In mathematics, especially differential topology and cobordism theory, a Kervaire–Milnor group is an abelian group defined as the h-cobordism classes of homotopy spheres with the connected sum as composition and the reverse orientation as inversion. It controls the existence of smooth structures on topological and piecewise linear (PL) manifolds. Concerning the related question of PL structures on topological manifolds, the obstruction is given by the Kirby–Siebenmann invariant, which is a lot easier to understand. In all but three and four dimensions, Kervaire–Milnor groups furthermore give the possible smooth structures on spheres, hence exotic spheres. They are named after the French mathematician Michel Kervaire and the American mathematician John Milnor, who first described them in 1962. (Their paper was originally only supposed to be the first part, but a second part was never published.)
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 higher dimensions is given by the description of topological, PL and smooth structures. Let Top n {\displaystyle \operatorname {Top} _{n}} be the topological group of homeomorphisms, PL n {\displaystyle \operatorname {PL} _{n}} the topological group of PL homeomorphisms and Diff n {\displaystyle \operatorname {Diff} _{n}} be the topological group of diffeomorphisms of euclidean space R n {\displaystyle \mathbb {R} ^{n}} . An inductive limit yields topological groups Top {\displaystyle \operatorname {Top} } , PL {\displaystyle \operatorname {PL} } and Diff {\displaystyle \operatorname {Diff} } (which is homotopy equivalent to the infinite orthogonal group O ( ∞ ) {\displaystyle \operatorname {O} (\infty )} ), for which classifying spaces can be regarded. For a topological manifold X {\displaystyle X} , its tangent bundle T X {\displaystyle TX} is also a topological manifold, which is classified by a continuous map X → BTop {\displaystyle X\rightarrow \operatorname {BTop} } . Analogous for a PL and a smooth manifold, there are classifying maps X → BPL {\displaystyle X\rightarrow \operatorname {BPL} } and X → BDiff {\displaystyle X\rightarrow \operatorname {BDiff} } respectively. The canonical inclusions BDiff ↪ BPL ↪ BTop {\displaystyle \operatorname {BDiff} \hookrightarrow \operatorname {BPL} \hookrightarrow \operatorname {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 {\displaystyle \operatorname {PL} /\operatorname {Diff} } and Top / Diff {\displaystyle \operatorname {Top} /\operatorname {Diff} } :
Θ n ≅ π n ( PL / Diff ) ≅ π n ( Top / Diff ) {\displaystyle \Theta _{n}\cong \pi _{n}\left(\operatorname {PL} /\operatorname {Diff} \right)\cong \pi _{n}\left(\operatorname {Top} /\operatorname {Diff} \right)}
for n ≥ 5 {\displaystyle n\geq 5} .
Examples Some low-dimensional Kervaire–Milnor groups are given by:
Θ 1 ≅ 1 {\displaystyle \Theta _{1}\cong 1}
Θ 2 ≅ 1 {\displaystyle \Theta _{2}\cong 1}
Θ 3 ≅ 1 {\displaystyle \Theta _{3}\cong 1}
Θ 4 ≅ 1 {\displaystyle \Theta _{4}\cong 1}
Θ 5 ≅ 1 {\displaystyle \Theta _{5}\cong 1}
Θ 6 ≅ 1 {\displaystyle \Theta _{6}\cong 1}
Θ 7 ≅ Z 28 {\displaystyle \Theta _{7}\cong \mathbb {Z} _{28}}
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