Thickenings
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Introduction
Let be a finite connected CW-complex of dimension
. For a given
we would like to know if there is a compact manifold
with boundary
such that:
- the map
is an isomorphism,
-
is homotopy equivalent to
.
In this case we say that thickens
. If there is such a manifold
, we would like to know how many up to homeomorphism or diffeomorphism if
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 or
denotes respectively the topological, piecewise linear or smooth categories.
Definition 1.1 [Wall1966a, Section 1].
Let be a finite connected CW complex. An
-dimensional
-thickening of
consists of
- a compact
-dimensional
-manifold
with connected boundary such that
- a basepoint
and an orientation of
,
- a simple homotopy equivalence
.
Two thickenings and
are isomorphic if there is a
-isomorphism
preserving
and the orientation of
and such that
is simple homotopic to
. 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}](/images/math/e/7/b/e7b93f90388fbe391b8ba5e78d5a0abe.png)
The set of isomorphism classes of -dimensional
-thickenings over
is denoted
![\displaystyle \mathcal{T}^n(K) := \{ [\phi: K \simeq M ] \}.](/images/math/6/f/2/6f2f2839b8992a717559a381a9424f72.png)
[edit] 2 Constructions and examples
The simplest examples of thickenings come from -disc bundles with sections over manifolds,
. Let
be a closed
-manifold of dimension
, let
be a bundle with fibre
and with section
. Then
- the pair
is an
-thickening of
.
- the pair
is an
-thickening of a point.
[edit] 3 Invariants
An important invariant of a thickening is the induced stable
-bundle over
:

where is the stable
-tangent bundle of
. Given that stable bundles over the space
are classified by maps to the classifying space
one equivalently thinks of
![\displaystyle T(M, \phi) = [f_{TM} \circ \phi]](/images/math/0/2/8/02809eaecaaddf353ed870a4aad4f046.png)
where is the classifying map. Clearly the isomorphism class of the bundle
is an invariant of the thickening class
. This if
denotes the set of isomorphism classes of stable
-bundles over
we obtain a map

[edit] 4 Classification
An extremely useful classification theorem in manifold theory is the classification of stable thickenings where originally due to Wall in the smooth catagory.
Theorem 4.1 [Wall1966a, Proposition 5.1] [Chazin1971, Theorem 1].
For all , 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}].](/images/math/1/8/d/18def848b04339f88703c6eb63e0310c.png)
[edit] 5 References
- [Chazin1971] R. L. Chazin, Stable thickenings in the topological category, Proc. Amer. Math. Soc. 29 (1971), 175–178. MR0296950 (45 #6009) Zbl 0214.22302
- [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501