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) |
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 |
Non-smoothable manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-smoothable_manifold&oldid=48005