From Manifold Atlas
Jump to: navigation, search

This page has not been refereed. The information given here might be incomplete or provisional.


[edit] 1 Introduction

Let K be a finite connected CW-complex of dimension k. For a given n = k + q > k we would like to know if there is a compact manifold M with boundary \partial M such that:

  1. the map \pi_1(\partial M) \to \pi_1(M) is an isomorphism,
  2. M is homotopy equivalent to K.

In this case we say that M thickens K. If there is such a manifold M, we would like to know how many up to homeomorphism or diffeomorphism if M is smooth.

In [Wall1966a] Wall introduced the notion of a thickening, defined below, to investigate the questions raised above. This page summarizes the basic results concerning thickenings.

Recall that \Cat = \Top, \PL or \Diff denotes respectively the topological, piecewise linear or smooth categories.

Definition 1.1 [Wall1966a, Section 1]. Let K be a finite connected CW complex. An n-dimensional \Cat-thickening of K consists of

  1. a compact n-dimensional \Cat-manifold M with connected boundary such that \pi_1(\partial M) \cong \pi_1(M)
  2. a basepoint m \in \partial M \subset M and an orientation of TM_m,
  3. a simple homotopy equivalence \phi : K \to M.

Two thickenings (M_0, \phi_0) and (M_1, \phi_1) are isomorphic if there is a \Cat-isomorphism f \colon M_0 \cong M_1 preserving m and the orientation of TM_m and such that f_0 is simple homotopic to \phi_1 \circ f. In particular there is a simple homotopy commutative diagram:

\displaystyle  \xymatrix{ M_0 \ar[dr]^{\phi_0} \ar[0,2]^{f} & & M_1 \ar[dl]_{\phi_1} \\ & K}

The set of isomorphism classes of n-dimensional \Cat-thickenings over K is denoted

\displaystyle  \mathcal{T}^n(K) := \{ [\phi: K \simeq M ] \}.

[edit] 2 Constructions and examples

The simplest examples of thickenings come from q-disc bundles with sections over manifolds, q > 2. Let X be a closed \Cat-manifold of dimension k, let M \to X be a bundle with fibre D^q and with section s \colon M \to X. Then

  • the pair (M, s) is an n-thickening of X.
  • the pair (D^n, pt) is an n-thickening of a point.

[edit] 3 Invariants

An important invariant of a thickening (M, \phi) is the induced stable \Cat-bundle over K:

\displaystyle  T(M, \phi) := \phi^* TM

where TM \to M is the stable \Cat-tangent bundle of M. Given that stable bundles over the space K are classified by maps to the classifying space B\Cat one equivalently thinks of

\displaystyle  T(M, \phi) = [f_{TM} \circ \phi]

where f_{TM} : M \to B\Cat is the classifying map. Clearly the isomorphism class of the bundle \phi^* TM \to K is an invariant of the thickening class [M, \phi]. This if Bun_\Cat(K) denotes the set of isomorphism classes of stable \Cat-bundles over K we obtain a map

\displaystyle  \tau_K : \mathcal{T}^n(K) \to Bun_\Cat(K).

[edit] 4 Classification

An extremely useful classification theorem in manifold theory is the classification of stable thickenings where n \geq 2k+1 originally due to Wall in the smooth catagory.

Theorem 4.1 [Wall1966a, Proposition 5.1] [Chazin1971, Theorem 1]. For all n \geq 2k+1, the stable classifying map gives rise to a set bijection

\displaystyle  \tau_K : \mathcal{T}^n(K) \equiv [K, B\Cat], \quad [M, \phi] \mapsto T(M, \phi) = [\phi \circ f_{TM}].

[edit] 5 References

Personal tools