In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence
of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion
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 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
The Whitehead group of a connected CW-complex or a manifold M is equal to the Whitehead group
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.
by the subgroup generated by elementary matrices, elements of G and
Notice that this is the same as the quotient of the reduced K-group
of finite based free R-chain complexes.
We can assign to the homotopy equivalence its mapping cone C* := cone*(h*) which is a contractible finite based free R-chain complex.
be any chain contraction of the mapping cone, i.e.,
of connected finite CW-complexes we define the Whitehead torsion
Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in
This is the Whitehead torsion τ(ƒ) ∈ Wh(π1(Y)).
be homotopy equivalences of finite connected CW-complexes.
is a homeomorphism of finite connected CW-complexes, then
be homotopy equivalences of finite connected CW-complexes.
The s-cobordism theorem states for a closed connected oriented manifold M of dimension n > 4 that an h-cobordism W between M and another manifold N is trivial over M if and only if the Whitehead torsion of the inclusion
Moreover, for any element in the Whitehead group there exists an h-cobordism W over M whose Whitehead torsion is the considered element.
There exists a homotopy theoretic analogue of the s-cobordism theorem.
Given a CW-complex A, consider the set of all pairs of CW-complexes (X, A) such that the inclusion of A into X is a homotopy equivalence.
The set of such equivalence classes form a group where the addition is given by taking union of X1 and X2 with common subspace A.