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Prof. Donald Stanley (University of Regina, Canada)
January 8, 10, 15, 17, 22 (Wednesday and Friday), 23 (Thursday), 2025
9:30 - 12:30
Room 202, Astronomy and Mathematics Building

Problems and some references (2025.1.10 updated)

The second set of problems (2025.1.22 updated)

The third set of problems (2025.1.23 updated)

Abstract: Homotopy colimits are a way of building objects out of smaller ones in a homotopy invariant way. We are interested in the cochain algebra C_*(X) of these spaces. The cochain algebra gives us the cohomology H^*(X) but contains much more information. For example C^*(X) determines the rational homotopy type of the space.

We will only consider a very restricted class of homotopy colimits where many of the proofs simplify. We will consider functors out of the poset categories associated to simplicial complexes. The functors will be have nice product decompositions. Working with these restrictions allows us to give explicit formulas for the cochains of the homotopy colimit.

Background: We will assume basic knowledge of categories and functors, up to colimits and limits. We assume knowledge of chain complexes and homology, and homological algebra at the level of the five lemma and the universal coefficient theorem. Students should have some familiarity singular (co)homology. The course will develop all the necessary theory beyond Hatcher Algebraic Topology Chapters 2 and 3. We will also present some of the necessary background from these chapters.