Difference between revisions of "Non-smoothable manifold"
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− | + | A piecewise-linear or topological [[Manifold|manifold]] that does not admit a smooth structure. | |
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+ | 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|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 $ \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|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|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(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 $ 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==== | ====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">[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> | ||
+ | <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> |
Latest revision as of 06:03, 18 April 2023
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) |
[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 |
Non-smoothable manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-smoothable_manifold&oldid=17941