Expository notes in English
The purpose of this note is to explain two strategies for proving homological stability for the usual inclusion sequence of symmetric groups. The exposition follows two amazing sources: Allen Hatcher's notes on the Madsen-Weiss theorem and Sander Kuper's notes for the eCHT minicourse. The only redeeming quality of my note is several illustrations that, hopefully, clarify the arguments a bit. Some sections are still under construction. In particular, I hope to add a meaningful comparison of the two approaches in the future. Some elementary proofs are also missing for now.
The note was prepared in partial fulfillment of the requirements for the "Homological Stability of Moduli Spaces" course at HSE.
In this note, we try to cover a circle of ideas related to smooth compact topological categories and their realization. These were first introduced in a foundational paper by Ralph Cohen, John Jones, and Graeme Segal, "Floer’s Infinite Dimensional Morse Theory and Homotopy Theory." These ideas were picked up over the last 20 years to be used for various low-dimensional invariants. In particular, a striking development was obtained by Robert Lipshitz and Sucharit Sarkar, who constructed a spectrum representing Khovanov homology using the ideas developed by Cohen--Jones--Segal. The text is largely still under construction but should be finished in the near future.
The note was prepared in partial fulfillment of the requirements for the "Floer homology" course at HSE.