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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
CommentsIt 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: 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 Retrieved from "https://encyclopediaofmath.org/index.php?title=Topology_of_manifolds&oldid=49630" Categories:
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