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'Â
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.
Full Notes (By Sections of the Textbook)
The main source of these notes is Loring Tu's "An Introduction to Smooth Manifolds" with the main goal of being an introduction to the Lie Derivative; a construction which is one answer the question: "How do we differentiate vector fields in the direction of other vector fields?". The main problem we need to deal with is the following; we need to be able to compare tangent vectors at points which are nearby each other, however, they are elements of different vector spaces... so we need a way to canonically identify tangent vectors in one tangent space to tangent vectors in a nearby tangent space in a geometrically meaningful way. One such way is to use a connection and the notion of parallel transport. The other classical way is the Lie derivative! Which uses the differential of the flow of a vector field to transport vectors between tangent spaces.
Though it is expected that the reader has a solid understanding of the tangent space of a smooth manifold and the differential of a smooth map between two smooth manifolds, we begin the notes with a coverage of these topics as in chapter 8 of \cite{Tu}. This establishes the notation and developments essential for the later chapters which may be different from the reader's previous reading/experience. In particular, the viewpoint of tangent vectors as derivations of the algebra of germs of smooth functions on the smooth manifold is extremely important to understand differential geometric constructions in general, not just the Lie Derivative. With this formalism we will be able to give a formula for the differential of a smooth function and interpret integral curves of smooth vector fields in particularly nice ways.
Chapter 12: The Tangent Bundle
We now recall the formalism of smooth vector bundles as to realize vector fields as smooth sections of the tangent bundle. In particular, the goal is to give a characterization of smoothness of sections in a smooth vector bundle via the smoothness of coefficients with respect to a smooth local frame. This section is technical, but its importance is clear. A deep understanding is not so important for the Lie Derivative perse, but answers many "mathy" questions which one may ask about vector fields.
Chapter 14: Smooth Vector Fields and Integral Curves
We use the fruits of the previous discussion to gloss over smoothness of vector fields. We then develop the notion of integral curves of a smooth vector field, which one can think of as the "flow" along the vector field when given an initial condition. We discuss some existence and uniqueness theorems about systems of first order ordinary differential equations to make clear conclusions about integral curves.
Chapter 20: Smooth Vector Fields and Integral Curves
We now have built up the necessary knowledge to give the definition of the Lie derivative of a vector field and of differential forms. As we mentioned before, the Lie derivative seeks to give one possible answer to the question: "How do we differentiate a vector field in the direction of another vector field?". We start with the formalism of one parameter families of vector fields and differential forms and the derivatives in the parameter direction. Then we can finally give the definition of the Lie Derivative!