Dror Bar-Natan: Classes: 2001-02: Fundamental Concepts in Algebraic Topology:

# The Final Exam

There will be a 3-hours long final exam. The material taught in this class was important and significant, and thus the main purpose of the exam, as I see it, is to encourage you to review that material and to verify that you've understood the main points. Ergo there will be emphasis on the deep understanding of direct class material. The bit about creativity will have a relatively small overall weight and will be there mostly to make happy those of you who really do need a challenge. The questions will be open-ended: "Say everything you know about the Chukumuku Theorem", for example.

A Sample is Worth a Thousand Words!

The exam may well be very hard, but as a whole I am happy with this class and unless the results will be appallingly dismal, they will be renormalized to an above-the-average scale. HW grades will only serve as "magen".

Moed Aleph will be on July 11th at 10:00-13:00 at Levin 8. Moed Beth will be on August 28th at 10:00-13:00 at Math. 2 (I will not be present on Moed Beth). There will be a review session before the exam, on July 7th at 10:00. The purpose of the review session will mostly be to read this page out loud and discuss whatever the students will raise.

In principle, all that was discussed in class is fair game. Topics that we discussed with precision, I expect you to know with precision. Topics on which I gave no details (and didn't indicate the details as "easy homework exercises"), you are only supposed to know to the same level. For your convenience, here is a list of the topics covered in the course, drawn from my own notes:

 Definition of retracts. Retracts and the Brouwer fixed point theorem. The basic idea of algebraic topology via the Brouwer fixed point theorem. Paths and homotopies. The fundamental group. The fundamental group of a convex set. The fundamental group of a circle. The lifting property for curves. The lifting property for homotopies. R->S1 is a covering. The fundamental theorem of algebra. The induced map f*: π1(X)->π1(Y). Functoriality of the induced map. Proof of the Brouwer fixed point theorem. Basepoint change and the fundamental group. The fundamental group of a product of two spaces. The fundamental group of a torus and of a ring. Homotopic maps between spaces. The fundamental group maps induced by homotopic maps of spaces are equal. Homotopy equivalent spaces and their fundamental groups. Free products of groups. The universal property of the free product. The Van Kampen theorem. The fundamental groups of spheres. The fundamental group of a figure eight. The fundamental group of a torus. The fundamental group of the complement of a (p,q) torus knot. Pushouts. Pushouts for vector spaces. Pushouts for groups. Pushouts for topological spaces. The fundamental group of a graph is free. The fundamental group of projective spaces. Proof of Van Kampen's theorem. Covering spaces. Examples of covering spaces. The homotopy lifting property. The fundamental group of a cover injects in the fundamental group of its base. The general algebraic lifting criterion. Uniqueness of liftings. Categories and functors. The category of covering spaces. The category of G-sets. Semi locally simply connected spaces (definition, examples). The main theorem of covering spaces: the category of covering spaces and the category of G-sets are equivalent. Connected components and orbits. The stabilizer and the fundamental group of a cover. Universal covering spaces. Uniqueness of the universal cover. Covers are isomorphic iff their groups are. Decks and cosets. Covers of covers and subgroups of subgroups. The automorphism group of a covering. The automorphism group and the fiber of a universal covering and the fundamental group of the base. Normal coverings. Normal coverings and normal subgroups. Algebraic description of the automorphism group of a cover. A space that has a universal covering is semi locally simply connected. The spelunking construction of the universal covering space. Proof of the main theorem of covering spaces. The intuitive idea of homology. Homological proof of Brouwer in arbitrary dimension. The expected homology of spheres. The expected homology of a torus. The standard simplex and general affine simplices. Chains. The boundary map. The square of the boundary map. Cycles, boundaries, homology. The homology of a point. The homology of disjoint unions. The zeroth homology and connected components. Reduced homology. The category of chain complexes. The assignment (spaces -> chain complexes) is a functor. The assignment (chain complexes -> homology groups) is a functor. Homotopies between morphisms between chain complexes. Homotopic morphisms on chain complexes induce equal maps on homology. The prism construction. Homotopic maps between spaces induce homotopic maps between chain complexes. Homotopic maps between spaces induce equal maps on homology. Homotopically equivalent spaces have the same homology. The homology of Euclidean spaces. Exact sequences. The meaning of the various short exact sequences. The long exact sequence of a topological quotient. The homology of spheres. Diagram chasing and the long exact sequence of an algebraic quotient. Relative homology - definition, functoriality, homotopy invariance. The long exact sequence of pair. The homology of a disk relative its boundary. Reduced relative homology. Reduced homology is homology relative to a single point. The statement of excision. Relative homology is the homology of the quotient space (using excision). Idea of the proof of excision. Barycentric subdivision. Barycentric subdivision shrinks diameters. Barycentric subdivision commutes with the boundary map. Barycentric subdivision is homotopic to the identity. Homology with "small simplices". Homology with "small simplices" is the same as the usual homology. Proof of excision. Explicit generators for the homologies of spheres. Naturality of the long exact sequence construction. Simplicial structures. Examples of simplicial structures. The simplicial boundary and simplicial homology. Relative simplicial homology. The five lemma (more diagram chasing). Simplicial homology is equal to the usual homology - the finite case. Simplicial homology is equal to the usual homology - the infinite case. Simplicial homology is equal to the usual homology - the relative case. The degree of a map f: Sn -> Sn. The degree of a map f: S1 -> S1. The degree of the identity map. The degree of a non-surjective map. Homotopy invariance of the degree. Multiplicativity of the degree. The degree of a reflection and of the antipodal map. The degree of a fixed-point-free map. You can't comb the sphere. The degree of an arbitrary rotation. Maps of arbitrarily high degree. Local degrees. The (global) degree is the sum of the local degrees in the vicinities of the preimage of a given point. A practical computation procedure for degrees. The degree of a map f: T2 -> S2. The linking number. Homotopical invariance of the linking number. The Hopf link is linked. Computation of the linking number in the general case. CW spaces. CW homology and its relation with singular homology. CW homology: the torus, genus g surfaces, projective spaces. The Euler characteristic - the cellular formula, the homological formula and their equality. Abelianization. The relation between the fundamental group and the first homology. "Normalized" simplices and "normalized" homology. "Topological homotopy" between maps between chain complexes. A "topological homotopy" induces an "algebraic homotopy" and hence equal maps between homologies. Pulling the corners to the basepoint and the equality of "normalized" homology and the usual homology of a space. The complement of an embedded disk in a Euclidean space is homologically trivial. The latter is false homotopically! Reconciliation of the last two statements. Homology doesn't see the topology of spheres embedded in spheres. The linking number from this perspective. Jordan's theorem (in arbitrary dimension). Invariance of domain. Manifolds. An embedding of a compact manifold in a connected manifold is surjective. Sn cannot be embedded in Rn. Rn+1 cannot be embedded in Rn. The Borsuk-Ulam theorem. (Follows from:) The degree of an odd map is odd. (Proven using:) The "half boundary" isomorphism between the various Z/2Z homologies of projective spaces. Formal construction of the "half boundary" map for a 2-cover, using the liftings-projection short exact sequence.

Good Luck!!!