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).

$$\begin{array}{}\text{(1)}& \begin{array}{ccc}\mathrm{P}& & \mathrm{P}(\mathrm{ANR})\\ \uparrow & & \uparrow \\ \mathrm{H}& & \mathrm{H}(\mathrm{ANR})\\ & \mathrm{TOP}& \\ \mathrm{TRI}& \uparrow & \mathrm{Lip}\\ & \mathrm{Handle}& \\ & \uparrow & \\ & \mathrm{PL}& \\ & \uparrow & \\ & \mathrm{Diff}& \end{array}\end{array}$$

Here $\mathrm{Diff}$ is the category of differentiable (smooth) manifolds; $\mathrm{PL}$ is the category of piecewise-linear (combinatorial) manifolds; $\mathrm{TRI}$ is the category of topological manifolds that are polyhedra; $\mathrm{Handle}$ is the category of topological manifolds admitting a topological decomposition into handles; $\mathrm{Lip}$ is the category of Lipschitz manifolds (with Lipschitz transition mappings between local charts); $\mathrm{TOP}$ is the category of topological manifolds (Hausdorff and with a countable base); $\mathrm{H}$ 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); $\mathrm{H}(\mathrm{ANR})$ 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 $H^{\ast }(X,X\setminus x;\mathbf{Z})$ is isomorphic to the group $H^{\ast }({\mathbf{R}}^n,{\mathbf{R}}^n\setminus 0;\mathbf{Z})$); $\mathrm{P}(\mathrm{ANR})$ 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 $r\ge n+1$, and the mapping $\mu \cap :H^r(X)\to H_{n-r}(X)$ is an isomorphism for all $r$); and $\mathrm{P}$ 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 $\mathrm{H}\to \mathrm{TOP}\to \mathrm{P}$, denote functors with the structure of forgetting functors. The arrow $\mathrm{Diff}\to \mathrm{PL}$ expresses Whitehead's theorem on the triangulability of smooth manifolds. In dimensions $<8$ this arrow is reversible (an arbitrary $\mathrm{PL}$- manifold is smoothable) but in dimensions $\ge 8$ there are non-smoothable $\mathrm{PL}$- manifolds and even $\mathrm{PL}$- manifolds that are homotopy inequivalent to any smooth manifold. The imbedding $\mathrm{PL}\subset \mathrm{TRI}$ is also irreversible in the same strong sense (there exist polyhedral manifolds of dimension $\ge 5$ that are homotopy inequivalent to any $\mathrm{PL}$- manifold). Here already for the sphere $S^n$, $n\ge 5$, there exist triangulations in which it is not a $\mathrm{PL}$- manifold.

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

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

The arrow $\mathrm{Handle}\to \mathrm{TOP}$ is reversible if $n\ne 4$ and irreversible if $n=4$( an arbitrary topological manifold of dimension $n\ne 4$ admits a handle decomposition and there exist four-dimensional topological manifolds for which this is not true).

Similarly, if $n\ne 4$ the arrow $\mathrm{Lip}\to \mathrm{TOP}$ is reversible (and moreover in a unique way).

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

The arrow $\mathrm{H}\to \mathrm{P}$ is irreversible in the strong sense (there exist Poincaré polyhedra that are homotopy inequivalent to any homology manifold).

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

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

The imbedding $\mathrm{TOP}\subset \mathrm{H}(\mathrm{ANR})$ expresses that an arbitrary topological manifold is an $\mathrm{ANR}$.

The similar question for the arrows $\mathrm{Diff}\to \mathrm{PL}\to \mathrm{TOP}\to \mathrm{P}$ 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

$$\begin{array}{}\text{(2)}& \mathrm{BO}\to \mathrm{BPL}\to \mathrm{BTOP}\to \mathrm{BG},\end{array}$$

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

$$\mathrm{PL}/\mathrm{O},\mathrm{TOP}/\mathrm{O},\mathrm{G}/\mathrm{O},\mathrm{TOP}/\mathrm{PL},\mathrm{G}/\mathrm{PL},\mathrm{G}/\mathrm{TOP}.$$

For every manifold $X$ from a category $\mathrm{Diff}$, $\mathrm{PL}$, $\mathrm{TOP}$, $\mathrm{P}$ 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 $H^{i+1}(X,{\pi }_i(\mathrm{PL}/\mathrm{O}))$, is solvable:

$$\begin{array}{}\text{(3)}& \begin{array}{lcc}& & \mathrm{BO}\\ & & \downarrow \\ X& \underset{{\tau }_X}{\to }& \mathrm{BPL}\end{array}\end{array}$$

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 $[X,\mathrm{PL}/\mathrm{O}]$ of homotopy classes of mappings $X\to \mathrm{PL}/\mathrm{O}$).

By replacing $\mathrm{PL}/\mathrm{O}$ by $\mathrm{TOP}/\mathrm{O}$, the same holds for the smoothability of topological manifolds $X$ of dimension $\ge 5$, and also (by replacing $\mathrm{PL}/\mathrm{O}$ by $\mathrm{TOP}/\mathrm{O}$) for their $\mathrm{PL}$- triangulations. The homotopy group ${\mathrm{\Gamma }}_k={\pi }_k(\mathrm{PL}/\mathrm{O})$ 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 $k\le 6$). Therefore, the number of non-equivalent smoothings of an arbitrary PL-manifold $X$ is finite and is bounded above by the number

$$\mathrm{ord}\sum _k{H}^k(X,{\pi }_k(\mathrm{PL}/\mathrm{O})).$$

The homotopy group ${\pi }_k(\mathrm{TOP}/\mathrm{PL})$ vanishes, with one exception: ${\pi }_3(\mathrm{TOP}/\mathrm{PL})=\mathbf{Z}/2$. Thus, the existence of a $\mathrm{PL}$- triangulation of a topological manifold $X$ of dimension $\ge 5$ is determined by the vanishing of a certain cohomology class $\mathrm{\Delta }(X)\in H^4(X,\mathbf{Z}/2)$, while the set of non-equivalent $\mathrm{PL}$- triangulations of $X$ is in bijective correspondence with the group $H^3(X,\mathbf{Z}/2)$.

The group ${\pi }_k(\mathrm{TOP}/\mathrm{O})$ coincides with the group ${\mathrm{\Gamma }}_k$ if $k\ne 3$ and differs from ${\mathrm{\Gamma }}_k$ for $k=3$ by the group $\mathbf{Z}/2$. The number of non-equivalent smoothings of a topological manifold $X$ of dimension $\ge 5$ is finite and is bounded above by the number $\mathrm{ord}\sum _k{H}^k(X,{\pi }_k(\mathrm{TOP}/\mathrm{O}))$.

Similar results are not valid for Poincaré polyhedra.

$$\begin{array}{}\text{(4)}& \begin{array}{lcc}& {}_{{\tau }_x^{\prime }}& \mathrm{BPL}\\ & & \downarrow \\ X& \underset{{\tau }_X}{\to }& \mathrm{BG}\end{array}\end{array}$$

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 $n\ge 5$) only the existence of a PL-manifold $M$ and a mapping $f:M\to X$ 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 $\ge 5$. 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 $\ge 5$ 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.

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It was found recently [a1] that the behaviour of smooth manifolds of dimension $4$ is radically different from those in dimensions $\ge 5$. 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 Kirby–Siebenmann theorem, the arrow $\mathrm{TOP}\to \mathrm{P}$, see also [a4].

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