Thickenings
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Contents |
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 basis results concerning thickenings.
Recall that or
denotes respectively the topological, piecewise linear or smooth categories.
Definition 0.1 [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)
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 $M \to W$ be a bundle with fibre $D^q$ and with section $s \colon M \to X$.
- the pair $(M, s)$ is an $n$-thickening of $X$.
- the pair $(D^, pt)$ is an $n$-thickening of a point.
3 Invariants
An important invariant of a thickening is the induced stable
-bundle over
:
![\displaystyle T(M, \phi) = \phi^* TM](/images/math/5/f/3/5f3a50b48423ad2d46fe7b13232c4863.png)
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.
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 3.1 \cite{} \cite{.}
For all , the stable classifying map gives rise to a set bijection
![\displaystyle \mathcal{T}^n(K) \equiv [K, B\Cat], \quad [M, \phi] \mapsto T(M, \phi) = [\phi \circ f_{TM}].](/images/math/e/0/3/e039e9b31f39a37e1a35602e33d50a7b.png)
5 References
- [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501