Whitehead torsion

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence ƒ: XY of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion τ(ƒ) which is an element in the Whitehead group Wh(π1(Y)). These 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 Kirby and Siebenmann. The restriction to manifolds of dimension >4 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 \hookrightarrow W vanishes.

The Whitehead group

The Whitehead group of a CW-complex or a manifold M is equal to the Whitehead group Wh(π1(M)) of the fundamental group π1(M) of M.

If G is a group, the Whitehead group Wh(G) is defined to be the cokernel of the map G × {±1} → K1(Z[G]) which sends (g,±1) to the invertible (1,1)-matrix (±g). Here 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) of a group G is the quotient of GL(Z[G]) by the subgroup generated by elementary matrices, elements of G and 1. Notice that this is the same as the quotient of the reduced K-group {\tilde K}_1(\mathbf{Z}[G]) by G.

Examples

The Whitehead torsion

At first we define the Whitehead torsion \tau(h_*) \in {\tilde K}_1(R) for a chain homotopy equivalence h_*: D_* \to E_* 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 \gamma_*: C_* \to C_{*+1} be any chain contraction of the mapping cone, i.e. c_{n+1} \circ \gamma_n + \gamma_{n-1} \circ c_n = \operatorname{id}_{C_n} for all n. We obtain an isomorphism (c_* + \gamma_*)_\mathrm{odd}: C_\mathrm{odd} \to C_\mathrm{even} with C_\mathrm{odd} := \oplus_{n \text{ odd}} \, C_n, C_\mathrm{even} := \oplus_{n \text{ even}} \, C_n. We define \tau(h_*) := [A] \in {\tilde K}_1(R), where A is the matrix of (c* + γ*)odd with respect to the given bases.

For a homotopy equivalence ƒ: XY of connected finite CW-complexes we define the Whitehead torsion τ(ƒ) ∈ Wh(π1(Y)) as follows. Let {\tilde f}: {\tilde X} \to {\tilde Y} be the lift of ƒ: XY to the universal covering. It induces Z1(Y)]-chain homotopy equivalences C_*({\tilde f}): C_*({\tilde X}) \to C_*({\tilde Y}). Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in {\tilde K}_1(\mathbf{Z}[\pi_1(Y)]) which we map to Wh(π1(Y)). This is the Whitehead torsion τ(ƒ) ∈ Wh(π1(Y)).

Properties

Homotopy invariance: Let ƒ, g: XY be homotopy equivalences of finite connected CW-complexes. If ƒ and g are homotopic then τ(ƒ) = τ(g).

Topological invariance: If ƒ: XY is a homeomorphism of finite connected CW-complexes then τ(ƒ) = 0.

Composition formula: Let ƒ: XY, g: YZ be homotopy equivalences of finite connected CW-complexes. Then \tau(g \circ f) = g_* \tau(f) + \tau(g).

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 \hookrightarrow 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 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 (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.

See also

References

External links

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