dan Wenk approved a plan to monitor the blaze, according to the statement, and two three-person crews are poised to respond to possible new starts in the park, due to thunderstorm activity.
Springhill community. Fire crews from the Forest Service and the Central Valley Fire Department reported to the fire on Aug. 6 and received some help in the form of rain over the weekend.
A smokejumper team made significant progress Saturday night, the Forest Service said in a statement Sunday, and were able to construct fireline around the blaze. A statement Monday from the Forest Service indicated that the Limestone Fire is now 100 percent contained.
An earlier version of this article reported that the Limestone Fire was on Forest Service land. The fire was on private property.
This article was updated at 4:30 p.m. Sunday. EBS will continue to update these stories as information becomes available.


Hello, everyone! So if you're an 80s child like me, today's post title brings back fond memories of one of our favorite movies- The Goonies! In fact, I won this polish in one of Fallen Angel Beauty Bar's Facebook live events because I knew where the name Chester Copper Pot came from. Fallen Angel Beauty Bar is so brand spankin' new, she doesn't have a site up yet, but you can see her products on her Facebook and Instagram pages and order directly from her. I'll have those links for you at the end of this post. Back to the polish!
Chester Copper Pot is a really complex polish and difficult to describe- thank god for pictures! It's sort of like a copper and hot pink duochrome with a teal flash. The effect is really beautiful. The formula is lovely and it goes on opaque in 2-3 coats, depending on thickness of each coat.
So, Angela, the owner of Fallen Angel Beauty Bar does more than polish, hence the reasoning behind her brand's name. So far, I have her Himalayan Pink Salt foot scrub bar, Citrus cuticle balm, lip scrub, and Whipped Body Butter in the scent Medicine Magick. Her products work so well. I'm sure enough of us are trying to get our feet sandal and flip flop ready. Her scrub bar worked so amazingly well I was stunned. She really has a knack for this stuff! If you're interested in checking her out, go to Fallen Angel Beauty Bar on Facebook to join the group and see her awesome products. You can also check her Insta out at http://instagram.com/fallen.angel.beauty.bar/. Her customer service is great and she's worth looking into!
I'll see you all tomorrow with some nail art I did with Chester Copper Pot!
I don't know this brand but new indies are always exciting!
I totally read your posts in reverse, but Chester Copper Pot by Fallen Angel Beauty Bar is beautiful!
Very pretty polish! I hope to hear more about this brand.


Replication stress results in various forms of aberrant replication intermediates that need to be resolved for faithful chromosome segregation. Structure-specific endonucleases (SSEs) recognize DNA secondary structures rather than primary sequences and play key roles during DNA repair and replication stress. Holliday junction resolvase MUS81 (methyl methane sulfonate (MMS), and UV-sensitive protein 81) and XPF (xeroderma pigmentosum group F-complementing protein) are a subset of SSEs that resolve aberrant replication structures. To ensure genome stability and prevent unnecessary DNA breakage, these SSEs are tightly regulated by the cell cycle and replication checkpoints. We discuss the regulatory network that control activities of MUS81 and XPF and briefly mention other SSEs involved in the resolution of replication intermediates.
The DNA replication fork is sensitive to a variety of intrinsic and extrinsic stresses (reviewed in [1,2]). Endogenous blocks include collisions with transcription apparatus, natural pausing sites, and unusual DNA structures or sequences (reviewed in ). Highly repetitive DNA sequences (e.g., ribosomal DNA, telomeres) or common fragile sites (CFS) are also more prone to replication stress (reviewed in [4,5]). External agents that disrupt replication include depletion of deoxyribonucleotide triphosphate (dNTP) by hydroxyurea (HU) and DNA lesions caused by ultraviolet (UV) radiation, alkylating agents such as methyl methane sulfonate (MMS), or the topoisomerase inhibitor camptothecin (CPT) (reviewed in ).
Replication stress can result in accumulation of single stranded DNA, chromosome breaks, and rearrangements, which are deleterious to the cell (reviewed in [1,2]). Additionally, it may generate aberrant intermediates including DNA secondary structures, DNA lesions, and protein-DNA complexes (reviewed in ). Not surprisingly, increased replication stress is now recognized as a contributor to oncogenesis (e.g., reviewed in [6,7]).
A subset of structure-specific endonucleases (SSEs) that recognize specific DNA structures rather than DNA sequences, plays a crucial role in processing these aberrant structures to ensure replication fork stability and progression (reviewed in ). These SSEs are essential to maintaining genome stability, coordinating with the cell cycle to ensure that cells do not enter mitosis with structures that would promote improper chromosome segregation and breakage (reviewed in ). In this review we describe SSEs involved in processing DNA replication intermediates directly or indirectly regulated by the replication checkpoint (Figure 1). We pay particular attention to two conserved, related SSEs: Mus81 (MMS and UV-sensitive protein 81) and XPF (xeroderma pigmentosum group F-complementing protein).
The Mus81 protein was identified for its role in processing complex branched DNA structures, including Holliday junctions, that form after complementary strand exchange between homologous sequences (reviewed in [27,28,29,30,31,32]). Mus81 can resolve synthetic Holliday junction structures in vitro [31,32] and has a high affinity for branched duplex DNA and replication fork substrates . Consistent with this, loss of mus81 leads to severed meiotic defects, resulting in abnormal chromosomal segregation defects in yeasts [31,34,35]. In fission yeast (Schizosaccharomyces pombe), it is essential to complete sister chromatid exchange at the mating locus [31,36].
Mus81-dependent resolution of entangled sister chromatids is essential for survival of cells that depend on homology-directed repair of collapsed replication forks [36,37]. In human cells, MUS81 is similarly needed for replication fork restart after replication stress inducing agents [38,39,40,41]. MUS81-deficient cells have decreased viability upon low-dose exposure to these replication inhibitors . Importantly, fork restart in BRCA2 (breast cancer–associated protein 2)-deficient cells requires MUS81-dependent cleavage of partially resected, regressed forks . In addition to resolving replication intermediates, compensatory DNA synthesis during mitosis and cleavage of mitotic interlinks to allow chromosomal segregation also require MUS81 . These and many more studies demonstrate Mus81 plays a critical role to resolve replication and recombination intermediates and ensure proper chromosome segregation during cell division.
But Mus81 is a double-edged sword. Unregulated activity can have deleterious effects. Mus81 causes replication stress-induced double stranded breaks (DSB) in mammalian cells  and promotes deletion mutations in polα mutant fission yeast . When an active replication fork converges on a collapsed fork, replication termination is prone to Mus81-dependent deletions between repetitive DNA sequences in fission yeast . In human cells, oncogene-induced chromosomal breakage involves MUS81 activity . These findings suggest that tight regulation of Mus81 is necessary to repair replication-associated DNA structures without inducing unnecessary DNA cleavage.
A key component of that regulation is cell cycle- and checkpoint-dependent regulation of Mus81. These restrict its activity to later in the cell cycle in unstressed cells. The Mus81 enzyme forms a complex with Eme1 (essential meiotic endonuclease 1) which creates a stable interaction with a DNA substrate for the complex . Phosphorylation of Eme1 by various cell cycle kinases provide one mechanism to regulate Mus81 activity. In budding yeast (Saccharomyces cerevisiae), Mus81 forms a complex with the Eme1 ortholog Mms4 (methyl methane sulfonate sensitivity protein 4) . Mus81-Mms4S.c. is activated in a cell cycle-dependent manner and depends on phosphorylation of Mms4S.c. by the cell cycle kinases Cdc28S.c. (CDK1 in human) and Cdc5S.c. (PLK1 in human) at the G2/M transition (Figure 2) [48,49,50]. This restricts Mus81-Mms4S.c. activity during S-phase to prevent unnecessary cleavage of DNA substrates while DNA replication is occurring [48,51]. Via the scaffold protein Rtt107S.c., Cdc7-Dbf4S.c. (Dbf4-dependent kinase, DDK) interacts with and phosphorylates Mus81-Mms4, which is required for Mus81 activation during mitosis .
In fission yeast, which spends of most of its lifetime in G2 phase, Mus81-Eme1S.p. activity is upregulated in response to DNA damage . Cdc2S.p. (CDK1 in human) phosphorylation of Eme1S.p. primes it for phosphorylation and activation by the DNA damage sensor and checkpoint activator Rad3S.p. (ATR in human) (Figure 2) . Mus81-Eme1 cleavage of replication intermediates in in turn may have a role in activation or propagation of checkpoint pathways. Deletion of Mus81S.p. in replication stress-induced, temperature-sensitive Mcm4 helicase mutant (mcm4-ts) results in failure to maintain the DNA damage checkpoint and in subsequent abnormal chromosomal segregation .
In human cells, MUS81 is up-regulated at the onset of mitosis and has two partners, EME1 and EME2 [8,55]. Approximately 80% of MUS81 is associated with EME1 while the remaining 20% is associated with EME2 (reviewed in ). It is not obvious whether EME1 or EME2 is responsible for S phase-specific functions of MUS81 [55,57]. Interestingly, MUS81-EME1 activity is needed for maintaining replication fork speed  while MUS81-EME2 activity promotes replication fork restart and chromosomal stability .
MUS81-EME1 activity in human cells peaks during M phase after hyperphosphorylation of EME1 by cell cycle kinases CDK1 and PLK1 (Figure 2) [12,58,59]. Uninhibited CDK1 activity results in chromosomal fragmentation following premature activation of MUS81 , further linking CDK to MUS81 activity. PLK1 promotes DNA repair protein BRCA1 recruitment to facilitate MUS81-mediated fork cleavage coupled with a break-induced replication . Moreover, PLK1 interaction with BRCA1 and CDK1 activation of RECQ5 DNA helicase promotes MUS81-EME1 recruitment to CFS . A recent study showed that the pleiotropic serine/threonine kinase CK2 kinase is able to phosphorylate MUS81 in late-G2/mitosis and upon mild replication stress to promote its association with EME1 and scaffold protein SLX4, another stimulator of MUS81 activity . These findings show that cell cycle-dependent kinases not only play a crucial role in restricting Mus81 activity to appropriate timing of the cell cycle but also contribute to Mus81-dependent DNA repair.
Other regulators down-regulate S-phase activity of MUS81. WEE1, a well-known inhibitor of CDKs, suppresses MUS81 activity during S-phase by: (1) Potentially phosphorylating MUS81, (2) by inhibiting CDK2 and thereby limiting origin firing and replication stress, and (3) by restraining CDK1 that phosphorylates and activates EME1 and scaffold protein SLX4 (Figure 2) (reviewed in ). In the absence of WEE1, MUS81-EME1 activity results in unnecessary replication fork cleavage, leading to accumulation of DNA damage [61,66]. Deletion of MUS81 in the absence of WEE1 reduces DSB  but does not prevent activation of ATR and CHK1 , suggesting that MUS81 activity is downstream of replication fork stalling and S-phase checkpoint. This is also evidenced by the detrimental MUS81-dependent processing of replication intermediates following CHK1 inhibition [68,69,70]. Although the mechanistic details are unknown, these findings indicate that CHK1 down-regulates MUS81 in human cells (Figure 3).
Unlike CDK1-and PLK1-regulated control of MUS81-EME1 activity, the control of MUS81-EME2 activity is not well-established despite the evidence that MUS81-EME2 is responsible for the DNA damage during premature entry to mitosis upon WEE1 inhibition . Because deletion of MUS81 or EME2 delays premature entry into mitosis induced by WEE1 inhibition, this suggests that regulating MUS81-EME2 activity may be the mechanism by which WEE1 prevents premature mitotic entry (Figure 2) [8,62].
During replication stress, Mus81 plays a crucial role in processing abnormal replication intermediates. It is recruited to sites of replication blockage to resolve replication intermediates and inhibits anaphase bridge formation, preventing chromosome mis-segregation and transmission of damaged DNA to daughter cells (reviewed in [71,72,73]). Loss of Mus81 attenuates recovery of stalled replication forks and makes cells hypersensitive to DNA damaging agents that obstruct replication fork progression [29,38,39,74,75,76,77]. Paradoxically, although Mus81 is required to resolve aberrant replication intermediates, it can also create DNA breaks that threaten genomic stability. This is why Mus81 regulation during replication stress is crucial. Upon replication disturbance, the replication checkpoint pathway is activated to resolve replication hindrances and to delay mitosis until the replication stress is relieved (reviewed in [27,78]).
Cds1S.p. is the fission yeast replication checkpoint effector (Figure 3). In budding yeast, the Cds1S.p. homolog Rad53S.c. is the effector of both the DNA damage checkpoint and the replication checkpoint. Fission yeast Cds1S.p. acts downstream of DNA-dependent protein kinase-like family Rad3S.p. (Mec1S.c./ATR in human) (reviewed in ). Upon replication stress, a conserved mediator protein Mrc1S.p. (CLASPIN in human) is phosphorylated by Rad3S.p. which then recruits Cds1S.p. to stalled replication forks to be activated (reviewed in [80,81]).
In fission yeast, Cds1S.p. is necessary to prevent accumulation of aberrant replication intermediates, indicating that Cds1S.p. regulates resolution of damaging DNA structures at replication forks (reviewed in ) [86,87,88,89,90,91]. Mus81S.p. is a key target downstream of Cds1S.p. [29,31,32,44]. This S-phase checkpoint kinase tightly regulates Mus81S.p. to prevent uncontrolled nuclease activity during DNA replication. Mus81S.p. associates with 2–5% of the Cds1S.p. protein through a forkhead-associated (FHA) domain on Cds1S.p. and is phosphorylated in Cds1S.p.-dependent manner upon replication stress [29,92]. Acute HU treatment results in phosphorylated Mus81S.p. dissociating from the chromatin to prevent extensive cleavage of replication intermediates . Loss of Cds1-Mus81S.p. interaction via mutations in the FHA domain of Cds1S.p. or FHA-binding site on Mus81S.p. abolishes HU-induced Mus81S.p. phosphorylation, so that Mus81S.p. remains associated with chromatin with deleterious effects . On the other hand, chronic low dose HU-treatment reduces Cds1S.p.-mediated inhibition of Mus81S.p. activity and this allows processing of DNA secondary structures that form during extended replication block . Mus81S.p. is also required for Cds1S.p.-mediated slowing of replication upon MMS treatment .
In addition to phosphorylating Mus81S.p., Cds1S.p. may also indirectly modulate Mus81S.p. activity by regulating proteins that function closely with Mus81S.p. to respond to damage during replication (Figure 3). For instance, the DNA repair protein Rad60S.p. and RecQ family DNA helicase Rqh1S.p. function coordinately with Mus81S.p in recombinational repair (reviewed in ). Cds1S.p. down-regulates nuclear localization of Rad60S.p. in replication-arrested cells [82,83] and small ubiquitin-related modifier (SUMO)-like domains of Rad60S.p homolog Esc2S.c. in budding yeast are critical for stimulation of Mus81S.c. . Moreover, Rqh1S.p. contributes to the formation of Mus81S.p.-mediated DSB in Cds1S.p.-deleted cells . In budding yeast, Cds1S.p./Rad53S.c. activator Mrc1S.p./S.c. protein level regulates recruitment of Rqh1S.p. homolog Sgs1S.c. to chromatin . Cds1S.p./Rad53S.c. regulation of Rad60S.p./Esc2S.c., Rqh1S.p./Sgs1S.c., and possibly other proteins involved in resolving replication fork stress may be an important coordinator of Mus81S.p activity.
In mammalian cells, DNA damage checkpoint kinase CHK1 and Cds1S.p. -homolog CHK2 are activated downstream of ATM/ATR kinases in response to certain replication blocks and to DNA damage during S-phase (reviewed in ) . Although Cds1-Mus81S.p. interaction is conserved in human cells (CHK2-MUS81), it is unclear if CHK2 directly regulates MUS81 as in fission yeast (reviewed in ), although there is evidence that CHK2 up-regulates the protein level of MUS81; MUS81 in turn contributes to activation of CHK2 in Cisplatin-treated breast cancer cells (Figure 3) .
There is growing evidence that there are other regulators of Mus81 activity besides cell cycle and replication checkpoint kinases (Table 1). For example, the N-terminal fragment of DNA repair protein Rad52S.c. stimulates the endonuclease activity of Mus81-Mms4S.c. on homologous recombination intermediates in budding yeast . RAD52 also promotes MUS81-mediated break-induced replication repair of collapsed forks and mitotic DNA synthesis in human cells [95,96]. The small ubiquitin-related modifier (SUMO)-like domain of the adaptor protein establishment of silent chromatin 2 (Esc2S.c.) in budding yeast interacts with and stimulates Mus81-Mms4S.c. catalytic activity . The replication factor C (RFC) complex and the loading of proliferating cell nuclear antigen (PCNA) also enhances recruitment and activity of Mus81-Mms4S.c. .
Interestingly, the Structural Maintenance of Chromosomes (SMC) complexes are another modulator of Mus81S.p/S.c. activity. In yeast, for example, the Smc5-Smc6S.p/S.c. complex promotes Mus81S.p/S.c.-dependent resolution of Holliday junctions [99,100]. The positive genetic interactions between certain mutants affecting methylation of cohesin subunit Psm1S.p. and Mus81-Eme1S.p. mutants in fission yeast suggests that methylation of cohesin subunits may be important for Mus81 activity at the stalled replication fork. Alternatively, Mus81 may be required for recruitment of the cohesin to sites of DNA damage . In human cells, depletion of SMC2, which is required for chromosome condensation, or WAPL (Wings apart protein-like), which is required for release of sister-chromatid arm cohesin, results in failure to recruit MUS81 to chromatin .
In human cells, post-translational modification of MUS81 other than phosphorylation may be important for its activity during DNA repair. This is evidenced by compromised DNA damage response in cells with SUMOlyation-resistant MUS81 upon arsenic treatment that mimic metal carcinogenesis . Epigenetic modifications adjacent to replication forks may also contribute to regulation of MUS81 recruitment and activity. For instance, EZH2 (enhancer of zeste homologue 2) that methylates histone H3 on Lys27 (H3K27) at stalled replication forks has been shown to mediate recruitment of MUS81 .
Localization of MUS81 is another way its activity is modulated. In human cells, MUS81 accumulates in the nucleolus during S phase, suggesting that it is required to maintain highly repetitive nucleolar DNA (reviewed in ). MUS81 relocates from the nucleolus to localized regions of UV damage specifically in S-phase cells . Sub-localization of Mus81S.c. also occurs in budding yeast. Following DNA damage, Mus81-Mms4S.c. relocalizes to subnuclear foci and colocalizes with other endonucleases and with Cmr1S.c., a protein involved in genome stability maintenance . Subnuclear Mus81-Mms4S.c. foci persist until the resolution of accumulated DNA intermediates following DNA damage .
These findings demonstrate that cells are equipped with multiple means to tightly regulate Mus81 recruitment and activity. Investigating how these various modulators of Mus81 communicate with each other will further elucidate Mus81-dependent genome stability maintenance.
An additional SSE plays a related role to Mus81. Xeroderma pigmentosum group F complementing protein (XPF)-excision repair cross-complementing group 1 (ERCC1) heterodimer complex is a 5ˇä-3ˇä structure-specific endonuclease involved in a variety of DNA repair pathways including nucleotide excision repair (NER) and has important roles in interstrand crosslink (ICL) repair and DSB repair (reviewed in ). Rad16-Swi10S.p. is the fission yeast ortholog and Rad1-Rad10S.c. is the budding yeast ortholog (reviewed in ).
With MUS81-EME1, XPF-ERCC1 processes under-replicated DNA and replication intermediates at CFS and prevents anaphase bridges following recruitment by SLX4 [72,73]. However, MUS81 and XPF may differ in the timing of their activity (Figure 1). In human cells, the biological function of MUS8-EME1 is mostly during mitosis although MUS81 activity is present throughout the cell cycle, probably through its association with EME2 (reviewed in ). During S- and G2-phase, XPF-ERCC1 along with another endonuclease ARTEMIS, are responsible for replication stress-induced fork cleavage needed to resume DNA replication . Data from fission yeast indicates that Mus81S.p. and Rad16S.p./XPF may direct repair towards different templates, with Mus81S.p. using the sister chromatid and Rad16S.p./XPF using ectopic sequences .
Despite minimal overlap in substrate specificity in vitro , evidence suggests that XPF and MUS81 provide overlapping activity. Mammalian XPF is not required for viability [127,128], possibly due to overlap with MUS81 or other SSEs [129,130,131]. XPF becomes essential in chicken DT40 cells if MUS81 is missing . In fission yeast, a double mutant lacking both Rad16S.p./XPF and Mus81S.p. is inviable . Consistent with this, XPF-ERCC1 partially compensates for MUS81 loss during mild replication stress in mammalian cells . MUS81-EME1 also rescues the viability of XPF-deleted cells .
Association with different recruiting partners and stimulating proteins appears to determine in which repair pathway XPF-ERCC1 will function (Table 1). In fission yeast, a recently identified protein Pxd1S.p. (pombe XPF and Dna2) stimulates 3ˇä-endonuclease activity of Rad16-Swi10S.p. . In budding yeast, Saw1S.c. (Single-strand annealing weakened protein 1), a structure-specific DNA binding protein, recruits Rad1-Rad10S.c. to single-strand annealing repair sites  while damage recognition protein Rad14S.c. brings Rad1-Rad10S.c. to NER . In human cells, ERCC1 cannot enter the nucleus without XPF, demonstrating that XPF-ERCC1 heterodimer formation is critical . In NER, XPF-ERCC1 cleavage of the damaged stand is stimulated by RPA and Rad52 [110,112]. RPA is also required for XPF-ERCC1 endonuclease activity in replication-coupled ICL repair . In human cells, both XPF-ERCC1 and MUS81-EME1 are recruited to the replication fork stalled at ICL by the scaffold protein SLX4 and this depends on ubiquitylation of the FANCD2 (Fanconi anaemia complementation group D2) (reviewed in ) [134,135]. Independently of SLX4, the scaffold protein UHRF1 (ubiquitin-like PHD and RING finger domain-containing protein1) is needed to recruit FANCD2 and MUS81-EME1 and XPF-ERCC1 to DNA damage sites [107,108].
In human cells and in budding yeast, SLX4/Slx4S.c. forms a complex with its interacting partner SLX1/Slx1S.c. and, as previously mentioned, serves as a binding platform and catalytic stimulator for both MUS81-EME1/Mus81-Mms4S.c. and XPF-ERCC1/Rad1-Rad10S.c. (Figure 4) (reviewed in ) [12,13,14,15]).
In budding yeast, upon replication stress, Slx4S.c. forms a complex with another scaffold protein Rtt107S.c. (PTIP H.s.) and DNA replication initiation protein Dpb11S.c. (TOPBP1H.s.) and associates with Mus81-Mms4S.c. behind replication forks after Cdc28S.c.-and Cdc5S.c.-mediated phosphorylation of Mms4 S.c. [16,17,18,19,20]. In Mus81S.c.-deficient cells, Slx4S.c. play a critical role supporting replication-coupled ICL repair by Rad1-Rad10S.c. . In contrast to budding yeast, Slx4S.p. in fission yeast lacks the XPF-interacting region (Figure 4) and does not appear to affect Rad16S.p. as Slx4S.p. deletion has no sensitivity to UV or MMS and no synthetic growth defects with Rad16S.p. mutation .
In human cells, increased association of MUS81-EME1 with the scaffold protein SLX4 contributes to MUS81-dependent processing of DNA secondary structures . SLX4 deletion reduces MUS81-dependent formation of DSBs that occur after WEE1 inhibition [62,67]. SLX4 is phosphorylated by CDK1 in late G2 and M phase and interacts with MUS81-EME1 complex and SLX1, forming stable SLX-MUS complex (reviewed in [49,62]. SLX4 recruitment to chromatin and SLX4-mediated sister chromatid resolution requires TOPBP1 . In addition to recruiting MUS81-EME1 and XPF-ERCC1, nuclease activity of SLX4 is important for processing telomeric structures and oppose aberrant telomere synthesis observed in cancers (reviewed in ) [140,141]. SLX4 also suppresses chromatin association with another SSE, GEN1(Yen1), in the absence of MUS81 and prevent DSBs after pathological replication stress . Human SLX4 has ubiquitin-binding zinc finger (UBZ) motif and SUMO-interaction motif (SIMs) (Figure 4) [135,136]. The UBZ motif is required for SLX4 recruitment to sites of replication-dependent ICL repair while the SIMs is required for the function of SLX4 during replication stress and in suppressing CFS instability [143,144].
Although MUS81-EME1 and XPF-ERCC1, along with the scaffold protein SLX4, are the most well-characterized SSEs to be responsible for processing replication intermediates, a few other SSEs have been noted to be important in dealing with replication stress.
Flap endonuclease 1 (FEN1 in human, Rad2S.p. in fission yeast, Rad27S.c. in budding yeast) has an important role of removing 5ˇä flaps that form during Okazaki fragment maturation via its interaction with DNA processivity factor PCNA (Table 1) (reviewed in ). FEN1 is also involved in processing DNA secondary structures during replication fork impediment, especially in rDNA and telomeres [145,146,147]. This process requires FEN1 to undergo SUMOylation and subsequent interaction with the PCNA-like Rad9-Rad1-Hus1 complex [116,124]. FEN1 and MUS81 associate with each other and collaborate in removing various aberrant DNA structures, including regressed replication fork substrates [117,118,119]. FEN1 removes the 5ˇä-flap after MUS81 processes DNA junction structures (reviewed in ) . This process requires FEN1 to be stimulated by the helicase WRN (Werner syndrome ATP-dependent RecQ like helicase) [120,121,122]. This activity is especially critical for the fork restart at telomeres . Regulation of FEN1 activity is important in maintaining genome stability as overexpression of FEN1 is associated with poor prognosis in various cancers . FEN1 overexpression results in impediment in replication fork progression, mid-S phase arrest, and hypersensitivity to DNA damaging agents .
FAN1 (Fanconi-associated nuclease I) is another structure-dependent endonuclease that plays a critical role in ICLs repair (reviewed in ) and in promoting replication fork progression in response to replication stress induced by agents such as HU and MMS [123,149]. There is no apparent FAN1 homolog in the budding yeast. FAN1 exhibits endonuclease activity toward 5ˇä flaps and has 5ˇä-3ˇä exonuclease activity . A recent study suggests that FAN1 dimerizes to have optimal cleavage of a long 5ˇä flap strand . FAN1 nuclease activity at stalled replication forks is tightly regulated as FAN1 activity is needed for fork restart but excessive activity can result in fork degradation (reviewed in ) . Fan1−/− mice have repeat expansions in brain and other somatic tissues, demonstrating that FAN1 activity contributes to the maintenance of genome integrity . Like SLX4, FAN1 has a UBZ motif which allows its association with monoubiquitylated FANCD2 and subsequent recruitment to at replication forks (Table 1) . FAN1 can also be recruited to aphidicolin-stalled replication forks via FANCD2-BLM (Fanconi anemia group D2 protein- Bloom's helicase) complex independent of the UBZ domain . FAN1 also contains PCNA interacting peptide (PIP) motif that allows its association with ubiquitylated PCNA accumulated at stalled replication forks .
Yen1S.c. (crossover junction endodeoxyribonuclease 1) in budding yeast and GEN1 (XPG-like endonuclease 1) in humans are SSEs that belong to the XPG/Rad2 family and define another Holliday junction resolvase that can process replication intermediates (reviewed in ). In MUS81-deficient human cells, GEN1 can induce DSB following replication stress which is opposed by the presence of SLX4 . In budding yeast, Yen1S.c. is phosphorylated by Cdc28S.c. at G1/S transition which inactivates its nuclear localization signal (NLS), ensuring Yen1S.c. stays in the cytoplasm until anaphase (Table 1) [22,23,24]. Cdc14S.c. dephosphorylates Yen1S.c. at anaphase, allowing it to enter the nucleus . In human cells, GEN1 contains a nuclear export signal (NES) and cannot access chromatin until the nuclear envelope is broken down during mitosis . It is absent in fission yeast which may explain why meiosis is highly dependent on Mus81–Eme1 in fission yeast (reviewed in ).
It is important to remember that there may be nucleases that have not been previously implicated in replication stress that may also contribute to processing replication intermediates. For example, a recent study suggests that Artemis, a nuclease involved in non-homologous DNA end-joining (NHEJ) (reviewed in ), contributes to processing stalled DNA replication forks and prevent chromosomal segregation defect during mitosis . Artemis is not present in yeast.
We have summarized findings showing how SSEs, MUS81 and XPF in particular, are controlled during cell cycle and replication stress (Figure 5). Cell cycle kinase, replication checkpoint kinase, and the various interacting partners as well as inducers of post-translational and epigenetic modifiers work in consortium, allowing cells to quickly respond to replication stress but limit extraneous DNA damage. Teasing out the regulatory networks that control SSE activities and how they communicate with each other can help gain a more comprehensive understanding of how SSEs contribute to cancer. On one hand, SSEs are needed to maintain genome stability but on the other hand, DNA cleavage by SSEs can contribute to inducing DNA damage and chromosome rearrangement. For example, Mus81 cleavage of the displacement loop (D-loop), the initial recombination intermediate that form in broken replication forks, limits mutagenic template switches that propels genome instability in cancers . The ability of Mus81 to work with Rad27S.c. (FEN1 in human) and post-replication DNA repair protein, Rad18S.c., to suppress repeat-mediated chromosomal rearrangements has been suggested to inhibit large inverted duplications of chromosomal segments observed frequently in cancers . In other contexts, Mus81 activity can contribute to survivability of cancer cells. For instance, Mus81-mediated resolution of toxic intermediates resulting from break-induced replication in the absence of Srs2S.c. helicase increases cell viability .
There is somewhat conflicting evidence on how SSEs influence chemotherapy response. In various types of cancer cells, downregulation of XPF or MUS81 increases sensitivity to chemotherapeutic drugs cells via CHK1 pathway activation or stimulation of apoptosis [63,158,159]. However, there is also evidence that cytosolic DNA generated by MUS81 in prostate cancers stimulate immune response, potentially contributing to host rejection of cancer cells . More in-depth understanding of how SSE activities are controlled will help formulate better predictions about their involvement in carcinogenesis and in patient-response to anti-cancer therapeutics.
How do regulation and roles of MUS81 and XPF differ between mitosis and meiosis?
What molecular brakes exist that allow SSEs to process aberrant replication structures without deleterious DNA breakage?
How does chromatin structure or components influence SSE recruitment and activity?
How do SSEs coordinate or communicate with other SSEs and other DNA-remodeling enzymes?
Do SSE activities contribute to checkpoint activation and cell cycle arrest? If so, what is the molecular mechanism?
Exploring these questions and other uncharacterized aspects of SSEs will garner exciting and important insights needed to integrate our understanding of the replication process, genome stability and the cell cycle.
Both authors drafted and revised the manuscript.
We thank the members of the Forsburg Lab for helpful discussions.
Figure 1. Structure-specific endonucleases (SSEs) in different phases of the cell cycle. Mus81 (methyl methane sulfonate (MMS) and UV-sensitive protein 81) activity in Saccharomyces cerevisiae and human cells is stimulated during G2-M transition (reviewed in ). In Schizosaccharomyces pombe, Mus81 is activated by DNA damage. Xeroderma pigmentosum group F complementing protein (XPF)-excision repair cross-complementing group 1 (ERCC1) (orthologs Rad1-Rad10S.c. and Rad16-Swi10S.p.) is important for various DNA repair pathways and cleaves replication intermediates during S and G2 phases . Scaffold protein SLX4 with associating partner SLX1 interacts with MUS81-EME1 (essential meiotic endonuclease 1) and XPF-ERCC1 in human cells (reviewed in ) [12,13,14,15] and their orthologs in in S. cerevisiae (reviewed in ) [17,18,19,20]. In contrast, Slx4 does not affect Rad16-Swi10 in S. pombe . Activity of Yen1S.c. is prevented until anaphase by restricting its nuclear entry due to phosphorylation of nuclear localization signal (NLS) [22,23,24]. Due to nuclear export signal (NES), GEN1 in human cells is able to access chromosomes only after nuclear membrane breakdown during mitosis . S. pombe do not have Yen1 ortholog (reviewed in ). FEN1 (flap endonuclease 1) (orthologs Rad27S.c. and Rad2S.p.) and FAN1 (Fanconi-associated nuclease I) (missing in S. cerevisiae) contribute to processing replication intermediates but cell cycle-dependent regulation of these SSEs are not well characterized (reviewed in ). Mms4 (methyl methane sulfonate sensitivity protein 4).
Figure 2. Mus81 regulation by cell cycle kinases. In S. cerevisiae, Mus81-Mms4 is phosphorylated by Cdc28S.c. (CDK1 ortholog) and Cdc5S.c. (PLK ortholog) kinases at the G2/M transition [48,49,50]. Scaffold protein Rtt107S.c. (PTIP ortholog) associates with Dpb11S.c. (TOPBP1 ortholog) and interacts with DDK which also phosphorylates Mms4S.c. . Rtt107-Dpb11-Slx4S.c. complex associates with Mus81-Mms4S.c. behind replication forks. In S. pombe, Mus81-Eme1S.p. is phosphorylated by Cdc2S.p. (CDK1 ortholog) which primes Eme1S.p. for phosphorylation by Rad3S.p. (ATR ortholog) upon DNA damage . Mus81-Eme1S.p. may be contributing to Chk1 activation in fission yeast as Mus81-deleted cells with replication defect are able to bypass Chk1 checkpoint . In human cells, MUS81-EME1 activity peak during M phase after EME1 is phosphorylated by CDK1, PLK1 [12,58,59]. SLX4 phosphorylation by CDK1 and MUS81 phosphorylation by CK2 also promotes MUS81-EME1 activity . During S-phase, WEE1 downregulates MUS81-EME1 activity by inhibiting CDK1 and thereby limiting EME1 and SLX4 phosphorylation (reviewed in ). WEE1 inhibition of CDK2 reduces origin firing and subsequently the replication intermediate substrates of MUS81. WEE1 may also inhibit MUS81 directly . Residual MUS81 activity during S-phase comes from MUS81 that forms complex with EME2 which can promote premature entry to mitosis upon WEE1 inhibition .
Figure 3. Mus81 regulation by replication checkpoint. In yeast, replication stress induces Rad3S.p./Mec1S.c. (ATR in human) activation of Cds1S.p./Rad53S.c. by promoting its association with Mrc1S.p./S.c. (CLASPIN in human) (reviewed in ). Upon acute and severe replication stress such as hydroxyurea treatment, Cds1S.p. limits Mus81S.p. activity (indicated by solid red line) . Cds1S.p. inhibits Rad60S.p. activity (indicated by blue line) by promoting delocalization from the nucleus [82,83]. Mrc1S.p./S.c. protein level regulates recruitment of Rqh1S.p. homolog Sgs1S.c. to chromatin (indicated by blue arrow) . Both Rad60S.p./Esc2S.c. and Rqh1S.p./Sgs1S.c. contribute to Mus81 activity (indicated by dashed blue arrow) [75,76]. In human cells, DNA damage checkpoint CHK1 and Cds1-homolog CHK2 is activated downstream of ATM/ATR kinases (reviewed in [79,80]) . It is unclear whether MUS81 is directly regulated by these checkpoint kinases in human cells. However, there is evidence that CHK2 upregulates MUS81 protein levels and MUS81 in turn contributes to CHK2 activation upon DNA damage (indicated by dashed double-headed arrow) . Deleterious MUS81-dependent processing of replication intermediates following CHK1 inhibition suggests that CHK1 downregulates MUS81 activity (indicated by dashed red line) [68,69,70].
Figure 4. Important domains in SLX4. Across all three species (S.p., S.c., and human), scaffold protein SLX4 have SLX1 binding domain (SBD) and MUS81-EME1 binding region (SAP) . Slx4S.c. in budding yeast and SLX4 in human cells also have XPF-ERCC1 interacting region MLR. Slx4S.p. in fission yeast lack MLR. SLX4 in human cells have UBZ (ubiquitin-binding zinc finger domain) and SIM (SUMO-interaction motif )motifs that contributes to its recruitment and activity [135,136]. (SBD: SLX1 binding domain; SAP: SAF-A/B, Acinus and PIAS domain that interacts with MUS81-EME1; MLR: MEI9XPF-interaction-like region that interacts with XPF-ERCC1; ICL: interstrand crosslink; CFS: common fragile site) (S.c.: S. cerevisiae; S.p.: S. pombe).
Figure 5. Summary of types of regulation of SSEs (MUS81 and XPF in particular) involved in resolving replication intermediates.
Table 1. Regulators of structure-specific endonucleases (SSEs) besides cell cycle kinases and Cds1S.p. homologs. There are many other regulators of Mus81 and other SSE activities besides cell cycle-dependent kinases. Proteins involved in DNA repair or replication are the most common regulators. (S.c.: S. cerevisiae; S.p.: S. pombe).


text:
\section{Introduction}\label{intro}
In \cite{DW1}, Dwyer and Wilkerson constructed a $2$-complete space $BDI(4)$,
so named because its $F_2$-cohomology groups form an algebra isomorphic to the
ring of rank-4 mod-2 Dickson invariants. Its loop space, called $DI(4)$, has
$H^*(DI(4);F_2)$ finite. In \cite{DW2}, they then defined a $p$-compact group to be
a pair $(X,BX)$, such that $X=\Omega BX$ (hence $X$ is redundant), $BX$ is connected
and $p$-complete, and $H^*(X;F_p)$ is finite. In \cite{Gro}, Andersen and Grodal proved that
$(DI(4), BDI(4))$ is the only simple 2-compact group not arising as the 2-completion of
a compact connected Lie group.

The $p$-primary $v_1$-periodic homotopy groups of a topological space $X$, defined in \cite{DM}
and denoted
$v_1^{-1}\pi_*(X)_{(p)}$ or just $v_1^{-1}\pi_*(X)$ if the prime is clear, are a first approximation
to the $p$-primary homotopy groups. Roughly, they are a localization of the portion of the
actual homotopy groups detected by $p$-local $K$-theory. In \cite{DHHA}, the second author
completed a 13-year project, often in collaboration with the first author, of determining
$v_1^{-1}\pi_*(X)_{(p)}$ for all compact simple Lie groups and all primes $p$

In this paper, we determine the 2-primary groups $v_1^{-1}\pi_*(DI(4))$. Here and throughout,
$\nu(-)$ denotes the exponent of 2 in the prime factorization of an integer.
\begin{thm}\label{main}
For any integer $i$, let $e_i=\min(21,4+\nu(i-90627))$. Then
$$v_1^{-1}\pi_{8i+d}(DI(4))\approx\begin{cases}{\Bbb Z}/2^{e_i}\oplus{\Bbb Z}/2&d=1\\
{\Bbb Z}/2^{e_i}&d=2\\
0&d=3,4\\
{\Bbb Z}/8&d=5\\
{\Bbb Z}/8\oplus{\Bbb Z}/2&d=6\\
{\Bbb Z}/2\oplus {\Bbb Z}/2\oplus{\Bbb Z}/2&d=7,8.\end{cases}$$
\end{thm}

Since every $v_1$-periodic homotopy group is a subgroup of some actual
homotopy group, this result implies that $\on{exp}_2(DI(4))\ge21$, i.e., some homotopy group
of $DI(4)$ has an element of order $2^{21}$. It would be interesting to know
whether this bound is sharp.

Our proof involves studying the spectrum $\Phi_1DI(4)$
which satisfies $\pi_*(\Phi_1DI(4))\approxv_1^{-1}\pi_*(DI(4))$. We will relate $\Phi_1DI(4))$
to the 2-completed $K$-theoretic pseudosphere $T_{K/2}$ discussed in \cite[8.6]{BoTop}.
We will prove the following surprising result, which was pointed out by Pete Bousfield.
\begin{thm}\label{spherethm} There is an equivalence of spectra $$\Phi_1DI(4)\simeq\Sigma^{725019}T_{K/2}\wedge M(2^{21}),$$
where $M(2^{21})$ is a mod $2^{21}$ Moore spectrum.
\end{thm}
In Section \ref{psspsec}, we will give the easy deduction of
Theorem \ref{main} from Theorem \ref{spherethm}.
As an immediate corollary of \ref{spherethm}, we deduce that the $2^{21}$ bound on
$\pi_*(\Phi_1DI(4))$ is induced from a bound on the spectrum itself.
\begin{cor} The exponent of the spectrum $\Phi_1DI(4)$ is $2^{21}$; i.e.,
$2^{e}1_{\Phi_1DI(4)}$ is null if and only if $e\ge21$.\end{cor}

In \cite{Bo}, Bousfield presented a framework that enables determination of the
$v_1$-periodic homotopy groups of many simply-connected $H$-spaces $X$ from their
united $K$-theory groups and Adams operations. The intermediate step is
$KO^*(\Phi_1 X)$. (All of our $K^*(-)$ and $KO^*(-)$-groups have coefficients in the 2-adic integers $\hat {\Bbb Z}_2$, which we omit
from our notation.) Our first proof of Theorem \ref{main} used Bousfield's exact sequence
\cite[9.2]{Bo} which relates $v_1^{-1}\pi_*(X)$ with $KO^*(\Phi_1X)$, but the approach via the
pseudosphere, which we present here, is stronger and more elegant. The insight for Theorem \ref{spherethm} is the observation that
the two spectra have isomorphic Adams modules $KO^*(-)$.

In several earlier e-mails,
Bousfield explained to the authors how the results of \cite{Bo} should
enable us to determine $KO^*(\Phi_1DI(4))$. In Section \ref{Bosec}, we present our account
of these ideas of Bousfield.  We thank him profusely for sharing his insights
with us.

The other main input is the Adams operations in $K^*(BDI(4))$.
In \cite{OS}, Osse and Suter showed that $K^*(BDI(4))$ is a power series algebra
on three specific generators, and gave some information toward the determination of the
Adams operations. In private communication in 2005, Suter expanded on this to give
explicit formulas for $\psi^k$ in $K^*(BDI(4))$. We are very grateful to him for
sharing this information. In Section \ref{compsec}, we will
explain these calculations and also how they then lead to the determination
of $KO^*(\Phi_1DI(4))$.

\section{Adams operations}\label{compsec}
In this section, we present Suter's determination of $\psi^k$ in $K^*(BDI(4))$ and
state a result, proved in Section \ref{Bosec}, that allows us to determine
$KO^*(\Phi_1DI(4))$ from these Adams operations.

Our first result, communicated by Suter, is the following determination of Adams operations
in $K^*(BDI(4))$. An element of $K^*(X)$ is called {\it real} if it is in the image of
$KO^*(X)@>c>> K^*(X)$.
\begin{thm}\label{psik} (Suter) There is an isomorphism of algebras
\begin{equation}\label{three}K^*(BDI(4))\approx \hat{\Bbb Z}_2[\![\xi_8,\xi_{12},\xi_{24}]\!]\end{equation}
such that the generators are in $K^0(-)$ and are real, $\psi^{-1}=1$, and the matrices of $\psi^2$ and $\psi^3$
on the three generators, mod decomposables, are
$$\Psi^2\equiv\begin{pmatrix}2^4&0&0\\ -2&2^6&0\\ 0&-2&2^{14}\end{pmatrix},\qquad\Psi^3\equiv
\begin{pmatrix}3^4&0&0\\-3^3&3^6&0\\ 36/527&-3^5\cdot41/17&3^{14}\end{pmatrix}.$$
\end{thm}
\begin{pf} The subscripts of the generators indicates their ``filtration," meaning
the dimension of the smallest skeleton on which they are nontrivial. A standard
property of Adams operations is that if $\xi$ has filtration $2r$, then $\psi^k(\xi)$
equals $k^r\xi$ plus elements of higher filtration.

The isomorphism (\ref{three}) is derived in \cite[p.184]{OS} along with the additional information
that $4\xi_{24}-\xi_{12}^2$ has filtration 28, and \begin{eqnarray}\label{l2}\xi_{12}&=&\lambda^2(\xi_8)+8\xi_8\\ \nonumber\xi_{24}&=&
\lambda^2(\xi_{12})+32\xi_{12}+c_1\xi_8^2+c_2\xi_8^3+c_3\xi_8\xi_{12},\end{eqnarray}
for certain explicit even coefficients $c_i$.

The Atiyah-Hirzebruch spectral sequence easily shows that $\xi_8$ is real, since the 11-skeleton
of $BDI(4)$ equals $S^8$. Since $\lambda^2(c(\theta))=c(\lambda^2(\theta))$, and products of real
bundles are real, we deduce from (\ref{l2}) that $\xi_{12}$ and $\xi_{24}$ are also real. Since $tc=c$, where $t$
denotes conjugation, which corresponds to $\psi^{-1}$, we obtain that the generators are invariant under
$\psi^{-1}$, and hence so is all of $K^*(BDI(4))$.

We compute Adams operations mod decomposables, writing $\equiv$ for equivalence mod decomposables.
Because $4\xi_{24}-\xi_{12}^2$ has filtration 28, we obtain \begin{equation}\label{strange}\psi^k(\xi_{24})\equiv k^{14}\xi_{24}.\end{equation}
Here we use, from \cite[p.183]{OS}, that all elements of $K^*(BDI(4))$ of filtration greater than 28
are decomposable. Equation (\ref{strange}) may seem surprising, since $\xi_{24}$ has filtration 24,
but there is a class $\xi_{28}$ such that $4\xi_{24}-\xi_{12}^2=\xi_{28}$, and we can have
$\psi^k(\xi_{24})\equiv k^{12}\xi_{24}+\alpha_k\xi_{28}$ consistently with (\ref{strange}).

Using (\ref{l2}) and that $\psi^2\equiv -2\lambda^2$ mod decomposables, we obtain
\begin{eqnarray}\label{psi2}\psi^2(\xi_8)&\equiv&2^4\xi_8-2\xi_{12}\\
\psi^2(\xi_{12})&\equiv&2^6\xi_{12}-2\xi_{24}\nonumber,\end{eqnarray}
yielding the matrix $\Psi^2$ in the theorem.

Applying $\psi^2\psi^3=\psi^3\psi^2$ to $\psi^3(\xi_{12})\equiv3^6\xi_{12}+\gamma\xi_{24}$
yields $-2\cdot3^6+2^{14}\gamma=2^6\gamma-2\cdot3^{14}$, from which we obtain $\gamma=-3^5\cdot41/17$.
Applying the same relation to $\psi^3(\xi_8)=3^4\xi_8+\alpha\xi_{12}+\beta\xi_{24}$, coefficients
of $\xi_{12}$ yield $-2\cdot3^4+\alpha\cdot2^6=2^4\alpha-2\cdot3^6$ and hence $\alpha=-3^3$.
Now coefficients of $\xi_{24}$ yield $-2\alpha+2^{14}\beta=2^4\beta-2\gamma$ and hence $\beta=36/527$.
\end{pf}

Let $\Phi_1(-)$ denote the functor from spaces to $K/2_*$-local spectra described in
\cite[9.1]{Bo}, which satisfies $v_1^{-1}\pi_*X\approx\pi_*\tau_2\Phi_1X$, where $\tau_2\Phi_1X$ is the 2-torsion
part of $\Phi_1X$.
 In Section \ref{Bosec}, we will use results of Bousfield in \cite{Bo} to prove the following result.
Aspects of Theorem \ref{psik}, such as $K^*(BDI(4))$ being a power
series algebra on real generators, are also used in proving this theorem.

Recall that $KO^*(-)$ has period 8.
\begin{thm}\label{KO} The groups $KO^i(\Phi_1DI(4))$ are $0$ if $i\equiv0,1,2\mod 8$, and $K^0(\Phi_1 DI(4))=0$. Let $M$ denote a
free $\hat{\Bbb Z}_2$-module on three generators, acted on by $\psi^2$ and $\psi^3$ by the matrices of Theorem
\ref{psik}, with $\psi^{-1}=1$. Let $\theta=\frac12\psi^2$ act on $M$. Then there are exact sequences
\begin{eqnarray*}0&\to& 2M@>\theta>>2M\to KO^3(\Phi_1DI(4))\to0\to0\to KO^4(\Phi_1DI(4))\to M/2\\
&@>\theta>>& M/2\to KO^5(\Phi_1DI(4))\to M/2@>\theta>>M/2\to KO^6(\Phi_1DI(4))\to M\\
&@>\theta>>& M\to KO^7(\Phi_1DI(4))\to 0
\end{eqnarray*}
and
$$0\to M@>\theta>> M\to K^1(\Phi DI(4))\to 0.$$
For $k=-1$ and $3$, the action of $\psi^k$ in $KO^{2j-1}(\Phi_1DI(4))$, $KO^{2j-2}(\Phi_1DI(4))$, and $K^{2j-1}(\Phi DI(4))$ agrees with $k^{-j}\psi^k$ in adjacent $M$-terms.
\end{thm}

In the remainder of this section, we use \ref{psik} and \ref{KO} to give explicit formulas for the Adams module $KO^i(\Phi_1 DI(4))$. A similar argument works for $K^*(\Phi_1 DI(4))$.
If $g_1$, $g_2$, and $g_3$ denote the three generators of $M$, then the action of $\theta$ is given by
\begin{eqnarray*}\theta(g_1)&=&8g_1-g_2\\
\theta(g_2)&=&2^5g_2-g_3\\
\theta(g_3)&=&2^{13}g_3.\end{eqnarray*}
Clearly $\theta$ is injective on $M$ and $2M$. We have $KO^7(\Phi_1DI(4))\approx\on{coker}(\theta|M)\approx{\Bbb Z}/2^{21}$ with generator $g_1$; note that $g_2=2^3g_1$ in this cokernel, and then $g_3=2^8g_1$. Similarly $KO^3(\Phi_1DI(4))\approx\on{coker}(\theta|2M)\approx{\Bbb Z}/2^{21}$.
Also $KO^4(\Phi_1DI(4))\approx\on{ker}(\theta|M/2)={\Bbb Z}/2$ with generator $g_3$, while
$KO^6(\Phi_1DI(4))\approx\on{coker}(\theta|M/2)={\Bbb Z}/2$ with generator $g_1$. There is a short exact sequence
$$0\to\on{coker}(\theta|M/2)\to KO^5(\Phi_1DI(4))\to\on{ker}(\theta|M/2)\to 0,$$
with the groups at either end being ${\Bbb Z}/2$ as before. To see that this short exact sequence is split,
we use the map $S^7@>f>> DI(4)$ which is inclusion of the bottom cell. The morphism $f^*$ sends the first
summand of $KO^5(\Phi_1DI(4))$ to one of the two ${\Bbb Z}/2$-summands of $KO^5(\Phi_1S^7)$, providing a splitting
homomorphism.
 Thus we have proved the first part of the following result.
\begin{thm}\label{gps} We have
\begin{eqnarray*}K^i(\Phi_1DI(4))&\approx&\begin{cases}0&i=0\\ {\Bbb Z}/2^{21}&i=1,\end{cases}\\
KO^i(\Phi_1DI(4))&\approx&\begin{cases}0&i=0,1,2\\
{\Bbb Z}/2^{21}&i=3,7\\
{\Bbb Z}/2&i=4,6\\
{\Bbb Z}/2\oplus{\Bbb Z}/2&i=5.\end{cases}\end{eqnarray*}
For $k=-1$ and $3$, we have $\psi^k=1$ on the ${\Bbb Z}/2$'s, and on $KO^{2j-1}(\Phi_1DI(4))$ with $j$ even and $K^{2j-1}(\Phi_1DI(4))$, $\psi^{-1}=(-1)^j$ and
$$\psi^3=3^{-j}(3^4-3^3\cdot2^3+\textstyle\frac{36}{527}2^8).$$\end{thm}

\begin{pf*}{Completion of proof}
To obtain $\psi^3$ on the ${\Bbb Z}/2$'s, we use the last part of Theorem \ref{KO} and the matrix $\Psi^3$ of
Theorem \ref{psik}. If $\psi^3$ is as in $\Psi^3$, then, mod 2, $\psi^3-1$ sends
$g_1\mapsto g_2$, $g_2\mapsto g_3$, and $g_3\mapsto0$. Thus  $\psi^3-1$ equals 0 on $KO^4(\Phi_1DI(4))$ and $KO^6(\Phi_1DI(4))$. Clearly $\psi^{-1}=1$ on these groups.

To see that $\psi^k-1$ is 0 on $KO^5(\Phi_1DI(4))$, we use the commutative diagram
$$\begin{CD} 0@>>>{\Bbb Z}/2@>i>> KO^5(\Phi_1DI(4)) @>\rho>>{\Bbb Z}/2@>>>0\\
@. @V\approx VV @V f^* VV @V0VV @.\\
0@>>>{\Bbb Z}/2 @>>> KO^5(\Phi_1S^7)@>>>{\Bbb Z}/2@>>>0.\end{CD}$$
We can choose generators $G_1$ and $G_2$ of $KO^5(\Phi_1DI(4))\approx{\Bbb Z}/2\oplus{\Bbb Z}/2$ so that $G_1\in\on{im}(i)$,
$\rho(G_2)\ne0$, and $f^*(G_2)=0$. Since $\psi^k-1=0$ on the ${\Bbb Z}/2$'s on either side of $KO^5(\Phi_1DI(4))$,
the only way that $\psi^k-1$ could be nonzero on $KO^5(\Phi_1DI(4))$ is if $(\psi^k-1)(G_2)=G_1$. However this yields the
contradiction
$$0=(\psi^k-1)f^*G_2=f^*(\psi^k-1)G_2=f^*(G_1)\ne0.$$

On $KO^{2j-1}(\Phi_1DI(4))$ with $j$ even and $K^{2j-1}(\Phi_1DI(4))$, $\psi^3$ sends the generator $g_1$ to
$$3^{-j}(3^4g_1-3^3g_2+\textstyle\frac{36}{527}g_3)=3^{-j}(3^4-3^3\cdot2^3+\textstyle\frac{36}{527}2^8)g_1,$$
and $\psi^{-1}(g_1)=(-1)^jg_1$ by Theorem \ref{KO}.
\end{pf*}

\section{Relationship with pseudosphere}\label{psspsec}
In this section, we prove Theorems \ref{spherethm} and \ref{main}.

Following \cite[8.6]{BoTop}, we let $T=S^0\cup_\eta e^2\cup_2 e^3$, and consider its
$K/2$-localization $T_{K/2}$. The groups $\pi_*(T_{K/2})$ are given in \cite[8.8]{BoTop}, while
the Adams module  is given by
\begin{eqnarray*}K^i(T_{K/2})&=&\begin{cases}\hat{\Bbb Z}_2&i\text{ even, with }\psi^k=k^{-i/2}\\
0&i\text{ odd;}\end{cases}\\
KO^i(T_{K/2})&=&\begin{cases}\hat{\Bbb Z}_2&i\equiv0\mod4,\text{ with }\psi^k=k^{-i/2}\\
{\Bbb Z}/2&i=2,3,\text{ with }\psi^k=1\\
0&i=1,5,6,7.\end{cases}
\end{eqnarray*}
Bousfield calls this the 2-completed $K$-theoretic pseudosphere. Closely related spectra
have been also considered in \cite{HMS} and \cite{BDM}.

Let $M(n)=S^{-1}\cup_n e^0$ denote the mod $n$ Moore spectrum. Then, for $e>1$ and $k$ odd,
\begin{eqnarray}\nonumber K^i(T_{K/2}\wedge M(2^e))&=&\begin{cases}{\Bbb Z}/2^e&i\text{ even, with }\psi^k=k^{-i/2}\\
0&i\text{ odd;}\end{cases}\\
\label{KY} KO^i(T_{K/2}\wedge M(2^e))&=&\begin{cases}{\Bbb Z}/2^e&i\equiv0\mod4,\text{ with }\psi^k=k^{-i/2}\\
{\Bbb Z}/2&i=1,3,\text{ with }\psi^k=1\\
{\Bbb Z}/2\oplus{\Bbb Z}/2&i=2,\text{ with }\psi^k=1\\
0&i=5,6,7.\end{cases}
\end{eqnarray}
\begin{pf} Let $Y=T_{K/2}\wedge M(2^e)$. Most of (\ref{KY}) is immediate from the exact sequence
$$@>2^e>>KO^i(T_{K/2})@>>> KO^i(Y)@>>> KO^{i+1}(T_{K/2})@>2^e>>.$$
To see that $KO^2(Y)={\Bbb Z}/2\oplus{\Bbb Z}/2$ and not ${\Bbb Z}/4$, one can first note that
\begin{equation}\label{Msplit}M(2^e)\wedge M(2)\simeq \Sigma^{-1}M(2)\vee M(2).\end{equation}
The exact sequence
\begin{equation}\label{KOY}KO^2(Y)@>2>>KO^2(Y)@>>>KO^2(Y\wedge M(2))@>>>KO^3(Y)@>2>>\end{equation}
implies that if $KO^2(Y)={\Bbb Z}/4$, then $|KO^2(Y\wedge M(2))|=4$. However, by (\ref{Msplit}),
\begin{equation}\label{KM}KO^2(Y\wedge M(2))\approx KO^2(T_{K/2}\wedge M(2))\oplus KO^3(T_{K/2}\wedge M(2)).\end{equation}
Also, there is a cofiber sequence
\begin{equation}\label{A1cof}T\wedge M(2)\to \Sigma^{-1}A_1 \to \Sigma^5 M(2),\end{equation}
where $H^*(A_1;F_2)$ is isomorphic to the subalgebra of the mod 2 Steenrod algebra generated
by $\on{Sq}^1$ and $\on{Sq}^2$, and satisfies $KO^*(A_1)=0$. Thus
$$KO^i(T_{K/2}\wedge M(2))\approx KO^i(\Sigma^4M(2))\approx\begin{cases}{\Bbb Z}/4&i=2\\
{\Bbb Z}/2&i=3,\end{cases}$$
so that $|KO^2(Y\wedge M(2))|=8$, contradicting a consequence of the hypothesis that $KO^2(Y)={\Bbb Z}/4$.

We conclude the proof by showing that, for odd $k$, $\psi^k=1$ on $KO^2(Y)$. First note that $\psi^k=1$
on $KO^*(M(2))$. This follows immediately from the Adams operations on the sphere, except for $\psi^k$ on $KO^{-2}(M(2))\approx{\Bbb Z}/4$. This is isomorphic
to $\widetilde{KO}(RP^2)$, where $\psi^k=1$ is well-known. Now use (\ref{MTM}) to deduce that $\psi^k=1$ on
$KO^*(T_{K/2}\wedge M(2))$, and then (\ref{KM}) to see that $\psi^k=1$ on $KO^2(Y\wedge M(2))$. Finally, use (\ref{KOY})
to deduce that $\psi^k=1$ on $KO^2(Y)$.
\end{pf}

Comparison of \ref{gps} and (\ref{KY}) yields an isomorphism of graded abelian groups
\begin{equation}\label{KOiso}KO^*(\Sigma^{8L+3}T_{K/2}\wedge M(2^{21}))\approx KO^*(\Phi_1DI(4))\end{equation}
for any integer $L$. We will show that if $L=90627$, then the Adams operations
agree too. By \cite[6.4]{BoKs}, it suffices to prove they agree for $\psi^3$ and $\psi^{-1}$.

Note that one way of distinguishing a $K$-theoretic pseudosphere from
a sphere is that in $KO^*(\text{sphere})$ (resp. $KO^*(\text{pseudosphere})$)
the ${\Bbb Z}/2$-groups are in dimensions 1 and 2 less than the dimensions in which
$\psi^3\equiv1$ mod 16 (resp. $\psi^3\equiv9$ mod 16), and similarly after smashing
with a mod $2^e$ Moore spectrum. Since
$3^4-6^3+\frac{36}{527}2^8\equiv9$ mod 16, the ${\Bbb Z}/2$-groups in $KO^*(\Phi_1DI(4))$
occur in dimensions 1, 2, and 3 less that the dimension in which $\psi^3\equiv9$ mod 16, and so
$\Phi_1DI(4)$ should be identified with a suspension of $T_{K/2}\wedge M(2^{21})$ and not
$S_{K/2}\wedge M(2^{21})$.

In $KO^{4t-1}(\Sigma^{8L+3}T_{K/2}\wedge M(2^{21}))$, $\psi^3=3^{-2(t-2L-1)}$ and $\psi^{-1}=1$.
Thus if $L$ satisfies
\begin{equation}\label{Leq} 3^{4L+2}\equiv 3^4-6^3+\textstyle\frac{36}{527}2^8\mod 2^{21},\end{equation}
then $KO^*(\Sigma^{8L+3}T_{K/2}\wedge M(2^{21}))$ and $KO^*(\Phi_1 DI(4))$ will be isomorphic
Adams modules. {\tt Maple} easily verifies that (\ref{Leq}) is satisfied for $L=90627$.

A way in which this number $L$ can be found begins with the mod $2^{18}$ equation
$$\sum_{i=1}^6\tbinom{2L-1}i 8^{i-1}\equiv\frac{3^{4L-2}-1}8\equiv\tfrac19(\textstyle\frac{2^7}{527}-3)\equiv192725,$$
where we use {\tt Maple} at the last step. This easily implies $L\equiv3$ mod 8, and so we let $L=8b+3$.
Again using {\tt Maple} and working mod $2^{18}$ we compute
$$\sum_{i=1}^6\tbinom{16b+5}i8^{i-1}-192725\equiv 2^{10}u_0+2^4u_1b+2^{10}u_2b^2+2^{17}b^3,$$
with $u_i$ odd. Thus we must have $b\equiv64$ mod 128, and so $L\equiv515$ mod $2^{10}$.
Several more steps of this type lead to the desired value of $L$.

Thus, in the terminology of \ref{CRexpl}, we have proved the following result.
\begin{prop} If $L=90627$, then there is an isomorphism of Adams modules
$$K_{CR}^*(\Sigma^{8L+3}T_{K/2}\wedge M(2^{21}))\approx K^*_{CR}(\Phi_1 DI(4)).$$
\end{prop}
Theorem \ref{spherethm} follows immediately from this using the remarkable
\cite[5.3]{BoTop}, which says, among other things, that 2-local spectra $X$
having some $K^i(X)=0$ are determined up to equivalence by their Adams module $K_{CR}^*(X)$.
Theorem \ref{main} follows immediately from Theorem \ref{spherethm} and the
following result.
\begin{prop}\label{piT} For all integers $i$,
$$\pi_{8i+d}(T_{K/2}\wedge M(2^{21}))\approx\begin{cases}{\Bbb Z}/2\oplus{\Bbb Z}/2^{\min(21,\nu(i)+4)}&d=-2\\
{\Bbb Z}/2^{\min(21,\nu(i)+4)}&d=-1\\
0&d=0,1\\
{\Bbb Z}/8&d=2\\{\Bbb Z}/2\oplus{\Bbb Z}/8&d=3\\
{\Bbb Z}/2\oplus{\Bbb Z}/2\oplus{\Bbb Z}/2&d=4,5.\end{cases}$$
\end{prop}
\begin{pf} For the most part, these groups are immediate from the groups $\pi_*(T_{K/2})$ given in
\cite[8.8]{BoTop} and the exact sequence
\begin{equation}\label{les}@>2^{21}>>\pi_{j+1}(T_{K/2})\to \pi_j(T_{K/2}\wedge M(2^{21}))\to \pi_j(T_{K/2})@>2^{21}>>.
\end{equation}
All that needs to be done is to show that the following short exact sequences, obtained from (\ref{les}), are split.
\begin{eqnarray}\label{4seq}0\ \to\ {\Bbb Z}/2\ \to&\pi_{8i+3}(T_{K/2}\wedge M(2^{21}))&\to\ {\Bbb Z}/8\ \to\ 0\\
0\ \to\ {\Bbb Z}/2\oplus{\Bbb Z}/2\ \to&\pi_{8i+4}(T_{K/2}\wedge M(2^{21}))&\to\ {\Bbb Z}/2\ \to\ 0\nonumber\\
0\ \to\ {\Bbb Z}/2\ \to&\pi_{8i+5}(T_{K/2}\wedge M(2^{21}))&\to\ {\Bbb Z}/2\oplus{\Bbb Z}/2\ \to\ 0\nonumber\\
0\ \to\ {\Bbb Z}/2^{\min(21,\nu(i)+4)}\ \to&\pi_{8i-2}(T_{K/2}\wedge M(2^{21}))&\to\ {\Bbb Z}/2\ \to\ 0.\nonumber\end{eqnarray}
Let $Y=T_{K/2}\wedge M(2^{21})$. We consider the exact sequence for $\pi_*(Y\wedge M(2))$,
\begin{equation}\label{YMseq}@>2>>\pi_{i+1}(Y)\to \pi_i(Y\wedge M(2))\to \pi_i(Y)@>2>>.\end{equation}
If the four sequences (\ref{4seq}) are all split, then by (\ref{YMseq}) the groups $\pi_{8i+d}(Y\wedge M(2))$
for $d=2,3,4,5,-2$ have orders $2^3$, $2^5$, $2^6$, $2^5$, and $2^3$, respectively, but if any of the sequences
(\ref{4seq}) fails to split, then some of the orders $|\pi_{8i+d}(Y\wedge M(2))|$ will have values smaller than those
listed here.

By (\ref{Msplit}),
$$\pi_i(Y\wedge M(2))\approx\pi_{i+1}(T_{K/2}\wedge M(2))\oplus \pi_i(T_{K/2}\wedge M(2)).$$
By (\ref{A1cof}), since localization preserves cofibrations and $(A_1)_{K/2}=*$, there is an equivalence
\begin{equation}\label{MTM}\Sigma^4M_{K/2}\simeq T_{K/2}\wedge M(2),\end{equation} and hence
\begin{equation}\label{twosum}\pi_i(Y\wedge M(2))\approx\pi_{i-3}(M_{K/2})\oplus\pi_{i-4}(M_{K/2}).\end{equation}
 By \cite[4.2]{Bo79},
$$\pi_{8i+d}(M_{K/2})=\begin{cases}0&d=4,5\\
{\Bbb Z}/2&d=-2,3\\
{\Bbb Z}/2\oplus{\Bbb Z}/2&d=-1,2\\
{\Bbb Z}/4\oplus{\Bbb Z}/2&d=0,1.\end{cases}$$
This is the sum of two ``lightning flashes," one beginning in $8d-2$ and the other in $8d-1$.
Substituting this information into (\ref{twosum}) yields exactly the orders which were shown in the previous
paragraph to be true if and only if all the exact sequences (\ref{4seq}) split. \end{pf}

\section{Determination of $KO^*(\Phi_1DI(4))$}\label{Bosec}
In this section, we prove Theorem \ref{KO}, which shows how
$\psi^k$ in $K^*(BDI(4))$ leads to the determination of $KO^*(\Phi_1DI(4))$.
Our presentation here follows suggestions in several e-mails from Pete Bousfield.

The first result explains how $KO^*(BDI(4))$ follows from $K^*(BDI(4))$.
\begin{thm}\label{And} There are classes $g_8$, $g_{12}$, and $g_{24}$ in
$KO^0(BDI(4))$ such that $c(g_i)=\xi_i$, with $\xi_i$ as in \ref{psik}, and
$$KO^*(BDI(4))\approx KO^*[\![g_8,g_{12},g_{24}]\!].$$
The Adams operations $\psi^2$ and $\psi^3$ mod decomposables on the basis of
$g_i$'s is as in \ref{psik}.\end{thm}
\begin{pf} In \cite[2.1]{And}, it is proved that if there is a torsion-free subgroup
$F^*\subset KO^*(X)$ such that $F^*\otimes K^*(pt)\to K^*(X)$ is an isomorphism, then
so is $F^*\otimes KO^*(pt)\to KO^*(X)$.
The proof
is a Five Lemma argument using exact sequences in \cite[p.257]{Kar}.
Although the result is stated for ordinary (not 2-completed) $KO^*(-)$,
the same argument applies in the 2-completed context.  If $F^*$ is a multiplicative
subgroup, then the result holds as rings. Our result then follows from \ref{psik},
since the generators there are real.
 A similar proof can be derived from \cite[2.3]{Bo}.
\end{pf}

Next we need a similar sort of result about $KO^*(DI(4))$. We could derive much of
what we need by an argument similar to that just used, using the result of \cite{JO}
about $K^*(DI(4))$ as input. However, as we will need this in a specific form in
order to use it to draw conclusions about $KO^*(\Phi_1 DI(4))$, we begin by introducing
much terminology from \cite{Bo}.

The study of united $K$-theory begins with two categories, which will then be endowed
with additional structure. We begin with a partial definition of each, and their
relationship. For complete details, the reader will need to refer to \cite{Bo} or
an earlier paper of Bousfield.
\begin{defin}\label{CRdef} $(\cite[2.1]{Bo})$ A {\it $CR$-module} $M=\{M_C,M_R\}$
consists of ${\Bbb Z}$-graded $2$-profinite abelian groups $M_C$ and $M_R$ with continuous additive
operations $M_C^*@>B>\approx>M_C^{*-2}$, $M_R^*@>B_R>\approx>M_R^{*-8}$, $M_C^*@>t>\approx>M_C^*$,
$M_R^*@>\eta>>M_R^{*-1}$, $M_R^*@>c>>M_C^*$, $M_C^*@>r>>M_R^*$, satisfying 15 relations, which we
will mention as needed.\end{defin}
We omit the descriptor ``2-adic," which Bousfield properly uses, just as we omit writing the 2-adic
coefficients $\hat{\Bbb Z}_2$ which are present in all our $K$- and $KO$-groups.
\begin{expl} \label{CRexpl}For a spectrum or space $X$, the united $2$-adic $K$-cohomology
$$K_{CR}^*(X):=\{K^*(X),KO^*(X)\}$$
is a $CR$-module, with complex and real Bott periodicity, conjugation, the Hopf map, complexification,
and realification giving the respective operations.\end{expl}
\begin{defin}\label{deldef} $(\cite[4.1]{Bo})$ A $\Delta$-module $N=\{N_C,N_R,N_H\}$ is a triple of
$2$-profinite abelian groups $N_C$, $N_R$, and $N_H$ with continuous additive operations
$N_C@>t>\approx>N_C$, $N_R@>c>>N_C$, $N_C@>r>>N_R$, $N_H@>c'>>N_C$, and $N_C@>q>>N_H$ satisfying
nine relations.\end{defin}
\begin{expl}\label{deltaexpl} For a $CR$-module $M$ and an integer $n$, there is a $\Delta$-module $\Delta^nM=\{M_C^n,M_R^n,M_R^{n-4}\}$
with $c'=B^{-2}c$ and $q=rB^2$. In particular, for a space $X$ and integer $n$, there is a $\Delta$-module
$K^n_{\Delta}(X):=\Delta^nK^*_{CR}(X)$.\end{expl}

Now we add additional structure to these definitions.
\begin{defin}\label{AdDeldef} $(\cite[4.3,6.1]{Bo})$ A $\theta\Delta$-module is a $\Delta$-module
$N$ together with homomorphisms $N_C@>\theta>>N_C$, $N_R@>\theta>>N_R$, and $N_H@>\theta>>N_R$
satisfying certain relations listed in \cite[4.3]{Bo}.
An {\it Adams $\Delta$-module} is a $\theta\Delta$-module $N$ together with
Adams operations $N@>\psi^k>\approx >N$ for odd $k$ satisfying the familiar properties.
\end{defin}
\begin{expl} In the notation of Example \ref{deltaexpl}, $K_{\Delta}^{-1}(X)$ is an Adams $\Delta$-module
with $\theta=-\lambda^2$.\end{expl}
\begin{defin} $(\cite[2.6,3.1,3.2]{Bo})$ A special $\phi CR$-algebra $\{A_C,A_R\}$ is a $CR$-module with
bilinear $A^m_C\times A_C^n\to A_C^{m+n}$ and $A^m_R\times A_R^n\to A_R^{m+n}$ and also $A_C^0@>\phi>> A_R^0$
and $A_C^{-1}@>\phi>> A_R^0$ satisfying numerous properties.\end{defin}
\begin{rmk} {\rm The operations $\phi$, which were initially defined in \cite{BoK}, are less familiar
than the others. Two properties are $c\phi a=t(a)a$ and $\phi(a+b)=\phi a+\phi b+r(t(a)b)$ for $a,b\in A_C^0$.
For a connected space $X$, $K_{CR}^*(X)$ is a special $\phi CR$-algebra.}\end{rmk}

The following important lemma is taken from \cite{Bo}.
\begin{lem} $(\cite[4.5,4.6]{Bo})$ For any $\theta\Delta$-module $M$, there is a universal
special $\phi CR$-algebra $\hat LM$. This means that there is a morphism $M@>\alpha>> \hat LM$ such that
any morphism from $M$ into a $\phi CR$-algebra  factors as $\alpha$ followed by a unique $\phi CR$-algebra morphism.
There is an algebra isomorphism $\hat\Lambda M_C\to(\hat LM)_C$, where $\hat\Lambda(-)$ is the 2-adic
exterior algebra functor.\end{lem}

In \cite[2.7]{Bo}, Bousfield defines, for a $CR$-algebra $A$, the indecomposable quotient
$\hat QA$. We apply this to $A=K^*_{CR}(BDI(4))$, and consider the $\Delta$-module $\hat QK^0_\Delta(BDI(4))$,
analogous to \cite[4.10]{Bo}. We need the following result, which is more delicate than the
$K^{-1}_\Delta$-case considered in \cite[4.10]{Bo}.
\begin{lem} With $\theta=-\lambda^2$, the $\Delta$-module $\hat QK^0_\Delta(BDI(4))$ becomes
a $\theta\Delta$-module.\end{lem}
\begin{pf} First we need that $\theta$ is an additive operation. In \cite[3.6]{BoK}, it is shown
that $\theta(x+y)=\theta(x)+\theta(y)-xy$ if $x,y\in KO^n(X)$ with $n\equiv0\mod4$. The additivity
follows since we are modding out the product terms. (In the case $n\equiv-1\mod 4$ considered in
\cite[4.10]{Bo}, the additivity of $\theta$ is already present before modding out indecomposables.)

There are five additional properties which must be satisfied by $\theta$.
That $\theta cx=c\theta x$ and $\theta tz=t\theta z$ are easily obtained from \cite[3.4]{BoK}.
That $\theta c'y=c\theta y$ follows from \cite[6.2(iii),6.4]{BoTop}. That $\theta qz=\theta rz$ follows
from preceding \cite[3.10]{BoK} by $c$, which is surjective for us. Here we use that $rc=2$ and $q=rB^2$.
Finally, $\theta rz=r\theta z$ for us, since $c$ is surjective; here we have used the result $\bar\phi cx=0$
given in \cite[4.3]{Bo}.\end{pf}

Now we obtain the following important description of the $CR$-algebra $K^*_{CR}(DI(4))$.
\begin{thm}\label{DI4} There is a morphism of $\theta\Delta$-modules
$$\hat QK_\Delta^0(BDI(4))\to \tilde K_\Delta^{-1}(DI(4))$$
which induces an isomorphism of special $\phi CR$-algebras
$$\hat L(\hat QK_\Delta^0(BDI(4)))\to K^*_{CR}(DI(4)).$$
\end{thm}
\begin{pf} The map $\Sigma DI(4)=\Sigma\Omega BDI(4)\to BDI(4)$ induces a morphism
$$K_\Delta^0(BDI(4))\to K^{-1}_\Delta(DI(4))$$ which factors through the indecomposable quotient
$\hat QK_\Delta^0(BDI(4))$.
In \cite[1.2]{JO}, a general result is proved which implies that $K^*(DI(4))$ is an exterior algebra
on elements of $K^1(DI(4))$ which correspond to the generators of the power series algebra
$K^*(BDI(4))$ under the above morphism followed by the Bott map. Thus our result will follow
from \cite[4.9]{Bo}, once we have shown that the $\theta\Delta$-module $M:=\hat QK_\Delta^0(BDI(4))$
is robust(\cite[4.7]{Bo}).  This requires that $M$ is profinite, which follows as in the remark following
\cite[4.7]{Bo}, together with two properties regarding $\bar\phi$, where $\bar\phi z:=\theta rz-r\theta z$
for $z\in M_C$. In our case, $c$ is surjective, and so $\bar\phi=0$ as used in the previous proof.

One property is that $M$ is torsion-free and exact. This follows from the Bott exactness of
the $CR$-module $K_{CR}^*(BDI(4))$ noted in \cite[2.2]{Bo}, and \cite[5.4]{Bo}, which states that,
for any $n$, the $\Delta$-module $\Delta^nN$ associated to a Bott exact $CR$-module $N$ with $N^n_C$
torsion-free and $N^{n-1}_C=0$ is torsion-free. The other property is $\on{ker}(\bar\phi)=cM_R+c'M_H+2M_C$.
For us, both sides equal $M_C$ since $c$ is surjective and $\bar\phi=0$.\end{pf}

Our Theorem \ref{KO} now follows from \cite[9.5]{Bo} once we have shown that the Adams $\Delta$-module
$M:=\hat QK_\Delta^0(BDI(4))$ is ``strong." (\cite[7.11]{Bo}) This result (\cite[9.5]{Bo}) requires that
the space (here $DI(4)$) be an $H$-space (actually $K/2_*$-durable, which is satisfied by $H$-spaces)
and that it satisfies the conclusion of our \ref{DI4}. It then deduces that $KO^*(\Phi_1DI(4))$ fits into an
exact sequence which reduces to ours provided $M_R=M_C$ and $M_H=2M_C$. These equalities are implied by
$\hat QK_\Delta^0(BDI(4))$ being exact, as was noted to be true in the previous proof, plus $t=1$ and $c$ surjective, as were noted to be true in \ref{psik}. Indeed, the exactness
property, (\cite[4.2]{Bo}), includes that $cM_R+c'M_H=\on{ker}(1-t)$ and $cM_R\cap c'M_H=\on{im}(1+t)$. Another
perceptible difference is that Bousfield's exact sequence is in terms of $\bar M:=M/\bar\phi$, while ours involves
$M$, but these are equal since, as already observed, $\bar\phi=0$ since $c$ is surjective.

Note also that the Adams operations in $\bar M$ in the exact sequence of \cite[9.5]{Bo}, which reduces to that
in our \ref{KO}, are those in the Adams $\Delta$-module $\hat QK_\Delta^0(BDI(4))$, which are given
in our \ref{psik}. The morphism $\theta$ in \cite[9.5]{Bo} or our \ref{KO} is $\frac12\psi^2$, since
this equals $-\lambda^2$ mod decomposables.

Finally, we show that our $M$ is strong. One of the three criteria for being strong is to be robust,
and we have already discussed and verified this. The second requirement for an Adams $\Delta$-module
to be strong is that it be ``regular."  This rather technical condition is defined in \cite[7.8]{Bo}.
In \cite[7.9]{Bo}, a result is proved which immediately implies that $\tilde K_\Delta^{-1}(DI(4))$ is
regular. By  \ref{DI4}, our $M$ injects into
$\tilde K_\Delta^{-1}(DI(4))$, and so by \cite[7.10]{Bo}, which states that a submodule of a regular
module is regular, our $M$ is regular.

The third requirement for $M$ to be strong is that it be $\psi^3$-splittable (\cite[7.2]{Bo}), which
means that the quotient map $M\to M/\bar\phi$ has a right inverse. As we have noted several times,
we have $\bar\phi=0$, and so the identity map serves as a right inverse to the identity map.
This completes the proof that our $M$ is strong, and hence that \cite[9.5]{Bo} applies to $DI(4)$
to yield our Theorem \ref{KO}.

\def\rule{.6in}{.6pt}{\rule{.6in}{.6pt}}

redpajama_set_name: RedPajamaArXiv


text:
Q: Django sem visualização de models no admin de usuario comum mesmo tendo permissões e estando no mesmo grupo Boa tarde,
Então estou tentando dar permissões que já vem no django para um usuário comum que mesmo pertencendo ao grupo não consegue acessar os models que é concedido a ele. Alguém teve esse problema da não visualização de models no admin. versão do python 2.7 e django 1.10
já tenho as conf do settings como  'django.contrib.auth','django.contrib.auth.middleware.AuthenticationMiddleware'
no models eu tenho apenas:
class Paciente(models.Model):
    nome = models.CharField(max_length=60)
    data_nascimento = models.DateField()
    def __str__(self):
        return self.nome
    class Meta:
        verbose_name = "Paciente"
        verbose_name_plural = "Pacientes"

e no admin tenho:
class PacienteAdmin(ModelAdmin):
    list_display = ('nome','data_nascimento')

admin.site.register(Paciente, PacienteAdmin)

apenas isso e mesmo assim sem visualização do mesmo.
obrigada desde já.

A: Fiz um teste com seu codigo e funciona perfeitamente, o único erro que vejo no código que vc postou aqui é no fragmento (admin.py):
class PacienteAdmin(ModelAdmin):
   list_display = ('nome','data_nascimento')

Troque a linha:
class PacienteAdmin(ModelAdmin):

Para:
class Paciente(models.Model):

Se não funcionar, tente seguir os passos:
1) (linha de comando)
$ django-admin startproject myproject .

2) (linha de comando)
$ ./manage.py migrate

3) (linha de comando)
$ ./manage.py createsuperuser

4) (linha de comando)
$ ./manage.py startapp myapp

Após esses passos, a árvore do seu projeto devera parecer com a figura abaixo:

5) Edit o arquivo models.py no diretorio myapp para que fique assim:
from django.db import models

class Paciente(models.Model):
    nome = models.CharField(max_length=60)
    data_nascimento = models.DateField()
    def __str__(self):
        return self.nome
    class Meta:
        verbose_name = "Paciente"
        verbose_name_plural = "Pacientes"

6) Edit o arquivo admin em myapp, para que fique assim:
from django.contrib import admin
from .models import Paciente

class PacienteAdmin(admin.ModelAdmin):
    list_display = ('nome','data_nascimento')

admin.site.register(Paciente, PacienteAdmin)

7) Edite o arquivo settings.py para adiconar myapp, então a seção INSTALLED_APPS, deverá ficar assim:
INSTALLED_APPS = [
    'myapp',
    'django.contrib.admin',
    'django.contrib.auth',
    'django.contrib.contenttypes',
    'django.contrib.sessions',
    'django.contrib.messages',
    'django.contrib.staticfiles',
]

Após esses passos, execute o projeto com runserver, entre em admin, crie um grupo e um usuario, dê permissao para que o usuario possa incluir mas não possa deletar pacientes. Se eu não cometi nenhum erro ao postar aqui e voce seguiu todos os passos corretamente, o resultado será o esperado.
Obs.:
O exemplo expresso aqui foi testado (e aprovado :-)) em um ambiente Linux.

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