Below is a full set of solutions + notes to "Differential Topology" by Guillimen and Pollack, "Differential Geometry" by Do Carmo, and "Morse Theory" by Milnor with emphasis on material taught in MAT 239 in Fall 23' taught by Professor Laura Starkston, MAT 240A in Winter 25' taught by Professor Michael Kapovich, and 240B taught by Professor Roger Casals in Spring 23'
239: Introduction to Smooth Manifolds
Notes (By Sections of the Course of Laura Starkston)
Chapter 1.1-1.4: Smooth Manifolds and Maps + Induced Local Properties
In this chapter we start by defining and constructing smooth manifolds and understanding morphisms in the smooth category. Here we take on two equivalent, yet distinct approaches: the intrinsic (smooth atlases) approach and the extrinsic (embedded) approach. Both perspectives are useful in that they balance geometric intuition with mathematical formalisms. We then turn to the idea that information of the derivative of a smooth function between smooth manifolds translates to information about the original function. In particular starting with the Inverse function theorem, stating that non-degeneracy of the derivative of a smooth function between smooth manifolds implies the original function is a local diffeomorphism. Then turn to the local immersion and submersion theorems, which weaken the condition of non-degeneracy to just injectivity or surjectivity respectively, but give rise to the original smooth map in coordinates being a standard inclusion or projection. These theorems, although very general give rise to explicit useful calculation tools which allows us to show certain spaces are manifolds! In particular, images of injective, proper, immersions are submanifolds of the codomain and preimages of regular values of a smooth function are submanifolds of the domain.
Chapter 1.5-2.1: Genericity, Stability, and Transversality
In these sections we explore a very strong property from smooth topology, known as genericity and stability. In particular, due to Sard's Theorem -- which states that almost every point in the codomain of a smooth function is a regular value -- we can deduce many EXTREMELY useful theorems stating that many properties of smooth maps are indeed "generic" -- they occur very commonly -- and are "stable" -- if a property occurs, then small perturbations to the function do not change the property. These "properties" include local diffeomorphism, diffeomorphism, immersion, submersion, transversality. Some immediate geometric results of Sard's theorem are explored, namely the density of Morse functions and the weak Whitney Embedding theorem. We then end off the section discussing the motivation and properties of maps transverse to some submanifold in the codomain. In particular, we develope the traditional formalism of transversality to generalize the preimage theorem and more geometric formalism to begin an introductory motivation to intersection theory.
Chapter 2.2-2.4 and 3.1-3.3: Intersection Theory
In these sections we use the fairly difficult theory from the previous section to construct some invariants of intersecting manifolds which are fairly simple to compute. In particular, we start with the mod 2 intersection number for an arbitrary smooth map f: X \to Y and submanifold Z. Since we can homotope f to be transverse to Z via genericity of transverse intersections we get a well-defined count of points modulo 2 in the preimage g^{-1}(Z) for a map g homotopic to f which is transverse to Z provided that dimX + dimZ = dimY. Then we refine this idea in the case where X,Y,Z are orientable manifolds with the same set up as the mod 2 theory, in that we can improve our mod 2 invariant to an integer valued invariant which counts signed intersection points via induced orientation of the point from the appropriately defined normal bundles. These two theories are powerful in their own right and in certain cases give an easy way to differentiate between manifolds.
Chapter 4.1-4.7: Integration on Manifolds
In these sections we develop integration theory for manifolds. In particular, we start with the formalism of differential forms, which due to the properties of their pull back under parametrizations, allows us to go from integration on manifolds to integration on Euclidean space via partitions of unity. We then discuss that for calculation purposes most times it is efficient to find a parametrization which only misses a measure zero subset of your manifold. Finally we introduce deRham cohomology and Stoke's Theorem with particular examples keeping track of orientation.