I’ve been struggling lately with Kervaire’s paper “A manifold which does not admit any differentiable structure.” The paper defines the Kervaire invariant of a 4-connected combinatorial 10-manifold and shows that it is automatically zero on smooth manifolds. He then constructs an example of a 4-connected combinatorial 10-manifold whose Kervaire invariant is ${1}$, concluding that it can admit no smooth structure.

The definition of the Kervaire invariant given in the paper is a little complicated, and I’d like to work through it carefully in this post.

1. Generalities on framed manifolds

Let ${M}$ be a framed ${k}$-dimensional manifold. One of the consequences of being framed is that ${M}$ admits a fundamental class in stable homotopy. Stated more explicitly, there is a stable map

$\displaystyle \phi_M: \Sigma^\infty S^{k} \rightarrow \Sigma^\infty_+ M$

which induces an isomorphism in ${H_{k}}$. One can construct such a map by imbedding ${M}$ inside a large euclidean space ${\mathbb{R}^{N+k}}$. The Thom-Pontryagin collapse map then runs

$\displaystyle S^{N+k} \rightarrow \mathrm{Th}( \nu),$

for ${\nu}$ the normal bundle of ${M}$ in ${\mathbb{R}^{N+k}}$. A choice of trivialization of this normal bundle (that is, a framing) allows us to identify ${\mathrm{Th}(\nu)}$ with ${S^{N} \wedge M_+}$, and this gives a map

$\displaystyle S^{N+ k}\rightarrow S^N \wedge M_+,$

which is an isomorphism in top homology. This gives the desired stable map.

A consequence of this observation is that, in a cell decomposition of ${M}$, the top cell stably splits off. That is, we can take the stable map ${\phi_M : S^{k} \rightarrow M }$ and compose it with the crushing map ${M \rightarrow S^{k}}$ (for instance, identifying ${S^{k}}$ with ${M/ M \setminus \mathrm{disk}}$) such that the composite

$\displaystyle S^{k} \stackrel{\phi_M}{\rightarrow} M \rightarrow S^{k}$

is an equivalence. (more…)

In earlier posts, we analyzed the Hurewicz homomorphism

$\displaystyle \pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq \mathbb{Z}[b_1, b_2, \dots ]$

in purely algebraic terms: ${\pi_* MU}$ was the Lazard ring, and the Hurewicz map was the map classifying the formal group law ${\exp( \exp^{-1}(x) + \exp^{-1}(y))}$ for ${\exp(x)}$ the “change of coordinates”

$\displaystyle \exp(x) = x + b_1 x + b_2 x^2 + \dots.$

We can also express the Hurewicz map in more geometric terms. The ring ${\pi_* MU}$ is, by the Thom-Pontryagin construction, the cobordism ring of stably almost-complex manifolds. The ring ${H_*(MU; \mathbb{Z})}$ is isomorphic to the Pontryagin ring ${H_*(BU; \mathbb{Z})}$ by the Thom isomorphism. In these terms, we can describe the Hurewicz map explictly:

Theorem 1 The Hurewicz map ${\pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq H_*(BU; \mathbb{Z})}$ sends the cobordism class of a stably almost complex manifold ${M}$ to the element ${f_* [M]}$ where

$\displaystyle f_* : M \rightarrow BU$

classifies the stable normal bundle of ${M}$ (together with its complex structure), and where ${[M]}$ is the fundamental class of ${M}$.

In particular, we will be able to work out explicitly where a given complex manifold representing a cobordism class goes under this map. As an application, we’ll show using Lagrange inversion that the complex projective spaces determine the logarithm of the formal group law on ${\pi_* MU}$. (more…)

The signature ${\sigma(M)}$ of a ${4k}$-dimensional compact, oriented manifold ${M}$ is a classical cobordism invariant of ${M}$; the so-called Hirzebruch signature formula states that ${\sigma(M)}$ can be computed as a complicated polynomial in the Pontryagin classes of the tangent bundle ${TM}$ (evaluated on the fundamental class of ${M}$). When ${M}$ is four-dimensional, for instance, we have

$\displaystyle \sigma(M) = \frac{p_1}{3}.$

This implies that the Pontryagin number ${p_1}$ must be divisible by three.

There are various further divisibility conditions that hold in special cases. Here is an important early example:

Theorem 1 (Rohlin) If ${M}$ is a four-dimensional spin-manifold, then ${\sigma(M)}$ is divisible by ${16}$ (and so ${p_1}$ by ${48}$).

I’d like to describe the original proof of Rohlin’s theorem, which relies on a number of tools from the 1950s era of topology. At least, I have not gotten a copy of Rohlin’s paper; the proof is sketched, though, in Kervaire-Milnor’s ICM address, which I’ll follow.  (more…)