Let $X$ be a simply connected space of dimension $4k, k \geq 2$ for which the Poincaré duality theorem holds. In the previous post, I stated a theorem of Browder and Novikov which provides necessary and sufficient conditions for $X$ to be homotopy equivalent to a smooth manifold.

• $X$ must admit a candidate $\xi$ for a stable normal bundle (a lift of the Spivak normal fibration).
• Hirzebruch’s signature theorem, with this proto-normal bundle fed in, should accurately compute the signature of $X$.

The goal of this post is to sketch a proof of this theorem. The proof proceeds in two steps. The first step is to produce a degree one normal map $f: M \to X$ from a smooth manifold, i.e. a map which pulls $\xi$ back to the stable normal bundle of $M$ and which preserves the fundamental class. The second (and harder) step is to do surgery on $f$, to make it a homotopy equivalence. This surgery can be done “formally” before the middle dimension, but at the middle dimension a more careful bookkeeping process is required. It is here that the signature theorem, together with facts about quadratic forms over the integer, become necessary.

These are essentially the second half of the notes I prepared for the Kan seminar, for my second talk. My second talk was officially on Browder’s paper “Homotopy type of differentiable manifolds,” which announces (without proof) this result. In practice, I found the Kervaire-Milnor paper “Groups of homotopy spheres I” and the Milnor paper “A procedure for killing the homotopy groups of differentiable manifolds” very helpful in learning about this material.