Let ${C}$ be an algebraic curve over ${\mathbb{C}}$. A theta characteristic on ${C}$ is a (holomorphic or algebraic) square root of the canonical line bundle ${K_C}$, i.e. a line bundle ${L \in \mathrm{Pic}(C)}$ such that

$\displaystyle L^{\otimes 2} \simeq K_C.$

Since the degree of ${K_C}$ is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian ${J(C) = \mathrm{Pic}^0(C)}$, which is isomorphic to ${H^1(C; \mathbb{Z}/2\mathbb{Z}) \simeq (\mathbb{Z}/2\mathbb{Z})^{2g}}$.

One piece of geometric motivation for theta characteristics comes from the following observation: theta characteristics form an algebro-geometric approach to framings. By a theorem of Atiyah, holomorphic square roots of the canonical bundle on a compact complex manifold are equivalent to spin structures. In complex dimension one, a choice of a spin structure is equivalent to a framing of ${M}$. On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod ${2}$ homology (with its intersection pairing), which gives an important invariant of the framed manifold known as the Kervaire invariant. (See for instance this post on the paper of Kervaire that introduced it.)

It turns out that the mod ${2}$ function ${L \mapsto \dim H^0(C, L)}$ on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least ${\neq 2}$) approach to the Kervaire invariant, for surfaces!

Most of the material in this post is from two papers: Atiyah’s Riemann surfaces and spin structures and Mumford’s Theta characteristics of an algebraic curve. (more…)

This is the second post in a series on Kervaire’s paper “A manifold which does not admit any differentiable structure.” In the previous post, we described a form on the middle cohomology of a $k-1$-connected $2k$-dimensional manifold, for $k \neq 1, 3, 7$. In this post, we can define the Kervaire invariant of such a framed manifold, by showing that this defines a form. I’ll try to sketch the proof that there is no framed manifold of Kervaire invariant one in dimension 10.

1. The form $q$ is a quadratic refinement

Let’s next check that the form ${q: H^k(M; \mathbb{Z}) \rightarrow \mathbb{Z}/2}$ defined in the previous post (we’ll review the definition here) is actually a quadratic refinement of the cup product. Precisely, this means that for ${x, y \in H^k(M; \mathbb{Z})}$, we want

$\displaystyle q(x+y) - q(x) - q(y) = (x \cup y)[M].$

In particular, this implies that ${q}$ descends to a function on ${H^k(M; \mathbb{Z}/2)}$, as it shows that ${q}$ of an even class is zero in ${\mathbb{Z}/2}$. The associated quotient map ${q: H^k(M; \mathbb{Z}/2) \rightarrow \mathbb{Z}/2}$ is, strictly speaking, the quadratic refinement.

In order to do this, let’s fix ${x, y \in H^k(M; \mathbb{Z}/2)}$. As we saw last time, these can be obtained from maps

$\displaystyle M \rightarrow \Omega \Sigma S^k$

by pulling back the generator in degree ${k}$. Let ${f_x}$ be a map associated to ${x}$, and let ${f_y}$ be a map associated to ${y}$. We then have that

$\displaystyle q(x) = f_x^*(u_{2k}) [M], \quad q(y) = f_y^*(u_{2k})[M],$

for ${u_{2k}}$ the generator of ${H^{2k}(\Omega \Sigma S^k; \mathbb{Z}/2)}$. As we saw, this was equivalent to the definition given last time. (more…)

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…)