# Topology of manifolds

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.

How to Cite This Entry:
Topology of manifolds. Encyclopedia of Mathematics. URL: http://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