Algebraic Topology

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Tethers and homology stability for surfaces" (with Karen Vogtmann). Alg. & Geom. Topology 17 (2017), 1871-1916. pdf file.

diffeomorphism groups of smooth manifolds. A full history would of course be impossible in an hour talk. Content: Algebraic topology is concerned with the construction of algebraic invariants (usually groups) associated to topological spaces which serve to distinguish between them. Most of these invariants are ``homotopy'' invariants. In essence, this means that they do not change under continuous deformation of the space and homotopy is a precise way of formulating the idea of continuous deformation. This module will concentrate on constructing the most basic family of such invariants, homology groups, and the applications of these homology groups.how the Madsen-Weiss theorem follows similarly, as do a couple analogs in dimension three involving handlebodies.

Give the definitions of simplicial complexes and their homology groups and a geometric understanding of what these groups measure Hatcher goes to great lengths to avoid category theory. I understand this is a choice, and I can see why one might make that choice. However, I think category theory has done wonders for intuition for me, and as algebraic topology is in some sense the birthplace of category theory, I think it just makes sense to lean into it a little bit more (or a lot more). expository talk at the 2004 Cornell Topology Festival. Also available is a pdf file of the transparencies for the talk itself.The starting point will be simplicial complexes and simplicial homology. An n-simplex is the n-dimensional generalisation of a triangle in the plane. A simplicial complex is a topological space which can be decomposed as a union of simplices. The simplicial homology depends on the way these simplices fit together to form the given space. Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex. For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes. Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space. It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes. This result means that we can combine the theoretical power of singular homology and the computability of simplicial homology to get many applications. These applications will include the Brouwer fixed point theorem, the Lefschetz fixed point theorem and applications to the study of vector fields on spheres.