View source for Topology of manifolds

The branch of the theory of manifolds (cf. [[Manifold|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).

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932302.png" /> is the category of differentiable (smooth) manifolds; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932303.png" /> is the category of piecewise-linear (combinatorial) manifolds; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932304.png" /> is the category of topological manifolds that are polyhedra; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932305.png" /> is the category of topological manifolds admitting a topological decomposition into handles; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932306.png" /> is the category of Lipschitz manifolds (with Lipschitz transition mappings between local charts); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932307.png" /> is the category of topological manifolds (Hausdorff and with a countable base); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932308.png" /> 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); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t0932309.png" /> is the category of generalized manifolds (finite-dimensional absolute neighbourhood retracts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323010.png" /> that are homology manifolds without boundary, i.e. with the property that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323011.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323012.png" /> is isomorphic to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323013.png" />); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323014.png" /> is the category of Poincaré spaces (finite-dimensional absolute neighbourhood retracts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323015.png" /> for which there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323016.png" /> and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323018.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323019.png" />, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323020.png" /> is an isomorphism for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323021.png" />); and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323023.png" />, denote functors with the structure of forgetting functors. The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323024.png" /> expresses Whitehead's theorem on the triangulability of smooth manifolds. In dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323025.png" /> this arrow is reversible (an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323026.png" />-manifold is smoothable) but in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323027.png" /> there are non-smoothable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323028.png" />-manifolds and even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323029.png" />-manifolds that are homotopy inequivalent to any smooth manifold. The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323030.png" /> is also irreversible in the same strong sense (there exist polyhedral manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323031.png" /> that are homotopy inequivalent to any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323032.png" />-manifold). Here already for the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323034.png" />, there exist triangulations in which it is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323035.png" />-manifold.

The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323036.png" /> expresses the fact that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323037.png" />-manifold has a handle decomposition.

The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323038.png" /> expresses the theorem on the existence of a Lipschitz structure on an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323039.png" />-manifold.

The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323040.png" /> is reversible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323041.png" /> and irreversible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323042.png" /> (an arbitrary topological manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323043.png" /> admits a handle decomposition and there exist four-dimensional topological manifolds for which this is not true).

Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323044.png" /> the arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323045.png" /> is reversible (and moreover in a unique way).

The question on the reversibility of the arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323046.png" /> gives the classical unsolved problem on the triangulability of arbitrary topological manifolds.

The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323047.png" /> is irreversible in the strong sense (there exist Poincaré polyhedra that are homotopy inequivalent to any homology manifold).

The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323048.png" /> expresses a theorem on the homotopy equivalence of an arbitrary homology manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323049.png" /> to a topological manifold.

The arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323050.png" /> expresses the theorem on the homotopy equivalence of an arbitrary topological manifold to a polyhedron.

The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323051.png" /> expresses that an arbitrary topological manifold is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323052.png" />.

The similar question for the arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323053.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323054.png" /> into the corresponding classifying spaces BO, BPL, BTOP, BG.

There exist canonical composition mappings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323056.png" /></td> </tr></table>

For every manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323057.png" /> from a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323061.png" /> there exists a normal stable bundle, i.e. a canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323062.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323063.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323064.png" /> is composed with the corresponding mappings (2). Hence, for example, for a PL-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323065.png" /> there exists a smooth manifold PL-homeomorphic to it (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323066.png" /> is said to be smoothable) only if the lifting problem (3), the homotopy obstruction to the solution of which lies in the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323067.png" />, is solvable:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Here it turns out that the solvability of (3) is not only necessary but also sufficient for the smoothability of a PL-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323069.png" /> (and all non-equivalent smoothings are in bijective correspondence with the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323070.png" /> of homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323071.png" />).

By replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323072.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323073.png" />, the same holds for the smoothability of topological manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323074.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323075.png" />, and also (by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323076.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323077.png" />) for their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323078.png" />-triangulations. The homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323079.png" /> is isomorphic to the group of classes of oriented diffeomorphic smooth manifolds obtained by glueing the boundaries of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323080.png" />-dimensional spheres. This group is finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323081.png" /> (and is even trivial for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323082.png" />). Therefore, the number of non-equivalent smoothings of an arbitrary PL-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323083.png" /> is finite and is bounded above by the number

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323084.png" /></td> </tr></table>

The homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323085.png" /> vanishes, with one exception: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323086.png" />. Thus, the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323087.png" />-triangulation of a topological manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323088.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323089.png" /> is determined by the vanishing of a certain cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323090.png" />, while the set of non-equivalent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323091.png" />-triangulations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323092.png" /> is in bijective correspondence with the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323093.png" />.

The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323094.png" /> coincides with the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323095.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323096.png" /> and differs from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323097.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323098.png" /> by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t09323099.png" />. The number of non-equivalent smoothings of a topological manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230100.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230101.png" /> is finite and is bounded above by the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230102.png" />.

Similar results are not valid for Poincaré polyhedra.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230104.png" />, but, generally speaking, it ensures (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230105.png" />) only the existence of a PL-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230106.png" /> and a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230107.png" /> of degree 1 such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230108.png" />. The transformation of this manifold into a manifold that is homotopy equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230109.png" /> requires the technique of [[Surgery|surgery]] (reconstruction), initially developed by S.P. Novikov for the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230110.png" /> is a simply-connected smooth manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230111.png" />. For simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230112.png" /> this technique has been generalized to the case under consideration. Thus, for a simply-connected Poincaré polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230113.png" /> a PL-manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230114.png" /> 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Novikov,   "On manifolds with free abelian group and their application"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30'''  (1966)  pp. 207–246  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Madsen,   R.J. Milgram,   "The classifying spaces for surgery and cobordism of manifolds" , Princeton Univ. Press  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Latour,   "Double suspension d'une sphere d'homologie [d'après R. Edwards]" , ''Sem. Bourbaki Exp. 515'' , ''Lect. notes in math.'' , '''710''' , Springer  (1979)  pp. 169–186</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.H. Freedman,   "The topology of four-dimensional manifolds"  ''J. Differential Geom.'' , '''17'''  (1982)  pp. 357–453</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F. Quinn,   "Ends of maps III. Dimensions 4 and 5"  ''J. Differential Geom.'' , '''17'''  (1982)  pp. 503–521</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Mandelbaum,   "Four-dimensional topology: an introduction"  ''Bull. Amer. Math. Soc.'' , '''2'''  (1980)  pp. 1–159</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R. Lashof,   "The immersion approach to triangulation and smoothing"  A. Liulevicius (ed.) , ''Algebraic Topology (Madison, 1970)'' , ''Proc. Symp. Pure Math.'' , '''22''' , Amer. Math. Soc.  (1971)  pp. 131–164</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.D. Edwards,   "Approximating certain cell-like maps by homeomorphisms"  ''Notices Amer. Math. Soc.'' , '''24''' :  7  (1977)  pp. A649</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  F. Quinn,   "The topological characterization of manifolds"  ''Abstracts Amer. Math. Soc.'' , '''1''' :  7  (1980)  pp. 613–614</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.W. Cannon,   "The recognition problem: what is a topological manifold"  ''Bull. Amer. Math. Soc.'' , '''84''' :  5  (1978)  pp. 832–866</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M. Spivak,   "Spaces satisfying Poincaré duality"  ''Topology'' , '''6'''  (1967)  pp. 77–101</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  N.H. Kuiper,   "A short history of triangulation and related matters"  P.C. Baayen (ed.)  D. van Dulst (ed.)  J. Oosterhoff (ed.) , ''Bicentennial Congress Wisk. Genootschap (Amsterdam 1978)'' , ''Math. Centre Tracts'' , '''100''' , CWI  (1979)  pp. 61–79</TD></TR></table>



====Comments====
It was found recently [[#References|[a1]]] that the behaviour of smooth manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230115.png" /> is radically different from those in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230116.png" />. Among very numerous recent results one has:

i) There is a countably infinite family of smooth, compact, simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230117.png" />-manifolds, all mutually homeomorphic but with distinct smooth structure.

ii) There is an uncountable family of smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230118.png" />-manifolds, each homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230119.png" /> but with mutually distinct smooth structure.

iii) There are simply-connected smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230120.png" />-manifolds which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230121.png" />-cobordant (cf. [[H-cobordism|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230122.png" />-cobordism]]) but not diffeomorphic.

For the lifting problem (3) see [[#References|[a2]]]–[[#References|[a3]]].

For the Kirby–Siebenmann theorem, the arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093230/t093230123.png" />, see also [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.M. Donaldson,   "The geometry of 4-manifolds"  A.M. Gleason (ed.) , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , Amer. Math. Soc.  (1987)  pp. 43–54</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,   B. Mazur,   "Smoothings of piecewise-linear manifolds" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Lashof,   M. Rothenberg,   "Microbundles and smoothing"  ''Topology'' , '''3'''  (1965)  pp. 357–388</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. Kirby,   L.C. Siebenmann,   "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR></table>