Whitehead Torsion - Geometric Interpretation

Geometric Interpretation

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 M W vanishes. Moreover, for any element in the Whitehead group there exists an h-cobordism W over M whose Whitehead torsion is the considered element. The proofs use handlebody decompositions.

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. Two pairs (X₁, A) and (X₂, A) are said to be equivalent, if there is a simple homotopy equivalence between X₁' and X₂ relative to A. The set of such equivalence classes form a group where the addition is given by taking union of X'₁ and X₂ with common subspace A. This group is natural isomorphic to the Whitehead group Wh(A) of the CW-complex A. The proof of this fact is similar to the proof of s-cobordism theorem.