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Difference between revisions of "Non-smoothable manifold"

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Example of a non-smoothable manifold. Let  $  W  ^ {4k} $,  
 
Example of a non-smoothable manifold. Let  $  W  ^ {4k} $,  
 
$  k > 1 $,  
 
$  k > 1 $,  
be a  $  4 k $-
+
be a  $  4 k $-dimensional Milnor manifold (see [[Dendritic manifold|Dendritic manifold]]). In particular,  $  W  ^ {4k} $
dimensional Milnor manifold (see [[Dendritic manifold|Dendritic manifold]]). In particular,  $  W  ^ {4k} $
 
 
is parallelizable, its [[Signature|signature]] is 8, and its boundary  $  M = \partial  W  ^ {4k} $
 
is parallelizable, its [[Signature|signature]] is 8, and its boundary  $  M = \partial  W  ^ {4k} $
is homotopy equivalent to the sphere  $  S  ^ {4k-} 1 $.  
+
is homotopy equivalent to the sphere  $  S  ^ {4k- 1} $.  
 
Glueing to  $  W $
 
Glueing to  $  W $
 
a cone  $  C M $
 
a cone  $  C M $
Line 32: Line 31:
 
is a piecewise-linear disc, so that  $  P $
 
is a piecewise-linear disc, so that  $  P $
 
is a piecewise-linear manifold. On the other hand,  $  P $
 
is a piecewise-linear manifold. On the other hand,  $  P $
is non-smoothable, since its signature is 8, while that of an almost-parallelizable (that is, parallelizable after removing a point)  $  4 $-
+
is non-smoothable, since its signature is 8, while that of an almost-parallelizable (that is, parallelizable after removing a point)  $  4 $-dimensional manifold is a multiple of a number  $  \sigma _ {k} $
dimensional manifold is a multiple of a number  $  \sigma _ {k} $
 
 
that grows exponentially with  $  k $.  
 
that grows exponentially with  $  k $.  
 
The manifold  $  M $
 
The manifold  $  M $
is not diffeomorphic to the sphere  $  S  ^ {k-} 1 $,  
+
is not diffeomorphic to the sphere  $  S  ^ {k- 1} $,  
 
that is,  $  M $
 
that is,  $  M $
 
is a [[Milnor sphere|Milnor sphere]].
 
is a [[Milnor sphere|Milnor sphere]].
  
A criterion for a piecewise-linear manifold to be smoothable is as follows. Let  $  \textrm{ O } _ {n} $
+
A criterion for a piecewise-linear manifold to be smoothable is as follows. Let  $  \mathop{\rm O} _ {n} $
 
be the orthogonal group and let  $  \mathop{\rm PL} _ {n} $
 
be the orthogonal group and let  $  \mathop{\rm PL} _ {n} $
 
be the group of piecewise-linear homeomorphisms of  $  \mathbf R  ^ {n} $
 
be the group of piecewise-linear homeomorphisms of  $  \mathbf R  ^ {n} $
preserving the origin (see [[Piecewise-linear topology|Piecewise-linear topology]]). The inclusion  $  \textrm{ O } _ {n} \rightarrow  \mathop{\rm PL} _ {n} $
+
preserving the origin (see [[Piecewise-linear topology|Piecewise-linear topology]]). The inclusion  $  \mathop{\rm O} _ {n} \rightarrow  \mathop{\rm PL} _ {n} $
induces a fibration  $  B \textrm{ O } _ {n} \rightarrow B  \mathop{\rm PL} _ {n} $,  
+
induces a fibration  $  B \mathop{\rm O} _ {n} \rightarrow B  \mathop{\rm PL} _ {n} $,  
 
where  $  B G $
 
where  $  B G $
 
is the [[Classifying space|classifying space]] of a group  $  G $.  
 
is the [[Classifying space|classifying space]] of a group  $  G $.  
 
As  $  n \rightarrow \infty $
 
As  $  n \rightarrow \infty $
there results a fibration  $  p :  B \textrm{ O } \rightarrow B  \mathop{\rm PL} $,  
+
there results a fibration  $  p :  B \mathop{\rm O} \rightarrow B  \mathop{\rm PL} $,  
the fibre of which is denoted by  $  M / \textrm{ O } $.  
+
the fibre of which is denoted by  $  M / \mathop{\rm O} $.  
 
A piecewise-linear manifold  $  X $
 
A piecewise-linear manifold  $  X $
 
has a linear stable normal bundle  $  u $
 
has a linear stable normal bundle  $  u $
 
with classifying mapping  $  v :  X \rightarrow B  \mathop{\rm PL} $.  
 
with classifying mapping  $  v :  X \rightarrow B  \mathop{\rm PL} $.  
 
If  $  X $
 
If  $  X $
is smoothable (or smooth), then it has a stable normal bundle  $  \overline{v}\; $
+
is smoothable (or smooth), then it has a stable normal bundle  $  \overline{v} $
with classifying mapping  $  \overline{v}\; :  X \rightarrow B \textrm{ O } $
+
with classifying mapping  $  \overline{v} :  X \rightarrow B \mathop{\rm O} $
and  $  p \circ \overline{v}\; = v $.  
+
and  $  p \circ \overline{v} = v $.  
 
This condition is also sufficient, that is, a closed piecewise-linear manifold  $  X $
 
This condition is also sufficient, that is, a closed piecewise-linear manifold  $  X $
 
is smoothable if and only if its piecewise-linear stable normal bundle admits a vector reduction, that is, if the mapping  $  v :  X \rightarrow B  \mathop{\rm PL} $
 
is smoothable if and only if its piecewise-linear stable normal bundle admits a vector reduction, that is, if the mapping  $  v :  X \rightarrow B  \mathop{\rm PL} $
can be  "lifted"  to  $  B \textrm{ O } $(
+
can be  "lifted"  to  $  B \mathop{\rm O} $ (there is a  $  \overline{v} :  X \rightarrow B \mathop{\rm O} $
there is a  $  \overline{v}\; :  X \rightarrow B \textrm{ O } $
+
such that  $  p \circ \overline{v} = v $).
such that  $  p \circ \overline{v}\; = v $).
 
  
 
Two smoothings  $  f :  M \rightarrow X $
 
Two smoothings  $  f :  M \rightarrow X $
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are said to be equivalent if there is a diffeomorphism  $  h :  M \rightarrow N $
 
are said to be equivalent if there is a diffeomorphism  $  h :  M \rightarrow N $
 
such that  $  h f ^ { - 1 } $
 
such that  $  h f ^ { - 1 } $
is piecewise differentiably isotopic to  $  g  ^ {-} 1 $(
+
is piecewise differentiably isotopic to  $  g  ^ {- 1} $ (see [[Structure(2)|Structure]] on a manifold). The sets  $  \mathop{\rm ts} ( X) $
see [[Structure(2)|Structure]] on a manifold). The sets  $  \mathop{\rm ts} ( X) $
+
of equivalence classes of smoothings are in a natural one-to-one correspondence with the fibre-wise homotopy classes of liftings  $  \overline{v} :  X \rightarrow B \mathop{\rm O} $
of equivalence classes of smoothings are in a natural one-to-one correspondence with the fibre-wise homotopy classes of liftings  $  \overline{v}\; :  X \rightarrow B \textrm{ O } $
 
 
of  $  v :  X \rightarrow B  \mathop{\rm PL} $.  
 
of  $  v :  X \rightarrow B  \mathop{\rm PL} $.  
 
In other words, when  $  X $
 
In other words, when  $  X $
is smoothable,  $  \mathop{\rm ts} ( X) = [ X ,  \mathop{\rm PL} / \textrm{ O } ] $.
+
is smoothable,  $  \mathop{\rm ts} ( X) = [ X ,  \mathop{\rm PL} / \mathop{\rm O} ] $.
  
 
====References====
 
====References====

Revision as of 01:32, 13 January 2022


A piecewise-linear or topological manifold that does not admit a smooth structure.

A smoothing of a piecewise-linear manifold $ X $ is a piecewise-linear isomorphism $ f : M \rightarrow X $, where $ M $ is a smooth manifold. Manifolds that do not admit smoothings are said to be non-smoothable. With certain modifications this is also applicable to topological manifolds.

Example of a non-smoothable manifold. Let $ W ^ {4k} $, $ k > 1 $, be a $ 4 k $-dimensional Milnor manifold (see Dendritic manifold). In particular, $ W ^ {4k} $ is parallelizable, its signature is 8, and its boundary $ M = \partial W ^ {4k} $ is homotopy equivalent to the sphere $ S ^ {4k- 1} $. Glueing to $ W $ a cone $ C M $ over $ \partial W $ leads to the space $ P ^ {4k} $. Since $ M $ is a piecewise-linear sphere (see generalized Poincaré conjecture), $ C M $ is a piecewise-linear disc, so that $ P $ is a piecewise-linear manifold. On the other hand, $ P $ is non-smoothable, since its signature is 8, while that of an almost-parallelizable (that is, parallelizable after removing a point) $ 4 $-dimensional manifold is a multiple of a number $ \sigma _ {k} $ that grows exponentially with $ k $. The manifold $ M $ is not diffeomorphic to the sphere $ S ^ {k- 1} $, that is, $ M $ is a Milnor sphere.

A criterion for a piecewise-linear manifold to be smoothable is as follows. Let $ \mathop{\rm O} _ {n} $ be the orthogonal group and let $ \mathop{\rm PL} _ {n} $ be the group of piecewise-linear homeomorphisms of $ \mathbf R ^ {n} $ preserving the origin (see Piecewise-linear topology). The inclusion $ \mathop{\rm O} _ {n} \rightarrow \mathop{\rm PL} _ {n} $ induces a fibration $ B \mathop{\rm O} _ {n} \rightarrow B \mathop{\rm PL} _ {n} $, where $ B G $ is the classifying space of a group $ G $. As $ n \rightarrow \infty $ there results a fibration $ p : B \mathop{\rm O} \rightarrow B \mathop{\rm PL} $, the fibre of which is denoted by $ M / \mathop{\rm O} $. A piecewise-linear manifold $ X $ has a linear stable normal bundle $ u $ with classifying mapping $ v : X \rightarrow B \mathop{\rm PL} $. If $ X $ is smoothable (or smooth), then it has a stable normal bundle $ \overline{v} $ with classifying mapping $ \overline{v} : X \rightarrow B \mathop{\rm O} $ and $ p \circ \overline{v} = v $. This condition is also sufficient, that is, a closed piecewise-linear manifold $ X $ is smoothable if and only if its piecewise-linear stable normal bundle admits a vector reduction, that is, if the mapping $ v : X \rightarrow B \mathop{\rm PL} $ can be "lifted" to $ B \mathop{\rm O} $ (there is a $ \overline{v} : X \rightarrow B \mathop{\rm O} $ such that $ p \circ \overline{v} = v $).

Two smoothings $ f : M \rightarrow X $ and $ g : N \rightarrow X $ are said to be equivalent if there is a diffeomorphism $ h : M \rightarrow N $ such that $ h f ^ { - 1 } $ is piecewise differentiably isotopic to $ g ^ {- 1} $ (see Structure on a manifold). The sets $ \mathop{\rm ts} ( X) $ of equivalence classes of smoothings are in a natural one-to-one correspondence with the fibre-wise homotopy classes of liftings $ \overline{v} : X \rightarrow B \mathop{\rm O} $ of $ v : X \rightarrow B \mathop{\rm PL} $. In other words, when $ X $ is smoothable, $ \mathop{\rm ts} ( X) = [ X , \mathop{\rm PL} / \mathop{\rm O} ] $.

References

[1] M. Kervaire, "A manifold which does not admit any differentiable structure" Comment. Math. Helv. , 34 (1960) pp. 257–270
[2] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974)

Comments

References

[a1] M.W. Hirsch, B. Mazur, "Smoothings of piecewise linear manifolds" , Princeton Univ. Press (1974)
[a2] L.C. Siebenmann, "Topological manifolds" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 2 , Gauthier-Villars (1971) pp. 133–163
[a3] S. Smale, "The generalized Poincaré conjecture in higher dimensions" Bull. Amer. Math. Soc. , 66 (1960) pp. 373–375
How to Cite This Entry:
Non-smoothable manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-smoothable_manifold&oldid=51871
This article was adapted from an original article by Yu.I. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article