[Math] Closed, simply connected manifolds which are not spheres

Question

In 2 or 3 dimensions, every closed simply connected manifold is a sphere. In the smooth category, I suppose you could take exotic smooth structures to give examples of closed simply connected manifolds which aren't spheres, but this doesn't really give a ton of insight for me. What about in the topological category? Is there an easy example here?

Answer 1

There are tons and tons of examples: $\mathbb{CP}^n$ for $n \ge 2$, $S^n \times S^m$ for $n,m \ge 2$, connected sums of these when they have the same dimensions, products of these... The list goes on and on.

It might help to review why a 2- or 3-dimensional closed manifold which is simply connected is a sphere to see why it fails in higher dimensions.

In dimension 2 it follows from the classification of closed surfaces, and there is no classification of manifolds in higher dimension that could tell us anything like that. This isn't really insightful.

Now in dimension 3. Hurewicz's theorem implies that $\require{cancel}H_1(M) \cong \cancel{\pi_1(M)}_{\mathrm{ab}} = 0$ because $M$ is simply connected. A simply connected manifold is always orientable, and so you can apply Poincaré duality $H^2(M) \cong H_1(M) = 0$, and the universal coefficients theorem implies $H_2(M) = 0$ (I'm glossing a bit over the details here, you also need the fact that every homology group of a closed manifold is finitely generated). So you're only left with $H_0(M) \cong H_3(M) \cong \mathbb{Z}$, in other words your manifold is a homology sphere.

Again Hurewicz's theorem implies $\pi_3(M) \cong \mathbb{Z}$, so there's a representative of a generator $f : S^3 \to M$ that induces an isomorphism on $\pi_3$. This $f$ then induces an isomorphism on $H_3$, and since there is no other nontrivial homology, $f$ is a homology equivalence. Since both spaces are simply connected, $f$ has to be a weak homotopy equivalence, and by Whitehead's theorem a homotopy equivalence. (Showing that the manifold is actually homeomorphic to a sphere would have earned you a million dollars back in the day.)

So what fails in higher dimension? Poincaré duality is not enough anymore to ensure that all homology beyond the top homology group vanishes. So you can have nontrivial $H_2$, and then all bets are off. But in dimension 3, there's just not enough room in (co)homology, so the manifold has to be a homology sphere; but a simply connected homology sphere is always homotopy equivalent to a sphere.