Non-smoothable manifold

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

A smoothing of a piecewise-linear manifold is a piecewise-linear isomorphism , where 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 , , be a -dimensional Milnor manifold (see Dendritic manifold). In particular, is parallelizable, its signature is 8, and its boundary is homotopy equivalent to the sphere . Glueing to a cone over leads to the space . Since is a piecewise-linear sphere (see generalized Poincaré conjecture), is a piecewise-linear disc, so that is a piecewise-linear manifold. On the other hand, is non-smoothable, since its signature is 8, while that of an almost-parallelizable (that is, parallelizable after removing a point) -dimensional manifold is a multiple of a number that grows exponentially with . The manifold is not diffeomorphic to the sphere , that is, is a Milnor sphere.

A criterion for a piecewise-linear manifold to be smoothable is as follows. Let be the orthogonal group and let be the group of piecewise-linear homeomorphisms of preserving the origin (see Piecewise-linear topology). The inclusion induces a fibration , where is the classifying space of a group . As there results a fibration , the fibre of which is denoted by . A piecewise-linear manifold has a linear stable normal bundle with classifying mapping . If is smoothable (or smooth), then it has a stable normal bundle with classifying mapping and . This condition is also sufficient, that is, a closed piecewise-linear manifold is smoothable if and only if its piecewise-linear stable normal bundle admits a vector reduction, that is, if the mapping can be "lifted" to (there is a such that ).

Two smoothings and are said to be equivalent if there is a diffeomorphism such that is piecewise differentiably isotopic to (see Structure on a manifold). The sets of equivalence classes of smoothings are in a natural one-to-one correspondence with the fibre-wise homotopy classes of liftings of . In other words, when is smoothable, .

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)

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