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The branch of the theory of manifolds (cf. [[Manifold|Manifold]]) concerned with the study of relations between different types 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).
 
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>
+
$$ \tag{1 }
  
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).
+
\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}
 +
 +
$$
  
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.
+
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 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 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 <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 $  \mathop{\rm PL} \rightarrow  \mathop{\rm Handle} $
 +
expresses the fact that every  $  \mathop{\rm PL} $-
 +
manifold has a handle decomposition.
  
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).
+
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.
  
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 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).
  
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.
+
Similarly, if  $  n \neq 4 $
 +
the arrow $  \mathop{\rm Lip} \rightarrow  \mathop{\rm TOP} $
 +
is reversible (and moreover in a unique way).
  
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 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 <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 $  \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 <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 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 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 arrow  $  \mathop{\rm TOP} \rightarrow  \mathop{\rm P} $
 +
expresses the theorem on the homotopy equivalence of an arbitrary topological manifold to a polyhedron.
  
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.
+
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
 
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>
+
$$ \tag{2 }
 +
\mathop{\rm BO}  \rightarrow  \mathop{\rm BPL}  \rightarrow  \mathop{\rm BTOP}  \rightarrow  \mathop{\rm BG} ,
 +
$$
  
 
of which the homotopy fibres and the homotopy fibres of their compositions are denoted, respectively, by the symbols
 
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>
+
$$
 +
\mathop{\rm PL} / \mathop{\rm O} ,  \mathop{\rm TOP} / \mathop{\rm O} ,  \mathop{\rm G} /
 +
\mathop{\rm O} ,  \mathop{\rm TOP} / \mathop{\rm PL} ,\
 +
\mathop{\rm G} / \mathop{\rm PL} ,  \mathop{\rm G} / \mathop{\rm TOP} .
 +
$$
  
For every manifold <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.
+
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 <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:
+
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:
  
<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>
+
$$ \tag{3 }
  
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" />).
+
\begin{array}{lcc}
 +
{}  &{}  & \mathop{\rm BO}  \\
 +
{}  &{}  &\downarrow  \\
 +
X  & \mathop \rightarrow \limits _ { {\tau _ {X} }}  & \mathop{\rm BPL}  \\
 +
\end{array}
  
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>
+
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} $).
  
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" />.
+
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
  
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" />.
+
$$
 +
\mathop{\rm ord}  \sum _ { k }
 +
H  ^ {k} ( X, \pi _ {k} (  \mathop{\rm PL} /  \mathop{\rm O} )).
 +
$$
 +
 
 +
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.
 
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>
+
$$ \tag{4 }
  
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.
+
\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|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====
 
====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>
 
<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====
 
====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:
+
It was found recently [[#References|[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 <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.
+
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 <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.
+
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 <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.
+
iii) There are simply-connected smooth $  4 $-
 +
manifolds which are $  h $-
 +
cobordant (cf. [[H-cobordism| $  h $-
 +
cobordism]]) but not diffeomorphic.
  
 
For the lifting problem (3) see [[#References|[a2]]]–[[#References|[a3]]].
 
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]]].
+
For the Kirby–Siebenmann theorem, the arrow $  \mathop{\rm TOP} \rightarrow  \mathop{\rm P} $,  
 +
see also [[#References|[a4]]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 14:56, 7 June 2020


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

[1] 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)
[2] J. Madsen, R.J. Milgram, "The classifying spaces for surgery and cobordism of manifolds" , Princeton Univ. Press (1979)
[3] 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
[4] M.H. Freedman, "The topology of four-dimensional manifolds" J. Differential Geom. , 17 (1982) pp. 357–453
[5] F. Quinn, "Ends of maps III. Dimensions 4 and 5" J. Differential Geom. , 17 (1982) pp. 503–521
[6] R. Mandelbaum, "Four-dimensional topology: an introduction" Bull. Amer. Math. Soc. , 2 (1980) pp. 1–159
[7] 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
[8] R.D. Edwards, "Approximating certain cell-like maps by homeomorphisms" Notices Amer. Math. Soc. , 24 : 7 (1977) pp. A649
[9] F. Quinn, "The topological characterization of manifolds" Abstracts Amer. Math. Soc. , 1 : 7 (1980) pp. 613–614
[10] J.W. Cannon, "The recognition problem: what is a topological manifold" Bull. Amer. Math. Soc. , 84 : 5 (1978) pp. 832–866
[11] M. Spivak, "Spaces satisfying Poincaré duality" Topology , 6 (1967) pp. 77–101
[12] 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

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 Kirby–Siebenmann theorem, the arrow $ \mathop{\rm TOP} \rightarrow \mathop{\rm P} $, see also [a4].

References

[a1] 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
[a2] M.W. Hirsch, B. Mazur, "Smoothings of piecewise-linear manifolds" , Princeton Univ. Press (1974)
[a3] R. Lashof, M. Rothenberg, "Microbundles and smoothing" Topology , 3 (1965) pp. 357–388
[a4] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977)
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