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A piecewise-linear or topological [[Manifold|manifold]] that does not admit a smooth structure.
 
A piecewise-linear or topological [[Manifold|manifold]] that does not admit a smooth structure.
  
A smoothing of a piecewise-linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673101.png" /> is a piecewise-linear isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673103.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673105.png" />, be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673106.png" />-dimensional Milnor manifold (see [[Dendritic manifold|Dendritic manifold]]). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673107.png" /> is parallelizable, its [[Signature|signature]] is 8, and its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673108.png" /> is homotopy equivalent to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n0673109.png" />. Glueing to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731010.png" /> a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731011.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731012.png" /> leads to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731013.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731014.png" /> is a piecewise-linear sphere (see generalized [[Poincaré conjecture|Poincaré conjecture]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731015.png" /> is a piecewise-linear disc, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731016.png" /> is a piecewise-linear manifold. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731017.png" /> is non-smoothable, since its signature is 8, while that of an almost-parallelizable (that is, parallelizable after removing a point) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731018.png" />-dimensional manifold is a multiple of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731019.png" /> that grows exponentially with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731020.png" />. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731021.png" /> is not diffeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731022.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731023.png" /> is a [[Milnor sphere|Milnor sphere]].
+
Example of a non-smoothable manifold. Let $  W  ^ {4k} $,  
 +
$  k > 1 $,  
 +
be a $  4 k $-
 +
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 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|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|Milnor sphere]].
  
A criterion for a piecewise-linear manifold to be smoothable is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731024.png" /> be the orthogonal group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731025.png" /> be the group of piecewise-linear homeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731026.png" /> preserving the origin (see [[Piecewise-linear topology|Piecewise-linear topology]]). The inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731027.png" /> induces a fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731029.png" /> is the [[Classifying space|classifying space]] of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731030.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731031.png" /> there results a fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731032.png" />, the fibre of which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731033.png" />. A piecewise-linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731034.png" /> has a linear stable normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731035.png" /> with classifying mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731037.png" /> is smoothable (or smooth), then it has a stable normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731038.png" /> with classifying mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731040.png" />. This condition is also sufficient, that is, a closed piecewise-linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731041.png" /> is smoothable if and only if its piecewise-linear stable normal bundle admits a vector reduction, that is, if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731042.png" /> can be  "lifted"  to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731043.png" /> (there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731045.png" />).
+
A criterion for a piecewise-linear manifold to be smoothable is as follows. Let $  \textrm{ 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|Piecewise-linear topology]]). The inclusion $  \textrm{ O } _ {n} \rightarrow  \mathop{\rm PL} _ {n} $
 +
induces a fibration $  B \textrm{ O } _ {n} \rightarrow B  \mathop{\rm PL} _ {n} $,  
 +
where $  B G $
 +
is the [[Classifying space|classifying space]] of a group $  G $.  
 +
As n \rightarrow \infty $
 +
there results a fibration $  p : B \textrm{ O } \rightarrow B  \mathop{\rm PL} $,  
 +
the fibre of which is denoted by $  M / \textrm{ 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 \textrm{ 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 \textrm{ O } $(
 +
there is a $  \overline{v}\; : X \rightarrow B \textrm{ O } $
 +
such that $  p \circ \overline{v}\; = v $).
  
Two smoothings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731047.png" /> are said to be equivalent if there is a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731049.png" /> is piecewise differentiably isotopic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731050.png" /> (see [[Structure(2)|Structure]] on a manifold). The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731051.png" /> of equivalence classes of smoothings are in a natural one-to-one correspondence with the fibre-wise homotopy classes of liftings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731053.png" />. In other words, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731054.png" /> is smoothable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067310/n06731055.png" />.
+
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(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 \textrm{ O } $
 +
of $  v : X \rightarrow B  \mathop{\rm PL} $.  
 +
In other words, when $  X $
 +
is smoothable, $  \mathop{\rm ts} ( X) = [ X ,  \mathop{\rm PL} / \textrm{ O } ] $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Kervaire,  "A manifold which does not admit any differentiable structure"  ''Comment. Math. Helv.'' , '''34'''  (1960)  pp. 257–270</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W. Milnor,  J.D. Stasheff,  "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Kervaire,  "A manifold which does not admit any differentiable structure"  ''Comment. Math. Helv.'' , '''34'''  (1960)  pp. 257–270</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W. Milnor,  J.D. Stasheff,  "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,  B. Mazur,  "Smoothings of piecewise linear manifolds" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.C. Siebenmann,  "Topological manifolds" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''2''' , Gauthier-Villars  (1971)  pp. 133–163</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Smale,  "The generalized Poincaré conjecture in higher dimensions"  ''Bull. Amer. Math. Soc.'' , '''66'''  (1960)  pp. 373–375</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,  B. Mazur,  "Smoothings of piecewise linear manifolds" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.C. Siebenmann,  "Topological manifolds" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''2''' , Gauthier-Villars  (1971)  pp. 133–163</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Smale,  "The generalized Poincaré conjecture in higher dimensions"  ''Bull. Amer. Math. Soc.'' , '''66'''  (1960)  pp. 373–375</TD></TR></table>

Revision as of 08:03, 6 June 2020


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 $ \textrm{ 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 $ \textrm{ O } _ {n} \rightarrow \mathop{\rm PL} _ {n} $ induces a fibration $ B \textrm{ 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 \textrm{ O } \rightarrow B \mathop{\rm PL} $, the fibre of which is denoted by $ M / \textrm{ 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 \textrm{ 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 \textrm{ O } $( there is a $ \overline{v}\; : X \rightarrow B \textrm{ 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 \textrm{ O } $ of $ v : X \rightarrow B \mathop{\rm PL} $. In other words, when $ X $ is smoothable, $ \mathop{\rm ts} ( X) = [ X , \mathop{\rm PL} / \textrm{ 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=48005
This article was adapted from an original article by Yu.I. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article