Topological and Homological Tools for the Study of Spaces from Infinitely Far Away

Speaker: Sergio Romero Alba

Georg-August-Universität Göttingen & University of Málaga

About the Series

This course provides an illustrative introduction to coarse geometry and progresses toward presenting currently open research problems related to the computation of large scale invariants, particularly through the framework of topological bornological spaces. From Coarse Geometry to Cohomology and Stone Spaces.

Schedule

Motivation. Groups as metric spaces

Part 1 of the series

Wednesday, November 5, 2025

05:00 PM IST

Abstract: We review, through examples, the foundational philosophy of Geometric Group Theory, where each group $G$ with a finite generating set $S$ is associated with a metric space $(G, dS)$, translating algebraic properties of the group into geometric features of the space. Although this assignment initially depends on the choice of $S$, such dependence vanishes when viewing $(G, dS)$ "from infinitely far away". This motivates the introduction of coarse structures.

Post-Lecture Resources:

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Coarse cohomology and the coarse category. Basic properties

Part 2 of the series

Wednesday, November 12, 2025

05:00 PM IST

Abstract: The coarse cohomology $HX^*(X)$ of a space $X$ is introduced as a functor from the coarse category, and its coarse invariance is discussed. We then particularize to metric spaces and study a few examples.

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Ideals of contractible DGAs. The Roe product in coarse cohomology

Part 3 of the series

Wednesday, November 19, 2025

05:00 PM IST

Abstract: We refine the structure of $HX^*(X)$, enhancing it to an algebra by endowing it with a product. Much like the cup product [3, §3.2] allows for a finer distinction between topological spaces, it is natural to seek products in coarse cohomology that sharpen the information it provides. A first strategy is to adapt the cup product to this context, but the resulting product in coarse cohomology turns out to be trivial. However, not all is lost: in such cases, one can define Massey triple products. In this session, we will introduce a closely related idea: the Roe product in coarse cohomology [5, §2.4]. This procedure can be abstracted to an algebraic setting, whose potential will later be exploited again.

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Topological bornological spaces and their cohomology

Part 4 of the series

Wednesday, November 26, 2025

05:00 PM IST

Abstract: The category $\mathrm{TopBorn}$ of topological bornological spaces is defined, as in [2, §7.1.1]. We propose an invariant $HB^*$ on these spaces, which we call bornological cohomology, and show that it can be determined from the Alexander-Spanier cohomology at infinity. We compute several examples and enrich $HB^*$ with a product, endowing it with the structure of a differential graded algebra.

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Low degree cohomology, extension spaces and boundaries at infinity. Open problems

Part 5 of the series

Wednesday, December 3, 2025

03:30 PM IST

Abstract: It is well known that the degree-zero singular homology of a topological space is determined by its path-connected components. Similarly, degree-zero Alexander-Spanier cohomology can be computed from its connected components. We ask whether, for a topological bornological space $X$, there exists a similar connection between $HB^{1}(X)$ and the ``components at infinity of $X$''.

This question naturally leads us to give meaning to notions such as rays in $X$, (path-)connected components at infinity, and ultimately to consider compactifications of $X$. After addressing this question, we generalize the tools involved by studying certain extensions of $X$ whose boundaries at infinity are Stone spaces [4], which can therefore be treated dually as topological spaces or as Boolean algebras (this is recent work in progress).

We will conclude the course by reviewing some questions that remain open.

Post-Lecture Resources:

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References

Books

  • Ulrich Bunke and Alexander Engel: Homotopy Theory with Bornological Coarse Spaces, Springer, 2020
  • Allen Hatcher: Algebraic Topology, Cambridge University Press, 2001
  • Peter T. Johnstone: Stone Spaces, Cambridge University Press, 1982
  • John Roe: Coarse Cohomology and Index Theory on Complete Riemannian Manifolds, AMS, 1993
  • John Roe: Lectures on Coarse Geometry, AMS, 2003

Papers

About the Speaker

Sergio Romero Alba

Georg-August-Universität Göttingen & University of Málaga

Sergio Romero Alba is affiliated with Georg-August-Universität Göttingen and University of Málaga. His research focuses on coarse geometry, homological algebra, and the study of spaces from infinitely far away using topological and homological tools. He works on coarse cohomology and its applications to geometric group theory and topological bornological spaces.

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