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…)