Automorphisms of T are easily constructed from automorphisms of the lattice Zn, which are classified by invertible integral matrices of size n with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on Rn in the usual way, one has the typical toral automorphism on the quotient. The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H•(Tn, Z) can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles. Configuration space
As the n-torus is the n-fold product of the circle, the n-torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, Tn = (S1)n. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold Tn/Sn, which is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates). For n = 2, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide.
Dysmelia (from the Greek dys (δυσ-), "bad" + mélos (μέλος), "limb" + English suffix -ia) is a congenital disorder of a limb resulting from a disturbance in embryonic development. Types
Dysmelia can refer to
missing (aplasia) limbs: amelia, oligodactyly, congenital amputation e.g. tibial or radial aplasia
malformation of limbs: shortening (micromelia, rhizomelia or mesomelia), ectrodactyly, phocomelia, meromelia, syndactyly, brachydactyly, club foot
too many limbs: polymelia, polydactyly, polysyndactyly
others: tetraamelia, hemimelia, symbrachydactyly

Occurrence rate
Birth defects involving limbs occur in 1 per 1000. Causes
Dysmelia can be caused by
Inheritance of abnormal genes, e.g. polydactyly, ectrodactyly or brachydactyly, symptoms of deformed limbs then often occur in combination with other symptoms (syndromes)
external causes during pregnancy (thus not inherited), e.g. via amniotic band syndrome
teratogenic drugs (e.g. thalidomide, which causes phocomelia) or environmental chemicals
ionizing radiation (nuclear weapons, radioiodine, radiation therapy)
infections
metabolic imbalance

Syndromes with dysmelia
References
External links
DysNet: An organisation for people affected by Dysmelia (congenital limb difference)
Reach: Association for Children with Upper Limb Deficiency)