Course.
21-752, Fall 202
MWF at 11:15am - 12:05pm ET at WEH 8201
Syllabus
Instructor.
Aristotelis Panagiotopoulos
Building WEH, office 7201
apanagio at andrew.cmu.edu
Office hours.
Monday 4-5pm (at my office)
Topology is a more flexible version of geometry, which ignores lengths and angles, and instead focuses only on those properties of shapes which stay invariant under continuous transformations. Algebraic topology is a branch of topology which uses powerful tools from abstract algebra in order to study these shapes. This is achieved by assigning to each topological space a collection of algebraic invariants (groups, rings, etc.) that keep track of the important features of its shape. There are various recipes for doing so and in this class we will study several of them, including:
- the fundamental group and covering spaces;
- simplicial and singular homology theories;
- the cohomology ring of a space (time permitting).
In the process of discussing these constructions we will also get a glimpse of several ideas from category theory and homological algebra which provide not only the modern language for doing algebraic topology, but also the right frameworks for transferring the "algebraic method" to other areas of mathematics.
Books.
There are many introductory books in algebraic topology and you should feel free to use whichever fits your style best. That being said, I will be using the following books:
- "Algebraic Topology" by Allen Hatcher
Cambridge University Press, 2002
- "A Concise Course in Algebraic Topology" by J. Peter May
Chicago Lectures in Mathematics Series, University of Chicago Press, 1999.
The first one is a standard introductory book. It has much more than we are going to cover in this class and can be used as a good reference book from anyone of you who decides to work in the future in areas related to algebraic topology. The second book develops the theory in a much more elegant (and concise) fashion but it lacks examples. It will be appreciated by folks who have some background or interest in category theory.
Some interesting/useful links:
- The most famous 3-dimensional "cavity" of the two dimensional sphere which cannot be filled with a "round filling" is provided by the Hopf fibration; see an animation and Niles Johnson's lecture.
- The correspondence between covering spaces of a space X and subgroups of its fundamental group resembles the Galois correspondence between field extensions and subgroups of the Galois group. As it turns out, this resemblance is rooted in something deeper. Indeed, Grothendieck developed an approach to Galois theory of field extensions via the study of the fundamental groups of appropriate spaces; see wiki and stackexchange for more information and references.
- The naive attempt to show that the 2-sphere S is simply connected is: given a loop f on the sphere based at *, choose a point x on S\{*} which does not lie in the image of f and use a contraction of S\{x} on * to show that f is nullhomotopic. However, the assumption that such x exists is wrong; see wiki on Peano curve.
- If it bothers you that the bonding maps in the definition of a CW-complex are allowed to be "arbitrarily wild", see the answer on this math stackexchange question.
- A translated version of Poincare's original Analysis Situs (1898), together with its Five Supplements.
Suggestions for Final Projects (I will keep adding).
- Choose some of Poincare's supplements in the Analysis Situs (1898) and write a complete account of the mistakes (in the original) he was trying to fix and if/how he achieved that.
- Write an expository article on the proof of Jordan-Schoenflies Theorem and use it to get as a corollary the classification of all closed surfaces. Note that the Jordan-Schoenflies Theorem does not hold in higher dimensions, see Alexander's horned sphere.
- Compute the fundamental group of the Hawaiian Earring and prove some of its interesting properties. For example show that while itselt is not a free group, every finitely generated subgroup of it is free. A good first reference which contains further references can be found here
- Compute the fundamental group of the Menger Sponge and prove some of its interesting properties. For example show that while itselt is not a free group, every finitely generated subgroup of it is free. Interestingly, one of the ways to compute this group goes through a variation on the classical puzzle known as the Towers of Hanoi.
- Universal covering spaces of 1-dimensional complexes turn out to be trees. Given a more general 1-dimensional space such as the Menger Sponge, or the Hawaiian Earring above, is there some similar tree-like object which acts as a universal covering space? Review the theory of "generized universal covers" from the paper Combinatorial R-trees as generalized Cayley graphs for fundamental groups of one-dimensional spaces and explain how it can be used to compute the fundamental group of a space such as the Menger Sponge or the Hawaiian Earring.
- Mycielski conjectured that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the continuum. This conjecture was proven by Shelah using set-theoretic techniques (absoluteness and Cohen's forcing method). Present the more "elementary" proof of Shelah's result which is given by Pawlikowski in a more recent paper.
Homework.
Calendar.
- (Nov 23) The category of compact metrizable pairs, inverse systems, inverse limits, the Cantor space, the Warsaw circle, Solenoids, Singular homology of solenoids; see pages 5-10 in Notes
- (Nov 22) Eilenberg-Steenrod Axioms, Singular Homology fails to detect local pathologies, see pages 1-5 in Notes
- (Nov 22) Applications of the homology: degree of an enodmorphism of an n-sphere; hairy ball theorem; the C(S)-module of all vector fields on the n-dimensional sphere S does not have a basis; there is no topological group structure on a sphere of positive even dimension
- (Nov 19) The inclusion of the singular chain complex relative to a cover is a "chain deformation retract" of the singular chain complex (part III), review of the theory we developed so far.
- (Nov 17) The inclusion of the singular chain complex relative to a cover is a "chain deformation retract" of the singular chain complex (part II).
- (Nov 15) The inclusion of the singular chain complex relative to a cover is a "chain deformation retract" of the singular chain complex (part I).
- (Nov 12) Equivalent forms of the Axiom of Excision, homology of X relative to a cover of X, "chain deformation retracts", if the inclusion of the singular chain complex relative to a cover is a "chain deformation retract" of the singular chain complex then singular homology satisfies Excision.
- (Nov 10) The Axiom of Excision, if singular homology satisfies excision then the relative singular homology H•(X,A) of a good pair coincides to the singular homology H•(X/A) of the quotient X/A .
- (Nov 8) Pairs of spaces and good pairs; relative cycles relative boundaries and homology of pairs (aka relative homology); the long exact sequence of homology of pairs; homology of pairs is equal to reduced homology of pairs, it recovers the singular homology (take H•(X,∅)), and the reduced singular homology (take H•(X,pt)).
- (Nov 3) A short exact sequence of chain complexes induces a long exact sequence of homology groups (part II).
- (Nov 1) A short exact sequence of chain complexes induces a long exact sequence of homology groups (part I).
- (Oct 29) Exact sequences; the long exact sequence which relates the homology of a space X, its subspace A, and the quotient space X/A; the homology groups of the n-sphere; fixed point theorem in all dimensions.
- (Oct 27) The (-1)-simplex, reduced homology, computing singular and reduced simgular homology groups in dimension -1 and 0.
- (Oct 25) Continuous maps between spaces induce chain maps of singular chain complexes, homotopies between maps induce chain homotopies between the associated chain maps; singular homology is a functor.
- (Oct 22) The category of chain complexes, chain maps, chain homotopies, chain maps induce homomorphisms of homology groups, chain homotopic chain maps induce the same homomorphisms.
- (Oct 20)The homology groups of the real projective plane, problems of simplicial homology, singular homology, singular and simplicial groups of the singleton .
- (Oct 18) Chain complexes, boundary maps of simplices, simplicial homology, the homology groups of the torus .
- (Oct 15) Problems with higher homotopy groups, introduction to homology theories, Delta-complexes.
- (Oct 13) Deck transformations, normal coverings, correspondence with normalizer quotients
- (Oct 11) Covering spaces: the full Galois correspondence, fundamental groups of graphs.
- (Oct 08) Covering spaces: uniqueness of lifting, existence of universal covering spaces.
- (Oct 06) Covering spaces: cosets correspond to sheets, universal cover, a general lifting criterion.
- (Oct 04) Covering spaces: a bridge between geometry and algebra, examples, injectivity of the induced map.
- (Oct 01) Groupoid version of the van Kampen Theorem, on the fundamental groups of the Hawaiian earring and the Menger sponge, n-nowhere dense sets.
- (Sep 29) The proof of the van Kampen Theorem (part II).
- (Sep 27) Computing the fundamental group of a 2-skeleton, the proof of the van Kampen Theorem (part I).
- (Sep 24) The van Kampen Theorem statement and applications, pushouts in arbitrary categories.
- (Sep 22) The fundamental group (up to iso) is invariant under homotopy equivalences, fundamental group of product of spaces, review of free groups and presentations.
- (Sep 20) Borsuk-Ulam theorem (part II), fundamental group of spheres, the fundamentantal group constructions extends to a functor from TOP to GROUPS.
- (Sep 17) Brouwer's fixed point theorem, the fundamental theorem of algebra, the Borsuk-Ulam theorem (part I).
- (Sep 15) Computation of the fundamental group of the circle (part II), The Homotopy lifting property.
- (Sep 13) Some words on n-connectedness, Computation of the fundamental group of the circle (part I).
- (Sep 10) The fundamental group(oid), change of basepoint homomorphism, simply connected spaces.
- (Sep 8) Real projective plane, group axioms, path equivalence, the reparametrization lemma, path concatenation.
- (Sep 3) Coproduct and quotient topology, mapping cylinders, map factorization, cell complexes (CW-complexes), Examples, Diagrams (fundamental polygons).
- (Sep 1) The history of algebraic topology, retracts and deformation retracts.
- (Aug 30) "What is topology?", homotopies, homeomorphism vs homotopy equivalence.
Home.
Back to my website.
