Connects the homology of the symmetric groups with mapping spaces of spheres
In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem (also referred to as Barratt–Priddy–Quillen theorem) expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem (named after Michael Barratt, Stewart Priddy, and Daniel Quillen) is also often stated as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction.
Statement of the theorem.
The mapping space formula_1 is the topological space of all continuous maps formula_2 from the n-dimensional sphere formula_3 to itself, under the topology of uniform convergence (a special case of the compact-open topology). These maps are required to fix a basepoint formula_4, satisfying formula_5, and to have degree 0; this guarantees that the mapping space is connected. The Barratt–Priddy theorem expresses a relation between the homology of these mapping spaces and the homology of the symmetric groups formula_6.
It follows from the Freudenthal suspension theorem and the Hurewicz theorem that the kth homology formula_7 of this mapping space is "independent" of the dimension n, as long as formula_8. Similarly, Minoru Nakaoka (1960) proved that the kth group homology formula_9 of the symmetric group formula_6 on n elements is independent of n, as long as formula_11. This is an instance of homological stability.
The Barratt–Priddy theorem states that these "stable homology groups" are the same: for formula_11, there is a natural isomorphism
formula_13
This isomorphism holds with integral coefficients (in fact with any coefficients, as is made clear in the reformulation below).
Example: first homology.
This isomorphism can be seen explicitly for the first homology formula_14. The first homology of a group is the largest commutative quotient of that group. For the permutation groups formula_6, the only commutative quotient is given by the sign of a permutation, taking values in {−1, 1}. This shows that formula_16, the cyclic group of order 2, for all formula_17. (For formula_18, formula_19 is the trivial group, so formula_20.)
It follows from the theory of covering spaces that the mapping space formula_21 of the circle formula_22 is contractible, so
formula_23. For the 2-sphere formula_24, the first homotopy group and first homology group of the mapping space are both infinite cyclic:
formula_25. 
A generator for this group can be built from the Hopf fibration formula_26. Finally, once formula_27, both are cyclic of order 2:
formula_28. 
Reformulation of the theorem.
The infinite symmetric group formula_29 is the union of the finite symmetric groups formula_30, and Nakaoka's theorem implies that the group homology of formula_29 is the stable homology of formula_30: for formula_33, 
formula_34.
The classifying space of this group is denoted formula_35, and its homology of this space is the group homology of formula_29: 
formula_37.
We similarly denote by formula_38 the union of the mapping spaces formula_39 under the inclusions induced by suspension. The homology of formula_38 is the stable homology of the previous mapping spaces: for formula_8, 
formula_42
There is a natural map formula_43; one way to construct this map is via the model of formula_44 as the space of finite subsets of formula_45 endowed with an appropriate topology. An equivalent formulation of the Barratt–Priddy theorem is that formula_46 is a "homology equivalence" (or "acyclic map"), meaning that formula_46 induces an isomorphism on all homology groups with any local coefficient system.
Relation with Quillen's plus construction.
The Barratt–Priddy theorem implies that the space "BΣ"∞+ resulting from applying Quillen's plus construction to "BΣ"∞ can be identified with Map0("S"∞,"S"∞). (Since π1(Map0("S"∞,"S"∞))≅"H"1("Σ"∞)≅Z/2Z, the map "φ": "BΣ"∞→Map0("S"∞,"S"∞) satisfies the universal property of the plus construction once it is known that φ is a homology equivalence.)
The mapping spaces Map0("S""n","S""n") are more commonly denoted by "Ω""n"0"S""n", where "Ω""n""S""n" is the n-fold loop space of the n-sphere Sn, and similarly Map0("S"∞,"S"∞) is denoted by "Ω"∞0"S"∞. Therefore the Barratt–Priddy theorem can also be stated as
formula_48 or
formula_49
In particular, the homotopy groups of "BΣ"∞+ are the stable homotopy groups of spheres:
formula_50
""K"-theory of F1".
The Barratt–Priddy theorem is sometimes colloquially rephrased as saying that "the "K"-groups of F1 are the stable homotopy groups of spheres". This is not a meaningful mathematical statement, but a metaphor expressing an analogy with algebraic "K"-theory.
The "field with one element" F1 is not a mathematical object; it refers to a collection of analogies between algebra and combinatorics. One central analogy is the idea that "GL""n"(F1) should be the symmetric group "Σ""n".
The higher "K"-groups "K""i"("R") of a ring "R" can be defined as
formula_51
According to this analogy, the K-groups "K""i"(F1) of F1 should be defined as π"i"("BGL"∞(F1)+)
π"i"("BΣ"∞+), which by the Barratt–Priddy theorem is:
formula_52