### topology

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…)

I’ve just uploaded to arXiv my paper “The homology of ${\mathrm{tmf}}$,” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod ${2}$ cohomology of the spectrum ${\mathrm{tmf}}$ of (connective) topological modular forms, as a module over the Steenrod algebra: one has

$\displaystyle H^*(\mathrm{tmf}; \mathbb{Z}/2) \simeq \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2,$

where ${\mathcal{A}}$ is the Steenrod algebra and ${\mathcal{A}(2) \subset \mathcal{A}}$ is the 64-dimensional subalgebra generated by ${\mathrm{Sq}^1, \mathrm{Sq}^2,}$ and ${ \mathrm{Sq}^4}$. This computation means that the Adams spectral sequence can be used to compute the homotopy groups of ${\mathrm{tmf}}$; one has a spectral sequence

$\displaystyle \mathrm{Ext}^{s,t}( \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2, \mathbb{Z}/2) \simeq \mathrm{Ext}^{s,t}_{\mathcal{A}(2)}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \pi_{t-s} \mathrm{tmf} \otimes \widehat{\mathbb{Z}_2}.$

Since ${\mathcal{A}(2) \subset \mathcal{A}}$ is finite-dimensional, the entire ${E_2}$ page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the ${E_2}$ page for ${\mathrm{tmf}}$ is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of ${H^*(\mathrm{tmf}; \mathbb{Z}/2)}$ in this paper is based on a certain eight-cell (2-local) complex ${DA(1)}$, with the property that

$\displaystyle \mathrm{tmf} \wedge DA(1) \simeq BP\left \langle 2\right\rangle,$

where ${BP\left \langle 2\right\rangle = BP/(v_3, v_4, \dots, )}$ is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum ${BP\left \langle 2\right\rangle}$ is a complex-orientable ring spectrum, so that computations with it (instead of ${\mathrm{tmf}}$) become much simpler. In particular, one can compute the cohomology of ${BP\left \langle 2\right\rangle}$ (e.g., from the cohomology of ${BP}$), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of ${\mathrm{tmf}}$. This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, connected Hopf algebras. (more…)

The topic of topological modular forms is a very broad one, and a single blog post cannot do justice to the whole theory. In this section, I’ll try to answer the question as follows: ${\mathrm{tmf}}$ is a higher analog of ${KO}$theory (or rather, connective ${KO}$-theory).

1. What is ${\mathrm{tmf}}$?

The spectrum of (real) ${KO}$-theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex ${K}$-theory. As a ring spectrum, ${K}$ is complex orientable, and it corresponds to the formal group ${\hat{\mathbb{G}_m}}$: the formal multiplicative group. Along with ${\hat{\mathbb{G}_a}}$, the formal multiplicative group ${\hat{\mathbb{G}_m}}$ is one of the few “tautological” formal groups, and it is not surprising that ${K}$-theory has a “tautological” formal group because the Chern classes of a line bundle ${\mathcal{L}}$ (over a topological space ${X}$) in ${K}$-theory are defined by

$\displaystyle c_1( \mathcal{L}) = [\mathcal{L}] - [\mathbf{1}];$

that is, one uses the class of the line bundle ${\mathcal{L}}$ itself in ${K^0(X)}$ (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying ${\hat{\mathbb{G}_m}}$,

$\displaystyle \mathrm{Spec} \mathbb{Z} \rightarrow M_{FG},$

from ${\mathrm{Spec} \mathbb{Z}}$ to the moduli stack of formal groups ${M_{FG}}$, is a flat morphism. (more…)

The purpose of this post, the third in a series on deformation theory and DGLAs, is to describe the obstruction theory for a formal moduli problem associated to a DGLA.

1. Tangent-obstruction theories

Standard problems in classical deformation theory usually have a “tangent-obstruction theory” parametrized by certain successive cohomology groups. For example, let’s consider the problem of deformations of a smooth variety ${X}$ over an algebraically closed field ${k}$, over finite-dimensional local ${k}$-algebras. Then:

• The “infinitesimal automorphisms” of ${X}$—that is, automorphisms of the trivial deformation over ${k[\epsilon]/\epsilon^2}$—are given by ${H^0( X, T_X)}$ where ${T_X}$ is the tangent bundle (i.e., vector fields).
• The isomorphism classes of deformations of ${X}$ over the dual numbers ${k[\epsilon]/\epsilon^2}$ are given by ${H^1(X, T_X)}$.
• There is an obstruction theory with ${H^1, H^2}$. Specifically, given a square-zero extension of finite-dimensional local ${k}$-algebras

$\displaystyle 0 \rightarrow I \rightarrow A' \rightarrow A \rightarrow 0,$

and given a deformation ${\xi}$ of ${X}$ over ${\mathrm{Spec} A}$, there is a functorial obstruction in ${H^2(X, T_X) \otimes_k I}$ to extending the deformation over the inclusion ${\mathrm{Spec} A \hookrightarrow \mathrm{Spec} A'}$.

• In the previous item, if the obstruction vanishes, then the isomorphism classes of extensions of ${\xi}$ over ${\mathrm{Spec} A'}$ are a torsor for ${H^1(X, T_X) \otimes_k I}$.

One has a similar picture for other deformation problems, for example deformations of vector bundles or closed subschemes. The “derived” approach to deformation theory provides (at least in characteristic zero) a general explanation for this phenomenon. (more…)

Let ${k}$ be a field of characteristic zero. In the previous post, we introduced the category (i.e., ${\infty}$-category) ${\mathrm{Moduli}_k}$ of formal moduli problems over ${k}$. A formal moduli problem over ${k}$ is a moduli problem, taking values in spaces, that can be evaluated on the class of “derived” artinian ${k}$-algebras with residue field ${k}$: this was the category ${\mathrm{CAlg}_{sm}}$ introduced in the previous post.

In other words, a formal moduli problem was a functor

$\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S} \ (= \text{spaces}),$

which was required to send ${k}$ itself to a point, and satisfy a certain cohesiveness condition: ${F}$ respects certain pullbacks in ${\mathrm{CAlg}_{sm}}$ (which corresponded geometrically to pushouts of schemes).

The main goal of the series of posts was to sketch a proof of (and define everything in) the following result:

Theorem 7 (Lurie; Pridham) There is an equivalence of categories between ${\mathrm{Moduli}_k}$ and the ${\infty}$-category ${\mathrm{dgLie}}$ of DGLAs over ${k}$.

4. Overview

Here’s a rough sketch of the idea. Given a formal moduli problem ${F}$, we should think of ${F}$ as something like a small space, concentrated at a point but with lots of “infinitesimal” thickening. (Something like a ${\mathrm{Spf}}$.) Moreover, ${F}$ has a canonical basepoint corresponding to the “trivial deformation.” That is, we can think of ${F}$ as taking values in pointed spaces rather than spaces.

It follows that we can form the loop space ${\Omega F = \ast \times_F \ast}$ of ${F}$, which is a new formal moduli problem. However, ${\Omega F}$ has more structure: it’s a group object in the category of formal moduli problems — that is, it’s some sort of derived formal Lie group. Moreover, knowledge of the original ${F}$ is equivalent to knowledge of ${\Omega F}$ together with its group structure: we can recover ${F}$ as ${B \Omega F}$ (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via ${B, \Omega}$) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality. (more…)

Let ${X}$ be a variety over an algebraically closed field ${k}$. ${X}$ is said to be rational if ${X}$ is birational to ${\mathbb{P}_k^n}$. In general, it is difficult to determine when a variety in higher dimensions is rational, although there are numerical invariants in dimensions one and two.

• Let ${X}$ be a smooth projective curve. Then ${X}$ is rational if and only if its genus is zero.
• Let ${X}$ be a smooth projective surface. Then ${X}$ is rational if and only if there are no global 1-forms on ${X}$ (i.e., ${H^0(X, \Omega_{X/k}) = 0}$) and the second plurigenus ${H^0(X, \omega_{X/k}^{\otimes 2}) }$ vanishes. This is a statement about the negativity of the cotangent bundle (or, equivalently, of the positivity of the tangent bundle) which is a birational invariant and which holds for ${\mathbb{P}^2_k}$. The result is a criterion of Castelnuovo, extended by Zariski to characteristic $p$.

In higher dimensions, it is harder to tell when a variety is rational. An easier problem is to determine when a variety is unirational: that is, when there is a dominant rational map

$\displaystyle \mathbb{P}_k^n \dashrightarrow X;$

or, equivalently, when the function field ${k(X)}$ has a finite extension which is purely transcendental. In dimensions one and two (and in characteristic zero), the above invariants imply that a unirational variety is rational. In higher dimensions, there are many more unirational varieties: for example, a theorem of Harris, Mazur, and Pandharipande states that a degree ${d}$ hypersurface in ${\mathbb{P}^N}$, ${N \gg 0}$ is always unirational.

The purpose of this post is to describe a theorem of Serre that shows the difficulty of distinguishing rationality from unirationality. Let’s work over ${\mathbb{C}}$. The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial ${\pi_1}$.

Theorem 1 (Serre) A unirational (smooth, projective) variety over ${\mathbb{C}}$ has trivial ${\pi_1}$.

The reference is Serre’s paper “On the fundamental group of a unirational variety,” in J. London Math Soc. 1959. (more…)

The purpose of this post is to describe Sullivan’s proof of the Adams conjecture via algebraic geometry; the conjecture and its motivation were described in the previous post (from where the notation is taken). The classical reference is Sullivan’s paper “Genetics of homotopy theory and the Adams conjecture,” and the MIT notes on “Geometric topology.”

1. First step: completion at a prime

Sullivan’s proof of the Adams conjecture  is based on interpreting the Adams operations via a surprising Galois symmetry in the (profinitely completed) homotopy types of classifying spaces. Let’s work in the complex case for simplicity. Our goal is to show that the composite

$\displaystyle BU(n) \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic. (The map $J$ was defined in the previous post.)

Since the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$ are finite, it will follow (by the Milnor exact sequence) that we can let ${n \rightarrow \infty}$ and conclude that the map

$\displaystyle BU \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic (i.e., there are no phantom maps into a spectrum with finite homotopy groups).

Using again the finiteness of the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$, we can get a splitting

$\displaystyle B \mathrm{gl}_1(S)[1/k] = \prod_{p \nmid k} \widehat{ B \mathrm{gl}_1(S)}_p$

into the respective profinite completions. There is a well-behaved theory of profinite completions for connective spectra, or for sufficiently nice (e.g. simply connected with finitely generated homology) spaces, which will be the subject of a different post. (more…)

The Sullivan conjecture on maps from classifying spaces originated in Sullivan’s work on localizations and completions in topology, which together with étale homotopy theory led him to a proof of the classical Adams conjecture. The purpose of this post is to briefly explain the conjecture; in the next post I’ll discuss Sullivan’s proof.

1. Vector bundles and spherical fibrations

Let ${X}$ be a finite CW complex. Given a real ${n}$-dimensional vector bundle ${V \rightarrow X}$, one can form the associated spherical fibration ${S(V) \rightarrow X}$ with fiber ${S^{n-1}}$ by endowing ${V}$ with a euclidean metric and taking the vectors of length one in each fiber. The Adams conjecture is a criterion on when the sphere bundles associated to vector bundles are fiber homotopy trivial. It will be stated in terms of the following definition:

Definition 1 Let ${J(X)}$ be the quotient of the Grothendieck group ${KO(X)}$ of vector bundles on ${X}$ by the relation that ${V \sim W}$ if ${V , W}$ have fiber homotopy equivalent sphere bundles (or rather, the group generated by it).

Observe that if ${V, W}$ and ${V', W'}$ have fiber homotopy equivalent sphere bundles, then so do ${V \oplus V', W \oplus W'}$; for example, this is because the sphere bundle of ${V \oplus V'}$ is the fiberwise join of that of ${V}$ and ${V'}$. It is sometimes more convenient to work with pointed spherical fibrations instead: that is, to take the fiberwise one-point compactification ${S^V}$ of a vector bundle ${V \rightarrow X}$ rather than the sphere bundle. In this case, the fiberwise join is replaced with the fiberwise smash product; we have

$\displaystyle S^{V \oplus W} \simeq S^V \wedge_X S^W,$

where ${\wedge_X}$ denotes a fiberwise smash product.

One reason is that this is of interest is that the group ${KO(X)}$ of vector bundles on a space ${X}$ is often very computable, thanks to Bott periodicity which identifies the ${KO}$-groups of a point. The set of spherical fibrations is much harder to describe: to describe the spherical fibrations over ${S^n}$ essentially amounts to computing a bunch of homotopy groups of spheres.

This post is part of a series (started here) of posts on the structure of the category ${\mathcal{U}}$ of unstable modules over the mod ${2}$ Steenrod algebra ${\mathcal{A}}$, which plays an important role in the proof of the Sullivan conjecture (and its variants).

In the previous post, we introduced some additional structure on the category ${\mathcal{U}}$.

• First, using the (cocommutative) Hopf algebra structure on ${\mathcal{A}}$, we got a symmetric monoidal structure on ${\mathcal{U}}$, which was an algebraic version of the Künneth theorem.
• Second, we described a “Frobenius” functor

$\displaystyle \Phi : \mathcal{U} \rightarrow \mathcal{U},$

which was symmetric monoidal, and which came with a Frobenius map ${\Phi M \rightarrow M}$.

• We constructed an exact sequence natural in ${M}$,

$\displaystyle 0 \rightarrow \Sigma L^1 \Omega M \rightarrow \Phi M \rightarrow M \rightarrow \Sigma \Omega M \rightarrow 0, \ \ \ \ \ (4)$

where ${\Sigma}$ was the suspension and ${\Omega}$ the left adjoint. In particular, we showed that all the higher derived functors of ${\Omega}$ (after ${L^1}$) vanish.

The first goal of this post is to use this extra structure to prove the following:

Theorem 39 The category ${\mathcal{U}}$ is locally noetherian: the subobjects of the free unstable module ${F(n)}$ satisfy the ascending chain condition (equivalently, are finitely generated as ${\mathcal{A}}$-modules).

In order to prove this theorem, we’ll use induction on ${n}$ and the technology developed in the previous post as a way to make Nakayama-type arguments. Namely, the exact sequence (4) becomes

$\displaystyle 0 \rightarrow \Phi F(n) \rightarrow F(n) \rightarrow \Sigma F(n-1) \rightarrow 0,$

as we saw in the previous post. Observe that ${F(0) = \mathbb{F}_2}$ is clearly noetherian (it’s also not hard to check this for ${F(1)}$). Inductively, we may assume that ${F(n-1)}$ (and therefore ${\Sigma F(n-1)}$) is noetherian.

Fix a subobject ${M \subset F(n)}$; we’d like to show that ${M}$ is finitely generated. (more…)

This is part of a series of posts intended to understand some of the basic structure of the category ${\mathcal{U}}$ of unstable modules over the (mod ${2}$) Steenrod algebra, to prepare for the proof of the Sullivan conjecture. Here’s what we’ve seen so far:

• ${\mathcal{U}}$ is a Grothendieck abelian category, with a set of compact, projective generators ${F(n)}$ (the free unstable module on a generator in degree ${n}$). (See this post.)
• ${F(n)}$ has a tautological class ${\iota_n}$ in degree ${n}$, and has a basis given by ${\mathrm{Sq}^I \iota_n}$ for ${I}$ an admissible sequence of excess ${\leq n}$. (This post explained the terminology and the proof.)
• ${F(1)}$ was the subspace ${\mathbb{F}_2\left\{t, t^2, t^4, \dots\right\} \subset \widetilde{H}^*(\mathbb{RP}^\infty; \mathbb{F}_2)}$.

Our goal in this post is to describe some of the additional structure on the category ${\mathcal{U}}$, which will eventually enable us to prove (and make sense of!) results such as ${F(n) \simeq (F(1)^{\otimes n})^{ \Sigma_n}}$. We’ll start with the symmetric monoidal tensor product and the suspension functor, and then connect this to the Frobenius maps (which will be defined below).

1. The symmetric monoidal structure

Our first order of business is to describe the symmetric monoidal structure on ${\mathcal{U}}$, which will be given by the ${\mathbb{F}_2}$-linear tensor product. In fact, recall that the Steenrod algebra is a cocommutative Hopf algebra, under the diagonal map

$\displaystyle \mathrm{Sq}^n \mapsto \sum_{i+j = n} \mathrm{Sq}^i \otimes \mathrm{Sq}^j.$

The Hopf algebra structure is defined according to the following rule: we have that ${\theta}$ maps to ${\sum \theta' \otimes \theta''}$ if and only if for every two cohomology classes ${x,y }$ in the cohomology of a topological space, one has

$\displaystyle \theta(xy) = \sum \theta'(x) \theta''(y).$

The cocommutative Hopf algebra structure on ${\mathcal{A}}$ gives a tensor product on the category of (graded) ${\mathcal{A}}$-modules, which is symmetric monoidal. It’s easy to check that if ${M, N}$ are ${\mathcal{A}}$-modules satisfying the unstability condition, then so does ${M \otimes N}$. This is precisely the symmetric monoidal structure on ${\mathcal{U}}$. (more…)

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