Meeting of the Swedsih, Spanish and Catalan Mathematical Societies
Umeå, June 12 – June 15, 2017
Special session: Homotopy Theory
Preliminary Program
Tuesday June 13
Time
Speaker
14.30–15.30 Alexander Berglund
15.30–16.00
16.00–17.00 Thomas Kragh
17.00–18.00 Cristina Costoya
Talk title
Rational homotopy theory of automorphisms of manifolds
COFFEE BREAK
Waldhausens K-theory of spaces and exact Lagrangian submanifolds
Polyhedral products in Kahn’s Realizability Problem
Wednesday June 14
Time
14.30–15.00
15.00–15.30
15.30–16.00
16.00–17.00
17.00–18.00
Speaker
Felix Wierstra
Kiko Belchı́
Antonio Viruel
Albert Ruiz
Talk title
Hopf invariants in rational homotopy theory
Persistent homology: from Stasheff to the hospital
COFFEE BREAK
On the realizability of group actions
On the classification of p-local compact groups over a fixed discrete p-toral group.
Thursday June 15
Time
14.00–14.30
14.30–15.30
15.30–16.00
16.00–16.30
16.30–17.30
Speaker
Carlos Sáez
Federico Cantero
Magnus Carlson
Ramón Flores
Talk title
Finite subgroups of diffeomorphisms of 4-manifolds
Rational homotopy theory of Thom spaces
COFFEE BREAK
Higher obstructions to the embedding problem in Galois theory
Burnside groups and idempotent transformations of groups
Abstracts for Tuesday June 13
Rational homotopy theory of automorphisms of manifolds
Alexander Berglund, Stockholms universitet
I will discuss some recent results on differential graded Lie algebra models, in the sense of
Quillen’s rational homotopy theory, for classifying spaces of automorphisms of simply connected
manifolds. In favorable situations, these models can be used to compute the rational cohomology
(in a stable range) of the classifying spaces in terms of certain decorated graph complexes. This is
joint work with Ib Madsen.
Waldhausens K-theory of spaces and exact Lagrangian sub-manifolds
Thomas Kragh, Uppsala universitet
In this talk I will define what an exact Lagrangian immersion is and what the notion of a generating
family for such is. I will shortly discuss how these generating families exists for some Lagrangian
embeddings. I will then discuss how this is related to Waldhausens K theory of spaces, and what
kind of results this relation produces for these embeddings. In particular I will discus the Lagrangian
Gauss map to U/O and why this is homotopy trivial in some interesting cases.
Polyhedral products in Kahn’s Realizability Problem
Cristina Costoya, Universidade da Coruña
Kahn’s realizability problem asks if every group G is the group of self-homotopy equivalences of
a simply-connected space X. Raised by D. Kahn in the 60’s, it is still an open problem. In the past,
we presented a general method to realize any finite group G by a space X, but since our techniques
came from Rational Homotopy Theory, the homology of X is of finite-type over Q, not over Z. In
this talk, we will introduce techniques from Toric Topology in the resolution of Kahn’s Problem to
show that any finite group G is realizable by an integral finite-type space X.
This is a joint work with A. Viruel.
Abstracts for Wednesday June 14
Hopf invariants in rational homotopy theory
Felix Wierstra, Stockholms universitet
The classical Hopf invariant is an important invariant of homotopy classes of maps f : S 4n−1 →
S 2 and can for example be used to show that the Hopf fibration is not null homotopic. In this
talk we will explain how the classical Hopf invariant is generalized to obtain a complete invariant
of rational homotopy classes of maps. Then we will explain how for manifolds this invariant can be
expressed as a sequence of integrals and that two maps are homotopic if and only if these integrals
are equal.
Persistent homology: from Stasheff to the hospital
Kiko Belchı̀, Southampton University
Persistent homology computes the (persistent) Betti numbers of a filtration of topological spaces.
Depending on the context, this can allow us to find highly non-linear structure in data or to compute
novel geometric descriptors of shapes. In this talk I will convince you that persistent homology is
useful and that you can apply your advanced knowledge in algebraic topology to enrich this theory.
To do so, I will explain how we use it to better understand Chronic Obstructive Pulmonary Disease
(COPD) and how we enhance the power of persistent homology by studying A∞ -structures which
encode information related to cup and Massey products. The practical section is a joint work with
Prof Jacek Brodzki’s EPSRC-funded research group Joining the dots and the Respiratory Unit at
Southampton General Hospital, and the theoretical part is a joint work with Prof Aniceto Murillo.
On the realizability of group actions
Antonio Viruel, Universidad de Málaga
If M is a ZG-module for a group G, we say that a simply-connected space X realize this action if,
for some k, πk (X) as a ZE(X)-module for the group E(X) of self-homotopy equivalences of X, is
isomorphic to M as a ZG-module. Which modules can be so realized? We obtain a positive answer
for any faithful finitely generated QG-module, where G is finite. Our proof relies on providing a
positive answer to Kahn’s problem for a large class of orthogonal groups of which, by using invariant
theory, our case is shown to be a particular one.
On the classification of p-local compact groups over a fixed discrete p-toral group
Albert Ruiz, Universitat Autónoma de Barcelona
p-local finite groups where defined by Broto-Levi-Oliver as a generalization of finite groups
studied at a prime p. Later on, the same authors, defined p-local compact groups as a generalization
of compact Lie Groups at a prime p and p-compact groups.
Examples of p-local finite groups which do not correspond to finite groups are known for every
prime number p. In the infinite case, very few cases which do not correspond to p-compact groups
have been studied. In this talk we will see a classification of p-local compact groups over some special discrete p-toral groups (joint work with Bob Oliver) which include a family of p-local compact
groups which are not p-compact groups (joint work with Alex González and Toni Lozano).
Abstracts for Thursday June 15
Finite subgroups of diffeomorphisms of 4-manifolds
Carlos Sáez, Universitat de Barcelona-Barcelona Graduate School of Mathematics
If X is a smooth manifold, Dif f (X) is said to be Jordan if there exists a constant C such that
every finite subgroup G of Dif f (X) has an abelian subgroup A satisfying [G : A] < C. I will
explain some recent progress on the problem of determining which 4-manifolds have Jordan groups
of diffeomorphims, and on the analogous problem obtained by substituting abelian by 2-nilpotent.
This is joint work with Ignasi Mundet i Riera
Rational homotopy theory of Thom spaces
Federico Cantero, Universitat de Barcelona-Barcelona Graduate School of Mathematics
In this talk we give rational homotopy models of Thom spaces of vector bundles, by showing
that the Thom isomorphism desdends to the level of commutative differential graded algebras, generalising work of Fèlix, Oprea and Tanrè. Then, we use this to study the cohomology of spaces of
submanifolds. This is joint work with Urtzi Buijs and Joana Cirici.
Higher obstructions to the embedding problem in Galois theory
Magnus Carlson, Kungliga tekniska högskolan, Stockholm
We will explain how one can apply obstruction theory for homotopy fixed points to study problems
from Galois theory. To be more precise, we will show how to use this obstruction-theoretic machinery
to prove that certain groups can not be realized as Galois groups of unramified extensions of certain
number fields. The obstructions we derive generalize the previously known obstructions to the
embedding problem in Galois theory, and we will mention ongoing work in the function field case.
This is joint work with Tomer Schlank.
Burnside groups and idempotent transformations of groups
Ramón Flores, Universidad de Sevilla
Burnside groups and idempotent transformations of groups. Abstract: In this talk we will describe
several interesting occurrences of Burnside groups of prime exponent in the context of localization
and cellular covers of groups. In particular, we show that groups in this family cannot we constructed
as colimits of copies of cyclic groups of order p, that their classifying spaces sometimes hide torsion in
other primes, and that they give counterexamples to a conjecture of Farjoun concerning idempotency
of composed functors. This is joint work with Fernando Muro and Jèróme Scherer