The goal of this post is to describe a small portion of the answer to the following question:

**Question 1:** When is a simply connected space homotopy equivalent to a (compact) -dimensional smooth manifold?

A compact manifold is homotopy equivalent to a finite CW complex, so must be one itself. Equivalently (since is simply connected), the homology must be finitely generated.

More interestingly, we know that a hypothetical compact -manifold homotopy equivalent to (which is necessarily orientable) satisfies *Poincaré duality.* That is, we have a *fundamental class* with the property that cap product induces an isomorphism

for all groups and for all . It follows that, if the answer to the above question is positive, an analogous condition must hold for . This motivates the following definition:

Definition 1A simply connected space is aPoincaré duality space (or Poincaré complex) of dimensionif there is a class such that for every group , cap product induces an isomorphism

A consequence is that the cohomology (and homology) groups of vanish above dimension , and is generated by . In other words, behaves like the fundamental class of an -dimensional manifold.

We can thus pose a refined version of the above question.

**Question 2: **Given a simply connected Poincaré duality space of dimension , is there a homotopy equivalence for a compact (necessarily -dimensional) manifold?

The answer to Question 2 is not always positive.

**Example 1** The Kervaire manifold is a topological 4-connected 10-manifold of (suitably defined) Kervaire invariant one. Since, as Kervaire showed, all smooth framed 10-manfolds have Kervaire invariant zero, he concluded that there was no smooth manifold in its homotopy type.

**Example 2** It is known that there exists a simply connected compact topological 4-manifold whose intersection form is even and has signature eight (the “ manifold”). Such a manifold cannot be homotopy equivalent to any smooth manifold. In fact, if it were homotopy equivalent to a smooth 4-manifold , then the evenness of the intersection form on shows that (and in fact all ) act trivially on . The Wu formulas imply that the Stiefel-Whitney classes of vanish. In particular, admits a spin structure, and a theorem of Rohlin asserts that the signature of a spin 4-manifold is divisible by . (more…)