Topology manifold

Topology of manifolds

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The branch of the theory of manifolds [cf. Manifold] concerned with the study of relations between different types of manifolds.

The most important types of finite-dimensional manifolds and relations between them are illustrated in [1].

$$ \tag{1 } \begin{array}{ccc} \mathop{\rm P} &{} & \mathop{\rm P} [ \mathop{\rm ANR} ] \\ \uparrow &{} &\uparrow \\ \mathop{\rm H} &{} & \mathop{\rm H} [ \mathop{\rm ANR} ] \\ {} &{} \mathop{\rm TOP} &{} \\ \mathop{\rm TRI} &\uparrow & \mathop{\rm Lip} \\ {} & \mathop{\rm Handle} &{} \\ {} &\uparrow &{} \\ {} & \mathop{\rm PL} &{} \\ {} &\uparrow &{} \\ {} & \mathop{\rm Diff} &{} \\ \end{array} $$

Here $ \mathop{\rm Diff} $ is the category of differentiable [smooth] manifolds; $ \mathop{\rm PL} $ is the category of piecewise-linear [combinatorial] manifolds; $ \mathop{\rm TRI} $ is the category of topological manifolds that are polyhedra; $ \mathop{\rm Handle} $ is the category of topological manifolds admitting a topological decomposition into handles; $ \mathop{\rm Lip} $ is the category of Lipschitz manifolds [with Lipschitz transition mappings between local charts]; $ \mathop{\rm TOP} $ is the category of topological manifolds [Hausdorff and with a countable base]; $ \mathop{\rm H} $ is the category of polyhedral homology manifolds without boundary [polyhedra, the boundary of the star of each vertex of which has the homology of the sphere of corresponding dimension]; $ \mathop{\rm H} [ \mathop{\rm ANR} ] $ is the category of generalized manifolds [finite-dimensional absolute neighbourhood retracts $ X $ that are homology manifolds without boundary, i.e. with the property that for any point $ x \in X $ the group $ H ^ {*} [ X, X \setminus x; \mathbf Z ] $ is isomorphic to the group $ H ^ {*} [ \mathbf R ^ {n} , \mathbf R ^ {n} \setminus 0; \mathbf Z ] $]; $ \mathop{\rm P} [ \mathop{\rm ANR} ] $ is the category of Poincaré spaces [finite-dimensional absolute neighbourhood retracts $ X $ for which there exists a number $ n $ and an element $ \mu \in H _ {n} [ X] $ such that $ H _ {r} [ X, \mathbf Z ] = 0 $ when $ r \geq n + 1 $, and the mapping $ \mu \cap : H ^ {r} [ X] \rightarrow H _ {n - r } [ X] $ is an isomorphism for all $ r $]; and $ \mathop{\rm P} $ is the category of Poincaré polyhedra [the subcategory of the preceding category consisting of polyhedra].

The arrows of [1], apart from the 3 lower ones and the arrows $ \mathop{\rm H} \rightarrow \mathop{\rm TOP} \rightarrow \mathop{\rm P} $, denote functors with the structure of forgetting functors. The arrow $ \mathop{\rm Diff} \rightarrow \mathop{\rm PL} $ expresses Whitehead's theorem on the triangulability of smooth manifolds. In dimensions $ < 8 $ this arrow is reversible [an arbitrary $ \mathop{\rm PL} $- manifold is smoothable] but in dimensions $ \geq 8 $ there are non-smoothable $ \mathop{\rm PL} $- manifolds and even $ \mathop{\rm PL} $- manifolds that are homotopy inequivalent to any smooth manifold. The imbedding $ \mathop{\rm PL} \subset \mathop{\rm TRI} $ is also irreversible in the same strong sense [there exist polyhedral manifolds of dimension $ \geq 5 $ that are homotopy inequivalent to any $ \mathop{\rm PL} $- manifold]. Here already for the sphere $ S ^ {n} $, $ n \geq 5 $, there exist triangulations in which it is not a $ \mathop{\rm PL} $- manifold.

The arrow $ \mathop{\rm PL} \rightarrow \mathop{\rm Handle} $ expresses the fact that every $ \mathop{\rm PL} $- manifold has a handle decomposition.

The arrow $ \mathop{\rm PL} \rightarrow \mathop{\rm Lip} $ expresses the theorem on the existence of a Lipschitz structure on an arbitrary $ \mathop{\rm PL} $- manifold.

The arrow $ \mathop{\rm Handle} \rightarrow \mathop{\rm TOP} $ is reversible if $ n \neq 4 $ and irreversible if $ n = 4 $[ an arbitrary topological manifold of dimension $ n \neq 4 $ admits a handle decomposition and there exist four-dimensional topological manifolds for which this is not true].

Similarly, if $ n \neq 4 $ the arrow $ \mathop{\rm Lip} \rightarrow \mathop{\rm TOP} $ is reversible [and moreover in a unique way].

The question on the reversibility of the arrow $ \mathop{\rm TRI} \rightarrow \mathop{\rm TOP} $ gives the classical unsolved problem on the triangulability of arbitrary topological manifolds.

The arrow $ \mathop{\rm H} \rightarrow \mathop{\rm P} $ is irreversible in the strong sense [there exist Poincaré polyhedra that are homotopy inequivalent to any homology manifold].

The arrow $ \mathop{\rm H} \rightarrow \mathop{\rm TOP} $ expresses a theorem on the homotopy equivalence of an arbitrary homology manifold of dimension $ n \geq 5 $ to a topological manifold.

The arrow $ \mathop{\rm TOP} \rightarrow \mathop{\rm P} $ expresses the theorem on the homotopy equivalence of an arbitrary topological manifold to a polyhedron.

The imbedding $ \mathop{\rm TOP} \subset \mathop{\rm H} [ \mathop{\rm ANR} ] $ expresses that an arbitrary topological manifold is an $ \mathop{\rm ANR} $.

The similar question for the arrows $ \mathop{\rm Diff} \rightarrow \mathop{\rm PL} \rightarrow \mathop{\rm TOP} \rightarrow \mathop{\rm P} $ has been solved using the theory of stable bundles [respectively, vector, piecewise-linear, topological, and spherical bundles], i.e. by examining the homotopy classes of mappings of a manifold $ X $ into the corresponding classifying spaces BO, BPL, BTOP, BG.

There exist canonical composition mappings

$$ \tag{2 } \mathop{\rm BO} \rightarrow \mathop{\rm BPL} \rightarrow \mathop{\rm BTOP} \rightarrow \mathop{\rm BG} , $$

of which the homotopy fibres and the homotopy fibres of their compositions are denoted, respectively, by the symbols

$$ \mathop{\rm PL} / \mathop{\rm O} , \mathop{\rm TOP} / \mathop{\rm O} , \mathop{\rm G} / \mathop{\rm O} , \mathop{\rm TOP} / \mathop{\rm PL} ,\ \mathop{\rm G} / \mathop{\rm PL} , \mathop{\rm G} / \mathop{\rm TOP} . $$

For every manifold $ X $ from a category $ \mathop{\rm Diff} $, $ \mathop{\rm PL} $, $ \mathop{\rm TOP} $, $ \mathop{\rm P} $ there exists a normal stable bundle, i.e. a canonical mapping $ \tau _ {X} $ from $ X $ into the corresponding classifying space.

In the transition from a "narrow" category of manifolds to a "wider" one, for example, from smooth to piecewise-linear, the mapping $ \tau _ {X} $ is composed with the corresponding mappings [2]. Hence, for example, for a PL-manifold $ X $ there exists a smooth manifold PL-homeomorphic to it [ $ X $ is said to be smoothable] only if the lifting problem [3], the homotopy obstruction to the solution of which lies in the groups $ H ^ {i + 1 } [ X, \pi _ {i} [ \mathop{\rm PL} / \mathop{\rm O} ]] $, is solvable:

$$ \tag{3 } \begin{array}{lcc} {} &{} & \mathop{\rm BO} \\ {} &{} &\downarrow \\ X & \mathop \rightarrow \limits _ { {\tau _ {X} }} & \mathop{\rm BPL} \\ \end{array} $$

Here it turns out that the solvability of [3] is not only necessary but also sufficient for the smoothability of a PL-manifold $ X $[ and all non-equivalent smoothings are in bijective correspondence with the set $ [ X, \mathop{\rm PL} / \mathop{\rm O} ] $ of homotopy classes of mappings $ X \rightarrow \mathop{\rm PL} / \mathop{\rm O} $].

By replacing $ \mathop{\rm PL} / \mathop{\rm O} $ by $ \mathop{\rm TOP} / \mathop{\rm O} $, the same holds for the smoothability of topological manifolds $ X $ of dimension $ \geq 5 $, and also [by replacing $ \mathop{\rm PL} / \mathop{\rm O} $ by $ \mathop{\rm TOP} / \mathop{\rm O} $] for their $ \mathop{\rm PL} $- triangulations. The homotopy group $ \Gamma _ {k} = \pi _ {k} [ \mathop{\rm PL} / \mathop{\rm O} ] $ is isomorphic to the group of classes of oriented diffeomorphic smooth manifolds obtained by glueing the boundaries of two $ k $- dimensional spheres. This group is finite for all $ k $[ and is even trivial for $ k \leq 6 $]. Therefore, the number of non-equivalent smoothings of an arbitrary PL-manifold $ X $ is finite and is bounded above by the number

$$ \mathop{\rm ord} \sum _ { k } H ^ {k} [ X, \pi _ {k} [ \mathop{\rm PL} / \mathop{\rm O} ]]. $$

The homotopy group $ \pi _ {k} [ \mathop{\rm TOP} / \mathop{\rm PL} ] $ vanishes, with one exception: $ \pi _ {3} [ \mathop{\rm TOP} / \mathop{\rm PL} ] = \mathbf Z /2 $. Thus, the existence of a $ \mathop{\rm PL} $- triangulation of a topological manifold $ X $ of dimension $ \geq 5 $ is determined by the vanishing of a certain cohomology class $ \Delta [ X] \in H ^ {4} [ X, \mathbf Z /2] $, while the set of non-equivalent $ \mathop{\rm PL} $- triangulations of $ X $ is in bijective correspondence with the group $ H ^ {3} [ X, \mathbf Z /2] $.

The group $ \pi _ {k} [ \mathop{\rm TOP} / \mathop{\rm O} ] $ coincides with the group $ \Gamma _ {k} $ if $ k \neq 3 $ and differs from $ \Gamma _ {k} $ for $ k = 3 $ by the group $ \mathbf Z /2 $. The number of non-equivalent smoothings of a topological manifold $ X $ of dimension $ \geq 5 $ is finite and is bounded above by the number $ \mathop{\rm ord} \sum _ {k} H ^ {k} [ X, \pi _ {k} [ \mathop{\rm TOP} / \mathop{\rm O} ]] $.

Similar results are not valid for Poincaré polyhedra.

$$ \tag{4 } \begin{array}{lcc} {} &{} _ {\tau _ {x} ^ \prime } & \mathop{\rm BPL} \\ {} &{} &\downarrow \\ X & \mathop \rightarrow \limits _ { {\tau _ {X} }} & \mathop{\rm BG} \\ \end{array} $$

Of course, the existence of a lifting, for example, in [4] is a necessary condition for the existence of a PL-manifold homotopy equivalent to the Poincaré polyhedron $ X $, but, generally speaking, it ensures [for $ n \geq 5 $] only the existence of a PL-manifold $ M $ and a mapping $ f: M \rightarrow X $ of degree 1 such that $ \tau _ {M} = f \circ \tau _ {x} ^ \prime $. The transformation of this manifold into a manifold that is homotopy equivalent to $ X $ requires the technique of surgery [reconstruction], initially developed by S.P. Novikov for the case when $ X $ is a simply-connected smooth manifold of dimension $ \geq 5 $. For simply-connected $ X $ this technique has been generalized to the case under consideration. Thus, for a simply-connected Poincaré polyhedron $ X $ a PL-manifold of dimension $ \geq 5 $ homotopy equivalent to it exists if and only a lifting [4] exists. The problem of the existence of topological or smooth manifolds that are homotopy equivalent to an [even simply-connected] Poincaré polyhedron is still more complicated.

References

Comments

It was found recently [a1] that the behaviour of smooth manifolds of dimension $ 4 $ is radically different from those in dimensions $ \geq 5 $. Among very numerous recent results one has:

i] There is a countably infinite family of smooth, compact, simply-connected $ 4 $- manifolds, all mutually homeomorphic but with distinct smooth structure.

ii] There is an uncountable family of smooth $ 4 $- manifolds, each homeomorphic to $ \mathbf R ^ {4} $ but with mutually distinct smooth structure.

iii] There are simply-connected smooth $ 4 $- manifolds which are $ h $- cobordant [cf. $ h $- cobordism] but not diffeomorphic.

For the lifting problem [3] see [a2][a3].

For the KirbySiebenmann theorem, the arrow $ \mathop{\rm TOP} \rightarrow \mathop{\rm P} $, see also [a4].

References

How to Cite This Entry:
Topology of manifolds. Encyclopedia of Mathematics. URL: //encyclopediaofmath.org/index.php?title=Topology_of_manifolds&oldid=49630

This article was adapted from an original article by M.A. Shtan'ko [originator], which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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