Craig Westerland: Topics in Topology

MATH 8360: Topics in Topology
(Fall 2015)

Lecturer: Craig Westerland, 459 Vincent Hall, 612-625-0523, cwesterl@umn.edu.

Lecture: 3:35 -- 4:50 Monday and Wednesday, Vincent Hall 209.

Office Hours: email me to make an appointment.

Goals and Objectives

This topics course will be focused on the algebraic topology of configuration, moduli, and function spaces. A background in basic algebraic topology (the fundamental group, homology, cohomology, some basic homotopy theory) will be assumed, as will an enthusiasm for spectral sequence computations. The tools will be very homotopy theoretic in nature, but the material should appeal to students working in algebraic/arithmetic geometry or geometric topology with some background in algebraic topology.

Topics will include (most likely a proper subset of): configuration spaces, operads, iterated loop spaces, the group completion theorem, moduli spaces and the Mumford conjecture on their stable cohomology, symmetric products, homological stability, knot spaces and other spaces of embeddings, the cohomology of the symmetric group and the Barratt-Priddy-Quillen theorem, and other examples of geometrically or arithmetically significant families of homologically stable spaces, such as automorphism groups of free groups or general linear groups of number rings.

Lecture notes

I will try to tex my notes for the lectures on a weekly basis:

Week 1: Configuration spaces and their many guises.
Week 2: Configuration spaces via iterated fibrations.
Week 3 is a composite of the end of the notes from Week 2, as well as an introduction to spectral sequences: Take a look at McCleary's "User's Guide to Spectral Sequences," most particularly chapters 2 and 5.
Week 4: Symmetric products and the Fox-Neuwirth cell decomposition.
Week 5: Operads and iterated loop spaces.

Rough plan of the course

Configuration spaces

Configuration spaces, their homotopy and homology via iterated fibrations, braid groups, Fox-Neuwirth cell decompositions, hyperplane arrangements, the Salvetti complex, Orlik-Solomon algebras.

Edward Fadell and Sufian Husseini: Geometry and Topology of Configuration Spaces.
Chad Giusti and Dev Sinha: Fox-Neuwirth cell structures and the cohomology of symmetric groups.
Alexandru Dimca and Sergey Yuzvinsky: Lectures on Orlik-Solomon Algebras.

Operads and function spaces

Operads, iterated loop spaces, infinite loop spaces and spectra, the James construction, the approximation and recognition theorems, the group completion theorem, the Barrat-Priddy-Quillen theorem, the definition of algebraic K-theory, symmetric products and the Dold-Thom theorem.

Peter May: The Geometry of Iterated Loop Spaces.
Martin Markl, Steve Shnider, Jim Stasheff: Operads in Algebra, Topology and Physics.
Frederick Cohen, Peter May, Thomas Lada: The Homology of Iterated Loop Spaces.
Allen Hatcher: Algebraic Topology.
Carl-Friedrich Bödigheimer, Frederick Cohen, Laurence Taylor: On the homology of configuration spaces.

Stability

The Segal-Dold splitting of the homology of configuration spaces, the Snaith splitting, the yoga of homological stability and examples: the symmetric and braid groups, Harer stability.

Dusa McDuff: Configuration spaces of positive and negative particles.
Victor Snaith: A stable decomposition of Ωⁿ Sⁿ X.
Nathalie Wahl: Homological stability for mapping class groups of surfaces and Homological stability for automorphism groups.

Moduli spaces

Definitions, the mapping class group, the Mumford conjecture, cobordism categories, genus 0, the Fulton-MacPherson and Deligne-Mumford compactifications.

Nathalie Wahl: The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces.
Søren Galatius, Ib Madsen, Ulrike Tillmann, Michael Weiss: The homotopy type of the cobordism category.
Yakov Eliashberg, Søren Galatius and Nikolai Mishachev: Madsen-Weiss for geometrically minded topologists.

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