# 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)