Homotopy classes of manifolds of dimension $2n$ which are $(n-1)$ connected

Question

In his essay "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres" Milnor states (in passing) "The manifold can be built up (up to homotopy type) by taking a union of finitely many n-spheres intersecting at a single point, and then attaching a $2n$-cell $e_{2n}$ by a mapping of the boundary to this union of spheres, so that $M_{2n} \cong S^n \vee \cdots \vee S^n \cup_f e_{2n}$." Does anybody know where I could look this up? Thanks

Answer 1

See Hatcher, proposition 4.C. Assuming $n>1$, the manifold is simply connected, and hence you can make it homotopy equivalent to a CW complex with one $n$-cell for each $\Bbb Z$ in $H_n$, and an $n$-cell and $(n+1)$-cell for each $\Bbb Z/p$.

Because your manifold is $(n-1)$-connected, Hurewicz says that the first nonzero homology is $H_n(M;\Bbb Z)$. Since your manifold is closed and orientable (because $H^1(M;\Bbb Z/2) = 0$), Poincare duality holds, and hence $H_n = H^n = \text{Hom}(H_n,\Bbb Z)$, and we see that $H_n$ is free. Inductively use Poincare duality and universal coefficients to see that $H_i = 0$ when $n<i<2n$, and then $H_{2n} = \Bbb Z$ because you're closed and orientable. Hence the desired result follows.