The course will cover important basic notions in geometry and topology and discuss relations between geometry and topology. Topics on the geometry side will include the basics of smooth manifolds and smooth maps, Sard's theorem and applications, submersions, immersions and embeddings, the notion of transversality. Topics on the topology side will include CW-complexes, the notion of homotopy, homotopy extension property, cellular maps and cellular approximation, the fundamental group and its computations and applications, the notion of higher homotopy groups, coverings, fiber bundles and fibrations, path lifting and homotopy lifting, topological classification of 1- and 2-manifolds. We will try to emphasize the connections between the two sides: given a smooth map, how can you tell if it defines a covering or a fiber bundle? What are the topological consequences? How can you define and compute the degree of a map topologically and geometrically?

A more detailed list of topics will be posted with the week-by-week schedule as the course progresses. We may need to make adjustments to the scope of the course as time permits.

Strong foundation in point-set topology will be assumed as a prerequisite, athough some finer point-set bits will be discussed as necessary (for example, to establish properties of CW-complexes). It will also be expected that the students have seen smooth manifolds and the fundamental group before, so this material is not entirely new, but we will develop the foundations in detail.

All course information will be posted on the course webpage.

Week 1 (Monday 8/26, Wednesday 8/28): Chapters 1 and 2 in Lee's book. We discussed the definitions and basic examples and properties of smooth manifolds. (Diagnostic point-set topology quiz on 8/28.)

Week 2 (Wednesday 9/4) : Please read the remaining parts of Chapter 1 (for manifolds with boundary) and Chapter 2 (partition of unity) in Lee's book. We discussed these topics and started Chapter 3, with two definitions of the tangent space to a smooth manifold at a point: via equivalence classes of smooth curves and via derivations. We haven't finished the discussion of derivations (will continue next week), but please read this material in Lee Chapter 3.

Week 3 (Monday 9/9, Wednesday 9/11) : We finished tangent spaces and differentials of smooth maps. (Lee Chapter 3.) We only discussed the case of manifolds with boundary briefly, so please read this material in Lee. We introduced the notion of vector bundles and fiber bundles (first definitions and examples only); vector bundles are featured in Homework 3. We defined regular and critical points and values of smooth maps, started Chapter 4, discussed local diffeomorphisms. As an application, proved the fundamental theorem of algebra (every non-constant complex polynomial has a root). For this, we followed Milnor's Topology from Differential Viewpoint, Chapter 1 (including an important lemma about regular values). We proved the rank theorem in Chapter 4. Will continue with Chapter 4 and Chapter 5 material next time.

Week 4 (Monday 9/16, Wednesday 9/18) : We discussed the key material from Chapters 4 and 5. These chapters contain too many details, and we will not need all of it. In particular, "submanifold" means "embedded submanifold" for us (not "immersed", even if sometimes we'll be interested in images of smooth immersions). We mostly only considered manifolds without boundary (although boundary will sometimes show up in the homework). We started Chapter 6 (sets of measure 0, moving towards Sard's theorem).

Week 5 (Monday 9/23, Wednesday 9/25): Proved Sard's theorem. Corollaries: Whitney embedding theorem (proved under the compactness assumption although the theorem holds in general, see textbook for the non-compact case). Approximations of continuous functions by smooth functions (Thm 6.21; thm 6.26 to be done next week). Corollaries of Sard's thm and smooth approximations: n-dimensional ball does not retract on its boundary (n-1)-sphere. Brouwer fixed point thm (will finish next week). The proof of non-existence of retraction of the ball onto the sphere can be found in Milnor's Differentiable Viewpoint book. (We modified that proof slightly to work with functions on manifolds with boundary. Alternatively, you can do the exercise from HW 5 about preimages of regular values in the boundary case and use it for the proof in Milnor's book.) Transversality (rest of Ch 6 in Lee) coming up next week.

Homework 5 due Oct 2, in class.

Week 6 (Monday 9/30, Wednesday 10/2): The notion of the normal bundle of a submanifold M in R^n. The notion of tubular neighborhood of M in R^n. Our dicussion was a bit impressionistic, with only sketches of proofs due to time constraints (and because you'll see better proofs later on in a more general context). However, it's important to understand the definitions and the general idea for these things. You are encouraged to read the details of the proofs in Lee Chapter 6. We used tubular neighborhood as a tool to finish the proof that any continuous map between smooth manifolds. is homotopic to a smooth map.
Transversality: explained the notion of transversality and its significance (the intersection of two transversely intersecting submanifolds is a submanifold and a similar statement for a map transverse to a submanifold.) Used Sard's theorem to prove parametric transversality theorem. There's one small bit left, a corollary for the next week: every smooth map is homotopic to map that is transverse to a given submanifold.

Homework 6 due Oct 9, in class.

Week 7 (Monday 10/7, Wednesday 10/9): We moved to the second part of the course (fundamental groups, basic homotopy theory). Recommended sourses for this part is Hatcher Chapters 0 and 1 and a bit of Chapter 4, and Fomenko-Fuchs Chapter 1. (Fomenko-Fuchs skip some of the proof details but sometimes make an overall picture clearer.) We discussed the notion of homotopy in detail, defined the fundamental group and higher homotopy groups, checked basic properties, showed that the higher homotopy groups are abelian, and discussed dependence on the basepoint. Before moving to this material, we also discussed some more examples of smooth manifolds, such as surfaces, the connected sum operation, and manifolds obtained as quotients by group actions. These will come up again very soon.

Homework 7 due Oct 17, at 2pm (submit into the box next to Mohammad's office 2-107).

Week 8 (Wednesday 10/16): We showed that continuous maps between topological spaces induce group homomorphisms on the fundamental groups and higher homotopy groups and established basic but important properties. (In fancy language, this gives a functor from the category of topological spaces to the category of groups.) Application: if A is a retract of X, then the inclusion of A into X induces an *injective* homomorphism from the fundamental group of A to the fundamental group of X (and the same is true for higher homotopy groups). Discussed the notion of homotopy equivalence of topological spaces; showed that a homotopy equivalence induces an isomorphism on homotopy groups. Deformation retractions are an important special case. We defined contractible spaces and showed that R^n and conves subspaces in R^n are all contractible. [We mostly focused on the case of the fundamental group but the proof are essentially the same for the higher homotopy groups.]

Note for Homework 8 Question 6: you can prove that the space of orbits of a "nice" group action on a smooth manifold is a smooth manifold under less restrictive hypotheses than compactness or finiteness of the group. See Lee, Lemma 21.11 and Theorem 21.13, and this stackexchange thread. Note that the point-set topology gets tricky though, and the Hausdorff property can fail without the additional hypotheses such as Lemma 21.11 (ii).

Week 9 (Monday 10/21, Wednesday 10/23): We defined covering spaces, discussed important examples, and proved two very important lemmas about path lifting and homotopy lifting. (These lemmas are in Hatcher Ch 1.) This gives an important corollary: if p:Y -> X is a covering, X is path connected, Y is simply connected, then homotopy classes of loops in X based at x are in one-to-one correspondence with the fiber over x. This gives you a tool to compute the fundamental group if you can find the group structure by computing products of the representatives of the homoltopy classes. the We applied this strategy to find the fundamental groups of the circle, of the projective spaces, and of the wedge of several circles (we drew pictures of the covering space of the wedge of two circles for concreteness, but the construction is the same for any wedge). An important result concerns the space of orbits under a "covering space" group action on a simply connected space X: under appropriate hypotheses on the action of the group G, the orbits are discrete, and X -> X/G is a covering. In that case, the fundamental group of X/G is the group G. At this point, we also know the fundamental groups of higher dimensional spheres (they are all zero, because every continuous loop is homotopic to a smooth loop which by Sard's can't be surjective, and therefore can be nullhomotoped), and the fundamental group of the torus (as a product of circles, see homework 7).

Week 10 (Monday 10/28, Wednesday 10/30): Midterm on Wednesday 10/30. On Monday, we defined CW-spaces, aka CW-complexes or cellular complexes (finite-dimensional case only for now, to avoid discussion of "weak topology" in the general case), and started developing a procedure for computing the fundamental group of a CW-complex (to be continued).

Week 10 (Monday 11/4, Wednesday 11/6): We proved a theorem about the effect of cell attachments on the fundamental group of a CW-complex. A cell of dim 3 or higher doesn't change the fundamental group because both loops and homotopies can be pushed out to the 2-skeleton. Attaching a cell of dimension 2 has the effect of factoring the fundamental group of the space by the normal subgroup generated by the loop given by the boundary of the 2-cell. Since we know that the 1-skeleton of a connected CW space is homotopy equivalent to a wedge of spheres, and its fundamental group is the free group on the corresponding generators, this enables us to compute the fundamental group of any CW-complex. Note that while one can obtain this theorem as an application of the van Kampen theorem, we took a different route, without van Kampen. (For the 1-skeleton, use covering spaces; for the higher-dim cells, use smooth approximations and Sard's theorem to prove that loops and homotopies can't be surjective onto cells of dim 3 or higher and so can be pushed out; for 2-cells, use the procedure described in Fomenko-Fuchs Chapter 1 section 11.1.) This latter procedure applies more generally (see the FF book and Question 4 of this homework).

We computed the fundamental groups of surfaces and (almost) proved the classification of surfaces theorem (to be continued). Some references for the classification of surfaces:
There are several common proofs, many of them assuming triangulation. One classical proof uses the representation of a surface as a polygon with the boundary edges identified in pairs; the proof proceeds by bringing this diagram to a standard form via a sequence of cut-and paste moves. This proof can be found in many textbooks, such as Massey's Algebraic Topology: an Introduction, Kinsey's Topology of Surfaces, etc. We are discussing another proof, which is is shorter and more illuminating, even if it requires filling in some details (see also this Mathoverflow thread).

Week 11 (Monday 11/11, Wednesday 11/13): We finished classification of surfaces (including compact surfaces with boundary). The upshot of the proof we discussed (the link above) is that the Euler characteristic is a topological invariant of a surface, at least when computed from a triangulation. (For a triangulation of a surface, the Euler characteristic is given by #(vertices) - #(edges) + #(triangles). For an arbitrary finite CW complex, the Euler characteristic is the alternating sum of the number of cells in each dimension, and this is a topological invariant, but we cannot prove it yet: you need singular homology to establish that result. Moreover, the Euler characteristic is a homotopy invariant, which we didn't prove.) We discussed how you can identify a given surface with one of the standard ones from the classification list by checking for orientability, the number of boundary components, and the homotopy type or the Euler characteristic.

Week 12 (Monday 11/18, Wednesday 11/20): We proved the standard theorems on the classification of coverings, existence of a simply-connected (universal) covering, hierarchy of coverings. All spaces are assumed to be "nice" (path connected, locally path connected, semilocally path connected). We discussed why these assumptions are necessary but they always hold for nice spaces such as manifolds or CW complexes. We described the construction of the universal covering and checked some of the properties but not all of them. Please spend some time reading though this proof on your own, you'll undertstand coverings better. The proof is in all standard books, Hatcher and Munkres and Fomenko--Fuchs, but these notes that I found online do a nice job explaining and motivating every step.

Week 14 + last lectures (Monday 11/25, Monday 12/2, Wednesday 12/4, Monday 12/9): CW complexes, their point-set topological properties. Infinite-dimensional case: weak topology. Theorem: a compact subset is contained in a finite-dimensional subcomplex. Continuity of functions and homotopies in weak topology. [Note that for homotopies, the fact is non-trivial and related to some complications of product topology on the product of CW complexes. See my notes with the summary of what we covered and the proof of the lemma about homotopies, which we didn't have time for in class.]

Homotopy properties of CW complexes: homotopy extension property for CW pairs, cellular approximation. We mostly followed Fomenko-Fuchs for this material. A key step of the proof of the cellular approximation is pushing the lower-dim cells out of the higher-dim cells. This is easy once we modify the map by a homotopy so that the image of the low-dim cell does NOT contain all of the higher-dim cell. We proved this fact ("the free-point lemma") using smooth approximation and Sard's theorem. Corollary of homotopy extension: if (X, A) is a CW pair, and A is contractible, then X is homotopy equivalent to X/A. [Contracting some contractible subcomplexes is a very useful trick if you're trying to show that a given space is homotopy equivalent to a simpler-looking space.]

At the very end of the course, we discussed fiber bundles and fibrations and the homotopy long exact sequence of a fibration. We applied this homotopy sequence for covering spaces to recover some of the familiar facts. We also applied it to the Hopf fibration (the fibration of S^3 over S^2 with fiber S^1) to prove that pi_2 (S^2) = Z and that pi_3 (S^2) = pi_3 (S^3)=Z [Note that we have previously proved, in the homework, that every map S^n to S^n is homotopic to a multiple of the standard generator, but you need a (co)homological invariant - degree of a map - to show that different multiples cannot be homotopic, so we didn't quite prove that pi_3 (S^3)=Z. This is coming up in MAT 531.]

To sketch the proof of the long exact sequence, we discussed the homotopy lifting property and defined Serre fibrations. We proved that a fiber bundle has a homotopy lifting property with respect to I^n and is therefore a Serre fibration. It is easy to lift the homotopy in the case of a *trivial* fiber bundle (a product); more generally, break I^n x I into small cubes mapping into charts corresponding to trivializations, and proceed in stages. Going back to the long exact sequence, we used homotopy lifting to define the connecting homomorphism. For lack of time, we didn't check all of the exactness statements; some of them are very easy but even the harder ones follow directly from unraveling the statement and using homotopy extension and/or lifting.

If you notice a typo or something looks weird, please let me know. I do make mistakes sometimes (often, it turns out). Any questions, feel free to email and ask!

The final exam is on Wednesday, December 18, 11:15am-1:45pm, in our usual classroom Physics P-122.

What's not included? We touched upon a number of extra topics that will not be on the test: vector bundles, fiber bundles, lifting properties for fibrations, the homotopy long exact sequence of fibration are not included. Higher homotopy groups are not on the test. For CW complexes, you need to know the inductive definition and what weak topology means for an infinite-dimensional CW space, you need to know CW structures for standard examples (surfaces, projective spaces, etc), and be able to work with CW complexes in simple situations (eg compute the fundamental group from a CW decomposition with just a few cells). You will not be asked to prove point-set topological properties of CW complexes or reproduce proofs of homotopy properties. However, you need to be able to state and apply the homotopy extension property and the cellular approximation in simple examples (such as HW 14 question 3). You will not be asked to prove any complicated statements about group actions, but you need to know that a quotient by a "nice" discrete group action gives a covering (and know properties of this covering). For smooth manifolds with boundary, you need to know the definitions and basic properties, but you don't need to know all the variants of theorems for manifolds with boundary.

The midterm exam was on Wednesday, Oct 30, in class.

The midterm topics All of the smooth manifolds material (Lee Chapters 1-6, excluding the bits that we didn't cover). The extra topics we touched upon (orientability, vector bundles) are not on the test. The basics of algebraic topology (homotopies and homotopy equivalences, retracts and deformation retracts, definitions and properties of the fundamental group) are also on the test. You should know the fundamental group of the circle, torus, wedge of circles, spheres, wedges of circles and spheres, projective spaces, and be able to compute some other fundamental groups by finding homotopy equvalences to familiar spaces. Higher homotopy groups are not on the test. Covering spaces are not on the midterm (but will be on the final exam). You should know all the definitions, basic proofs, statements of important theorems, and least the ideas of the more complex proofs. You will not be asked to reproduce any long and technical proofs of theorems (no need to memorize the proof of Sard's theorem). Refer to the week-by-week schedule on this page for the detailed description of the course topics.

Important: For each homework problem, please give a proof or detailed explanation as appropriate (unless otherwise stated). Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. Although you are welcome to work with others to understand how to solve the problems, you have to write all the solutions on your own, in your own words. (See Academic Integrity Statement at the bottom of the page.) You should not look for solutions online.

Lee, Introduction to smooth manifolds (GTM 218), 2nd edition. We'll cover chapters 1-6 and small bits of Chapters 8, 9, and 10.

Fomenko, Fuchs, Homotopical Topology (GTM 273). We will cover part of Chapter 1 only. This book has the advantage of discussing some important homotopy theory early on.

Hatcher, Algebraic Topology. We will cover the material of Chapters 0, 1, and a part of Chapter 4. This book is available online for free download.

Milnor, Topology from Differentiable Viewpoint. This is a classic, everyone should read it. You'll be glad you did. The book makes some simplyfing assumptions from the start (defining smooth manifolds as submanifolds in R^n) but gets to important theorems very quickly. Very illuminating read.

No make-up exams will be given for the midterm. If a student misses a midterm exam for a well-documented medical reason or other similar circumstances beyond the student's control, the student may be excused from the exam, with the final grade determined from the other exams, homework, and class participation. For the final exam, make-ups will be given ONLY in cases of properly documented medical reasons or other similar circumstances, at the instructor's discretion.

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