## What is Algebraic Topology?

It is rarely the case that the converse can be shown; i. Within the realms of algebraic topology, there are several basic concepts that underly the theory and serve as the building blocks and models for subsequent generalisation, the algebraic topology of today being a very broad and highly generalised area that has pervaded much of contemporary mathematics.

Such concepts include homotopy, homology and cohomology, and the course will be aimed at providing students with an introduction to these key ideas.

David Baraglia Lecturer for this course. Description The aim of Algebra ic Topology is to use algebraic structures and techniques to classify topological spaces up to homeomorphism.

Class notes for May 23rd the long exact sequence for a quotient space, assuming excision. Handout on the barycentric subdivision: BarycentricSubdivision. Class notes for May 28th proof of excision. Homework assignment 9: Ex9. Class notes for May 30th explicit generators, simplicial structures.

Class notes for June 4th homology of simplicial structures, definition of degrees. Class notes for June 6th theorem on degrees and local degrees, the linking number. Homework assignment Ex Print your own Rayman -Excision stickers!

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## Fundamental Concepts in Algebraic Topology

Get RaymanExcision. Print your own Euler characteristic stickers! Get EulerCharacteristic. Torus by Tom Longtin. Class notes for June 11th review of degrees, CW complexes, the Euler characteristic.

Handout on Topological Pathologies in R 3. Class notes for June 18th normalized simplices, theorems and pathologies in Euclidean spaces.

Class notes for June 20th the homology of the comlement of a sphere in another one, applications to the linking number, invariance of domain. Class notes for June 25th and 27th end of invariance of domain, the Borsuk-Ulam theorem.

Review class: Sprintzak , July 7th at See handout and sample test.