nLab Kervaire-Milnor group

Idea

Kervaire-Milnor groups are the oriented h-cobordism classes of homotopy spheres with the connected sum as group operation and the reverse orientation as inversion. It controls the existence of smooth structures on topological and piecewise linear (PL) manifolds. The analogue concept controlling the existence of PL structures on topological is the Kirby-Siebenmann invariant.

Definition

An important property of spheres is their neutrality with respect to the connected sum of manifolds. Expanding this monoid structure with a composition and a neutral element to a group structure requires the restriction on manifolds, for which a connected sum can result in a sphere, hence which intuitively doesn’t have holes. This is possible with homotopy spheres, which are closed smooth manifolds with the same homotopy type as a sphere, with restriction to h-cobordism classes being useful for application. Inversion is then given by changing their orientation, which results in a group structure.

An alternative definition in five and more dimensions is given by the description of topological, PL and smooth structures. Let $Top_n$ be the topological group of homeomorphisms, $PL_n$ the topological group of PL homeomorphisms? and $Diff_n$ be the topological group of diffeomorphisms of Euclidean space $\mathbb{R}^n$. There are canonical inclusions $Diff_n\hookrightarrow PL_n\hookrightarrow Top_n$. Taking the cartesian product with the identity furthermore gives inclusions $Top_n\hookrightarrow Top_{n+1}$, $PL_n\hookrightarrow PL_{n+1}$ and $Diff_n\hookrightarrow Diff_{n+1}$. An inductive limit yields topological groups:

$$Top \coloneqq\lim_{n\rightarrow\infty}Top_n;$$

$$PL \coloneqq\lim_{n\rightarrow\infty}PL_n;$$

$$Diff \coloneqq\lim_{n\rightarrow\infty}Diff_n.$$

($Diff$ is homotopy equivalent to the infinite orthogonal group $O(\infty)$). There are induced canonical inclusions $Diff\hookrightarrow PL\hookrightarrow Top$. Their classifying spaces can be now be regarded to study the different structures: For a topological manifold $X$, its tangent bundle $TX$ is also a topological manifold, which is classified by a continuous map $X\rightarrow BTop$. Analogous for a PL and a smooth manifold, there are classifying maps $X\rightarrow BPL$ and $X\rightarrow BDiff$, respectively. The canonical inclusions $BDiff\hookrightarrow BPL\hookrightarrow BTop$ show that every smooth is a PL and every PL is a topological structure.

The Kervaire–Milnor groups are then alternatively given by the homotopy groups of the quotient groups $PL/Diff$ and $Top/Diff$:

$$\Theta_n \cong\pi_n\left(PL/Diff\right) \cong\pi_n\left(Top/Diff\right)$$

for $n\geq 5$. (The quotient group $PL/Top$ is an Eilenberg-MacLane space $K(\mathbb{Z}_2,3)$ and its single non-trivial homotopy group leads to the Kirby-Siebenmann invariant.)

Examples

• $\Theta_1\cong\Theta_2\cong\Theta_3\cong\Theta_4\cong\Theta_5\cong\Theta_6\cong 1$

• $\Theta_7\cong\mathbb{Z}_{28}$

• $\Theta_8\cong\mathbb{Z}_2$

• $\Theta_9\cong\mathbb{Z}_2^2$

• $\Theta_{10}\cong\mathbb{Z}_6$

• $\Theta_{11}\cong\mathbb{Z}_{992}$

• $\Theta_{14}\cong\mathbb{Z}_2$

• $\Theta_{16}\cong\mathbb{Z}_2$

• $\Theta_{61}\cong 1$

Properties

Proposition

All Kervaire-Milnor groups are finite.

This was shown for $n\neq 3$ by Kervaire & Milnor 1963, with the remaining case $n=3$ being solved by the proof of the Poincaré conjecture by Perelman 2002, 03a, 03b.

Proposition

$\Theta_1$, $\Theta_3$, $\Theta_5$ and $\Theta_61$ are the only trivial Kervaire–Milnor groups in odd dimensions.

References

The eponymous original discussion:

• Michel A. Kervaire, John W. Milnor: Groups of Homotopy Spheres I, Annals of Mathematics 77 3 (1963) [doi:10.2307/1970128, jstor:1970128, pdf]

See also:

• Wikipedia: Kervaire–Milnor group