LMS/EPSRC Short Course: Homological Algebra and Equivariant Homology

Introduction

Homological algebra is at the heart of algebra, topology and geometry. The aim of the short course is to provide an introduction to this wide and classical field. More precisely, the goal is to provide a general understanding of the basic concepts involved and give an introduction to some applications in the cohomology of groups, algebraic topology and equivariant homology theories.

Programme

The lecture series will start on Monday morning and finish on Friday. On Monday, registration will begin at 9:00; the first lecture will start at 9:30. On Friday afternoon there will be a special lecture. All lectures and tutorials will take place in Lecture Theatre 4A, Building 54 (School of Mathematics) on Highfield Campus. Tutorial help will be given by Ruben Sanchez , University of Sheffield and by Srdjan Micic , ETH Zürich. A timetable is now available.

Course Overview

Course 1: Peter H. Kropholler   (Classical cohomology of groups and spaces)

This course will focus on the classical methods of homological algebra which are used to axiomatise homology and cohomology theories. Key topics to be covered are as follows:
1. Homological and cohomological functors, the long exact sequence axiom and coeffaceability.
2. Projective and injective modules over a ring; examples, their role. Sheaves over topological spaces; fine, soft and flasque sheaves; examples; their role.
3. Examples of homology and cohomology theories; homology and cohomology of groups; sheaf cohomology; de Rham cohomology.
The course will include concrete examples and exercises.

Course 2: Brita E.A. Nucinkis (Bredon cohomology and classifying spaces for proper actions)

This course gives an introduction to classifying spaces for proper actions and their algebraic counterpart, Bredon cohomology. The main focus will be on looking at finiteness conditions such as dimension and type of these spaces. A goal of the course is to discuss the behaviour of these finiteness conditions under finite extensions.
1. Introduction to classifying spaces for proper actions (G-CW-complexes, homology of chain complexes, examples for free actions, definition, examples, a first look at finiteness conditions).
2. Introduction to Bredon cohomology (Orbit category, functor category, exactness, free and projective Bredon functors, Bredon cohomology, finiteness conditions, connections with classifying spaces for proper actions, group extensions, spectral sequences).
3. Groups of type VF (Questions on finiteness condions for virtually torsion free groups, Some counterexamples, soluble groups).

Course 3: Holger Reich (Equivariant homology theories)

This course gives an introduction to equivariant generalized homology theories and the techniques that are used to construct, compute and investigate them. The main goal is to formulate the Farrell-Jones Conjecture in algebraic K-theory. The following topics will be discussed:
1. G-CW complex, orbit category, equivariant homology theory, spaces, modules and chain complexes over a category, homological algebra over a category, free objects, resolutions, classifying spaces for families, the fundamental theorem, Bredon homology
2. Definition K₀ of group rings, Hattori-Stallings rank, K₀ as a functor over the orbit category, Hattori-Stallings rank as a natural transformation between such functors.
3. Spectra, homology theories represented by spectra, spectra over a category, generalized Bredon homology, examples of theories: Hochschild homology, K-theory, the Dennis trace map, formulation of the Farrell-Jones conjecture

Special Lecture

Ian J. Leary: "Classifying spaces and curvature"

Accommodation 

Accommodation has been arranged from Sunday, 8 July, until Saturday 14 July 2007, in single ensuite rooms in the university's Glen Eyre halls of residence. This is a short walk from Highfield Campus. Reception will be open 24 hours.
Getting there: There is a bus from Southampton's two main train stations and the airport. From Southampton Airport/Parkway (Platform 2 exit) take the U1C to Highfield Interchange and from Southampton Central (Platform 4 exit) take the U1A to Highfield Interchange. National Express have services directly from Heathrow Airport to Southampton University (Highfield Interchange). Taxis from either train station to Glen Eyre should cost around £ 7.

Registration 

All research students registered at a UK university will be charged a registration fee of £100 (in the case of EPSRC funded research students, this fee should be paid by their departments from their DTA, for non-EPSRC research students, their department might be prepared to pay the fee). Overseas students, Postdocs and whose working in industry must pay the full subsistence costs of £379, plus a registration fee of £250, making a total of £629 for this course. All participants must pay their own travel costs. An online application form is available from the London Mathematical Society. The closing date for applications is Friday 18 May . All applicants will be contacted by the London Mathematical Society approximately one week after this deadline. Late applications are still being considered.

Course Material 

A list of prerequisites for the course on Bredon cohomology and Classifying spaces for proper actions can be found here.

Further Information 

Further information is available from: Peter Kropholler and Brita Nucinkis.