Non-smoothable manifold

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