Whitehead torsion

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence formula_1 of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion formula_2 which is an element in the Whitehead group formula_3. 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 formula_4 vanishes.
Whitehead group.
The Whitehead group of a connected CW-complex or a manifold "M" is equal to the Whitehead group formula_5 of the fundamental group formula_6 of "M".
If "G" is a group, the Whitehead group formula_7 is defined to be the cokernel of the map formula_8 which sends ("g", ±1) to the invertible (1,1)-matrix (±"g"). Here formula_9 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 formula_7 of a group "G" is the quotient of formula_11 by the subgroup generated by elementary matrices, elements of "G" and formula_12. Notice that this is the same as the quotient of the reduced K-group formula_13 by "G".
The Whitehead torsion.
At first we define the Whitehead torsion formula_21 for a chain homotopy equivalence formula_22 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. Let formula_23 be any chain contraction of the mapping cone, i.e., formula_24 for all "n". We obtain an isomorphism formula_25 with 
We define formula_27, where "A" is the matrix of formula_28 with respect to the given bases.
For a homotopy equivalence formula_29 of connected finite CW-complexes we define the Whitehead torsion formula_30 as follows. Let formula_31 be the lift of formula_32 to the universal covering. It induces formula_33-chain homotopy equivalences formula_34. Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in formula_35 which we map to Wh(π1("Y")). This is the Whitehead torsion τ(ƒ) ∈ Wh(π1("Y")).
Properties.
Homotopy invariance: Let formula_36 be homotopy equivalences of finite connected CW-complexes. If "f" and "g" are homotopic, then formula_37.
Topological invariance: If formula_38 is a homeomorphism of finite connected CW-complexes, then formula_39.
Composition formula: Let formula_38, formula_41 be homotopy equivalences of finite connected CW-complexes. Then formula_42.
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 formula_43 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 handle 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 ("X"1, "A") and ("X"2, "A") are said to be equivalent, if there is a simple homotopy equivalence between "X"1 and "X"2 relative to "A". The set of such equivalence classes form a group where the addition is given by taking union of "X"1 and "X"2 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.