Barcelona Topology Workshop 2013

An even unimodular lattice of dimension $n$ is a free $\mathbb{Z}$-module $L$ of dimension $n$, equipped with a quadratic form $\mathrm{q}:L\to\mathbb{Z}$, non-degenerate over $\mathbb{Z}$ (i.e. such that the associated bilinear form induces an isomorphism $L\cong\mathrm{Hom}_{\mathbb{Z}}(L,\mathbb{Z})$) and positive definite. Such an $L$ can be thought of as a lattice in the euclidean vector space $V:=\mathbb{R}\otimes_{\mathbb{Z}}L$, satisfying the two following properties:

• one has $x.x\in 2\hspace{1pt}\mathbb{Z}$ for all $x$ in $L$ (hence $x.y\in\mathbb{Z}$ for all $x$ and $y$ in $L$);
• the lattice $L$ has covolume $1$ in $V$.

This observation explains the terminology.
One denotes by $\mathrm{X}_{n}$ the set of isomorphism classes of $n$-dimensional even unimodular lattices. It is well-known that the set $\mathrm{X}_{n}$ is finite and that it is non-empty if and only if $n$ is divisible by $8$. The set $\mathrm{X}_{n}$ has been determined for $n\leq 24$:

• $\mathrm{X}_{8}$ has only one element $\mathrm{E}_{8}$ (related to the root system $\mathbf{E}_{8}$);
• $\mathrm{X}_{16}$ has two elements $\mathrm{E}_{16}$ (related to the root system $\mathbf{D}_{16}$) and $\mathrm{E}_{8}\oplus\mathrm{E}_{8}$;
• $\mathrm{X}_{24}$ was determined by Niemeier in 1968, it has 24 element (one of them is the famous Leech lattice, the $23$ other ones are again related to root systems).

One shows that $\mathrm{X}_{32}$ has more than $8\times 10^{7}$ elements (Serre says playfully in his cours d'arithmétique that they have not been listed!).
Let $p$ be a prime. Let $V$ be an euclidean vector space of dimension $n$. Two even unimodular lattices $L$ and $L'$ in $V$ are said to be $p$-neighbours (in the sense of M. Kneser) if $L\cap L'$ is of index $p$ in $L$ (and $L'$).
In this context, the Hecke operator $\mathrm{T}_{p}$ is the endomorphism of the free $\mathbb{Z}$-module $\mathbb{Z}[\mathrm{X}_{n}]$, generated by the set $\mathrm{X}_{n}$, defined by the formula $$ \hspace{24pt} \mathrm{T}_{p}[L]=\sum_{\text{$L'$ $p$-neighbour of $L$}}[L'] \hspace{24pt}. $$ Theorem. The matrix of the endomorphism $\mathrm{T}_{p}$ in the basis $(\mathrm{E}_{16},\mathrm{E}_{8}\oplus\mathrm{E}_{8})$ is $$ \hspace{18pt} \frac{p^{4}-1}{p-1}\hspace{4pt} (\hspace{2pt}(p^{11}+p^{7}+p^{4}+1) \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \frac{p^{11}-\tau(p)+1}{691} \begin{bmatrix} -286 & 405 \\ 286 & -405 \end{bmatrix}\hspace{2pt}) \hspace{18pt}, $$ $\tau$ denoting the Ramanujan function.
One will explain how to prove this theorem using the theory of Hecke operators for Siegel modular forms and one will describe the ingredients involved in the analog of the above formula in dimension $24$.

A polyhedral product is an old object in homotopy theory, which is recently focused on from toric topology. A new interesting discovery is a connection with combinatorial commutative algebra; the Stanley-Reisner rings and some of their derived algebras are realized as the cohomology of certain polyhedral products. I will talk about a recent result with Kouyemon Iriye on wedge decomposition of polyhedral products which implies the Golod property of the Stanley-Reisner rings.

Kac-Moody groups generalize Lie groups but are typically infinite dimensional. In this talk, we will construct a map from a discrete Kac-Moody group over the algebraic closure of the field with p elements to a complex topological Kac-Moody group of the same type. This map generalizes a map Friedlander and Mislin constructed in the Lie case and is a homology equivalence at any prime q different from p.

A quasi toric manifolds is the remainder of a smooth toric variety after stripping away all algebraic geometric structure. But how much of a complex structure can you still implement on quasi toric manifolds? We will answer this question for almost complex structures an will give a complete classification of these. The proof is based on ideas and methods used very successfully in the theory of classifying spaces, in particular on homology decomposition techniques.

There is a mysterious analogy between the Dold-Kan correspondance and Newton's finite difference calculus which I discovered some years ago. I recently made some progress in understanding the relation. I will discuss a few simple applications to algebraic topology.

Let \(G\) be an abstract group. Is there a space \(X\) such that the group of self homotopy equivalences of \(X\), \(\mathcal E (X)\), is isomorphic to \(G\)? This is the classical realizability problem for groups. It asks for characterization of those groups that appear as the group of self homotopy equivalences of a space. In a recent paper, the authors gave an affirmative answer to that question in the case of finite groups. In this talk we go one step further by considering the following question. Let \(G\) be a group acting faithfully on a finitely generated module \(M\). Is there a space \(X\) such that, for some \(n \geq 2\), the \(\mathcal E(X)\)-module \(\pi_n (X)\) is isomorphic to the \(G\)-module \(M\)? Ours is a realizability problem for group actions on modules. It asks for characterization of those actions that appear as the natural \(\mathcal E (X)\)-action on the homotopy groups of a space \(X\). This work fits within the framework of (a dual of) the classical Steenrod problem for \(G\)-Moore spaces.

An important problem in topology is to classify maps between topological spaces. The case where the spaces involved are classifying spaces of groups is a favourable one in which explicit results have been obtained. In the last years, the classifying spaces of certain algebraic objects (p-local compact groups) defined by Broto-Levi-Oliver play the role of homotopy analogues of the p-completion of classifying spaces of compact Lie groups. I will describe some recent results on the study of complex homotopy representations of p-local compact groups, and in particular, the existence of unitary embeddings of finite loop spaces at a prime p. This is a joint work with J. Cantarero.