Exotic Smooth Structures and Milnor Spheres — Examples

Assume the Axiom of Choice. The $(1,0)$ Milnor sphere bundle is diffeomorphic to the standard $S^7$, and its disk filling has $q=4$, $\sigma =1$ and $\lambda =0(\mathrm{m}\mathrm{o}\mathrm{d}7)$.

Assume the Axiom of Choice. For $(h,j)=(2,-1)$ the Euler class of ${\xi }_{2,-1}$ is $e=(2-1)u=u$ by the class formula (Euler and first Pontryagin classes of ${\xi }_{h,j}$). The integral Gysin sequence of the oriented $S^3$-bundle $S^3\to M_{2,-1}\to S^4$ is $\cdots \to H^{k-4}(S^4;\mathbb{Z})\stackrel{⌣u}{\to }{H}^k(S^4;\mathbb{Z})\to H^k(M_{2,-1};\mathbb{Z})\to H^{k-3}(S^4;\mathbb{Z})\to \cdots ,$ and outside degrees $0$ and $4$ the base cohomology vanishes. The only nonempty multiplication map is $H^0(S^4)=\mathbb{Z}\stackrel{⌣u}{\to }{H}^4(S^4)=\mathbb{Z}$, multiplication by $1$, which is an isomorphism; exactness therefore forces $H^0(M_{2,-1};\mathbb{Z})=H^7(M_{2,-1};\mathbb{Z})=\mathbb{Z}$ and $H^k(M_{2,-1};\mathbb{Z})=0$ for $1\le k\le 6$. Transferring to homology by the universal coefficient theorem and finite generation gives $H_k(M_{2,-1};\mathbb{Z})=H_k(S^7;\mathbb{Z})$ for every $k$, which is exactly the statement verified by the homology-seven-sphere theorem (Euler number $\pm 1$ implies the Milnor sphere bundle is a homology seven-sphere).

Assume the Axiom of Choice. For $(h,j)=(2,-1)$ the disk bundle $W=W_{2,-1}=D({\xi }_{2,-1})$ has Euler number $\epsilon =h+j=1$ and $k=h-j=3$. The middle form of Middle form and signature of the Milnor disk bundle is generated by the zero section $S^4\subseteq W$, whose normal bundle is ${\xi }_{2,-1}$ with Euler number $1$; hence the one-by-one matrix of the form is $[1]$ and the boundary signature is $\sigma (W_{2,-1})=1$. The normalized Thom evaluation $⟨U⌣x,[W,M]⟩=1$ gives the same sign in the rank-one computation.

Assume the Axiom of Choice and countable choice. For $(h,j)=(1,0)$ we have $h+j=1$ and $k=h-j=1$, so $\lambda (M_{1,0})=(h-j{)}^2-1=1-1=0(\mathrm{m}\mathrm{o}\mathrm{d}7),$ while for $(h,j)=(2,-1)$ we have $h+j=1$ and $k=h-j=3$, so $\lambda (M_{2,-1})=(h-j{)}^2-1=9-1=8\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}7)\mathrm{.}$ The displayed values follow from The Milnor sphere $M_{2,-1}$ is homeomorphic but not diffeomorphic to $S^7$. Both manifolds are homeomorphic to $S^7$ by The Milnor homotopy seven-spheres are homeomorphic to $S^7$, and the invariant is preserved under orientation-preserving diffeomorphism and negated under orientation reversal by The Milnor lambda invariant is well defined modulo seven, so no diffeomorphism between them exists in either orientation. Thus $M_{1,0}$ and $M_{2,-1}$ are two closed smooth seven-manifolds with distinct congruence invariants.

False claim: two closed smooth seven-manifolds that are homeomorphic have diffeomorphic smooth structures; equivalently, the homeomorphism type of a closed seven-manifold determines its smooth structure up to diffeomorphism.