In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f : X → Y {\displaystyle f\colon X\to Y} of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion τ ( f ) {\displaystyle \tau (f)} which is an element in the Whitehead group Wh ( π 1 ( Y ) ) {\displaystyle \operatorname {Wh} (\pi _{1}(Y))} . These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying surgery theory to non-simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first obtained in the early 1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and Laurent C. Siebenmann. The restriction to manifolds of dimension greater than four are due to the application of the Whitney trick for removing double points. In generalizing the h-cobordism theorem, which is a statement about simply connected manifolds, to non-simply connected manifolds, one must distinguish simple homotopy equivalences and non-simple homotopy equivalences. While an h-cobordism W between simply-connected closed connected manifolds M and N of dimension n > 4 is isomorphic to a cylinder (the corresponding homotopy equivalence can be taken to be a diffeomorphism, PL-isomorphism, or homeomorphism, respectively), the s-cobordism theorem states that if the manifolds are not simply-connected, an h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion M ↪ W {\displaystyle M\hookrightarrow W} vanishes.
Whitehead group The Whitehead group of a connected CW-complex or a manifold M is equal to the Whitehead group Wh ( π 1 ( M ) ) {\displaystyle \operatorname {Wh} (\pi _{1}(M))} of the fundamental group π 1 ( M ) {\displaystyle \pi _{1}(M)} of M. If G is a group, the Whitehead group Wh ( G ) {\displaystyle \operatorname {Wh} (G)} is defined to be the cokernel of the map G × { ± 1 } → K 1 ( Z [ G ] ) {\displaystyle G\times \{\pm 1\}\to K_{1}(\mathbb {Z} [G])} which sends (g, ±1) to the invertible (1,1)-matrix (±g). Here Z [ G ] {\displaystyle \mathbb {Z} [G]} is the group ring of G. Recall that the K-group K1(A) of a ring A is defined as the quotient of GL(A) by the subgroup generated by elementary matrices. The group GL(A) is the direct limit of the finite-dimensional groups GL(n, A) → GL(n+1, A); concretely, the group of invertible infinite matrices which differ from the identity matrix in only a finite number of coefficients. An elementary matrix here is a transvection: one such that all main diagonal elements are 1 and there is at most one non-zero element not on the diagonal. The subgroup generated by elementary matrices is exactly the derived subgroup, in other words the smallest normal subgroup such that the quotient by it is abelian. In other words, the Whitehead group Wh ( G ) {\displaystyle \operatorname {Wh} (G)} of a group G is the quotient of GL ( Z [ G ] ) {\displaystyle \operatorname {GL} (\mathbb {Z} [G])} by the subgroup generated by elementary matrices, elements of G and ± 1 {\displaystyle \pm 1} . Notice that this is the same as the quotient of the reduced K-group K ~ 1 ( Z [ G ] ) {\displaystyle {\tilde {K}}_{1}(\mathbb {Z} [G])} by G.
… excerpt ends here. Continue reading the full article.
