In January and February of 2023, me and Pahul, in our second year of BSMS at the time, audited (half of) a fourth year mathematics elective course on Algebraic topology instructed by Chetan Balwe.
We knew the definition of topological spaces and continuous maps between them. And a few more preliminary ideas. We did not know, for example, what free products of groups were, which is usual for people crediting this course anyways?
I present below the notes we took by hand, as is.
Homotopy
Missing notes
We missed the first few classes.
The first fundamental group
Simply connected iff any two paths with same endpoints are homotpic
Homotopic relative to
Functoriality of
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Dependence on basepoint
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Homotopy equivalence
Homotopy equivalence preserves first fundamental group
Hence in particular, deformation retract preserves first fundamental group.
Covering spaces, and lifting homotopies
Consequences of homotopy lifts
Fundamental group of
Van-Kampen theorem
factorization of loops
free product of groups
statement of Van-kampen
proof
example: Wedge sum of circles
Applications of Van-Kampen theorem
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Gluing disks to spaces
CW-complexes
Tutorial
Exercise
Lifting functions to covering spaces
motivation
statement and proof
When does a simply connected cover exist?
Deck Transformations and Group Actions
Missing notes
Homology
simplex, complex, homology
Reduced homology, relative homology
