Below is a full set of solutions + notes to "Algebraic Topology" by Allen Hatcher and "A Concise Course in Algebraic Topology" by May with emphasis on material taught in MAT 215A in Winter 22' taught by Professor Eugene Gorsky, 215B taught by Professor Roger Casals in Spring 24' and Professor Laura Starkston in Spring 22' and MAT 215C taught by Eugene Gorsky in Fall 22'.
215A: Introduction to Homotopy Theory and The Fundamental Group
Notes (By Sections of the Course of Eugene Gorsky)
Chapter 0: Some Underlying Geometric Notions
In this chapter we start by defining and constructing some natural interesting topological spaces including the cell complex. Cell complexes are especially nice topological spaces and serve as the motivation of homotopy groups which will be explored in the next chapter. What is Algebraic Topology? It's one of the strongest developments in mathematics which allows mathematicians to use algebra tools (Group Thry, Homological Algebra,...) to study topological spaces "up to equivalence". This "equivalence", i.e. deciding when do we consider two topologcal spaces "the same" is an interestingly subtle question which is not as simple as saying "up to homeomorphism" as one would naturally think when learning about the Topological Category. In fact, it's a slightly looser equivalence dubbed "Homotopy Equivalence"! Here we explore many useful theorems for calculational tools such as the Homotopy Extension property which sparks the entire motivation os studying embeddings of spheres into topological spaces (Homotopy Groups)!
Chapter 1.1-1.2: The Fundamental Groups and Calculation Tools
In this section we define homotopy or paths which fix endpoints and use this to define the Fundamental Group associated to a topological space. We prove functoriality of the fundamental group functor from the homotopy category to the category of groups as well as the homotopy invariance property. This gives the fundamental group the algebraic structure of a useful topological invariant! We then calculate many examples of the fundamental group for easier spaces and provide interesting topological results like the Borsak-Ulum Theorem, Brower Fixed Point Theorem, and the Fundamental Theorem of Algebra! We then wrap up by using the Cellular Approximation theorem to deduce how to "read" the fundamental group from the 2 skeleton of a CW complex and a statement of the Seifert Van Kampen theorem.
In this section we explore the theory of covering spaces, which as we discover, captures insight into the group structure of the underlying topological space of interest. In fact, the connection is much deeper and we find that covering spaces are in many ways just another perspective on topological information capturing the same information as the Fundamental Group! Covering spaces and the classification of connected covers of certain nice topological spaces (needs some conditions of connectedness) provides or us many different tools to calculate and probe properties of the fundamental group of the base. In particular, paths in the covering tell a deep story about the loops in the base -- an idea made rigorous via the Homotopy Lifting property of coverings.
Parts of Chapter 4: Higher Homotopy Theory and Fibrations
In this last section of the course we view some of the basic definitions of higher homotopy theory and set up for many amazing results (which we won't get to here) in more advanced algebraic topology. Despite this we see many cool properties of higher homotopy groups, in particular, they are all commutative (resembling homology groups) and given any covering space of a topological space, they have isomorphic higher homotopy groups. However, many of the nice calculation properties we have for the fundamental group no longer hold. We do have a major calculational tool via the fact that locally trivial fibrations induce long exact sequences of homotopy groups! We end the course by computing homotopy groups of special matrix groups.