Time and Place:
Friday 10-12, Endenicher Allee 60, SR 1.008
Topics covered:
Week 1 (April 12th):
Examples of 2-complexes (surfaces, one relator complexes, presentation complexes)
Trefoil group/3-strand braid group, Poincare dodecahedral group
Stable classification of 2-complexes (Tietze's theorem)
Week 2 (April 19th):
Review of topology (fundamental group, universal cover, Whitehead and Hurewicz theorems)
asphericity of one-relator complexes (some special cases)
locally indicable groups and the zero-divisor conjecture
Fox calculus
trefoil group as a semidirect product
Week 3 (April 26th):
asphericity of one-relator complexes/local indicability of one-relator groups
non-positive immersions/towers
Week 4 (May 3rd, on Zoom):
Whitehead conjecture (relation to asphericity of knot complements and ribbon disk complements, proof for locally indicable groups via tower argument)
Week 5 (May 10th):
asphericity and good stackings. Wise's conjecture on towers and L^2 Betti numbers.
Week 6 (May 17th): No lecture. (Homology growth conference at MPIM)
Week 7 (May 24rd): No lecture. (University holidays)
Week 8 (May 31st):
Non-planar graphs
van Kampen obstruction
Embedding theory for 2-complexes
Uni Bonn online course evaluations (open May 31st-June 7th).
Week 9 (June 7th):
Conway-Gordon theorem
second proof of van Kampen's theorem
products of non-planar graphs
Week 10 (June 14th):
More 2-complexes that do not embed in R^4 (Freedman-Krushkal-Teichner examples)
Week 11 (June 21):
Stallings' theorem on nilpotent quotients
Week 12 (June 28):
Homework problem review session
Week 13 (July 5th):
Magnus' theorem, topological applications
Scheduling for exam
Week 14 (July 12th):
Kervaire-Laudenbach conjecture: acyclic 2-complexes do not embed in contractible 2-complexes
Approach via equations over groups
Proof for spine of Poincare homology sphere
Week 15 (July 19th):
Exams
Class notes (Last updated July 18th ... still very much a work in progress)
Homework:
Exercises (April 12-July 12th)
Note also that exercise 3 for April 19th was wrong, and won't be on the exam.
Exam dates:
July 15-19, September 25-27.
Description of class:
Questions about two-dimensional spaces lurk in the background of three and four-dimensional manifold topology as well as hyperbolic geometry. The aim of this course is to push them to the foreground. More concretely, the course will
present examples and constructions in low dimensional topology with an emphasis on 2-dimensional complexes (one relator complexes, free-by-cyclic mapping tori, subcomplexes of a product of graphs, small cancellation constructions, exotic 2-complexes),
introduce various curvature conditions both classical (Gromov hyperbolic, CAT(K)) and modern (non-positive immersions) that have been used to successfully study such complexes, and
discuss conjectures (Whitehead, Eilenberg-Ganea, Kervaire-Laudenbach, Andrew-Curtis, Kaplanski, some without names) in the area in a coherent and connected manner.
Other topics we might cover include embedding theory of 2-complexes, classification of 2-complexes with a given fundamental group, L^2-Betti numbers of 2-complexes and relations to 3 and 4-dimensional topology.
Prerequisites:
Topology I (especially fundamental groups and covering spaces).
Relevant texts and readings:
A general reference for all kinds of algebraic topology (much more than we need, but occasionally useful to refer to and available freely online) is
``Algebraic Topology'' by A. Hatcher.
There is no textbook for some the material covered in the course, so (in addition to the class notes above) I will try to post references to relevant books/papers as we move along. (If they appear here, it means I probably consulted them while preparing for the lecture). Here are some to start:
``Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten.'' H. Tietze
``Introduction to knot theory'' Chapter IV by R. Crowell and R. Fox
``Two-dimensional homotopy and combinatorial group theory'' Chapter I by C. Hog-Angeloni and W. Metzler
``Combinatorial group theory'' R. Lyndon and P. Schupp
``Orderable groups minicourse notes'' A. Clay
``Group rings of infinite groups'' G. Gardam
``Free differential calculus I: Derivation in the free group ring'' R. Fox
``Stackings and the W-cycles Conjecture'' L. Louder and H. Wilton
``Counting cycles in labeled graphs: The nonpositive immersion property for one-relator groups'' J. Helfer and D. Wise
3-manifold notes ``Chapter 1: Combinatorial foundations'', especially sections 1 and 2 (Sphere theorem) D. Calegari
``A proof of the generalized Schoenflies theorem'' M. Brown
``Injective labeled oriented trees are aspherical'' J. Harlander and S. Rosebrock
``Rational curvature invariants for 2-complexes'' H. Wilton
``Kuratowski's theorem'' Y. Xu
``The spherical genus and virtually planar graphs'' S. Negami
``Knots and links in spatial graphs'' J.H. Conway and C. McA. Gordon
``Komplexe in euklidischen Raumen'' E. R. van Kampen
``Embedding products of graphs into Euclidean spaces'' M. Skopenkov
``van Kampen's embedding obstruction is incomplete for 2-complexes in R^4'' M. H. Freedman, V. S. Krushkal, and P. Teichner
``Homology and central series of groups'' J. Stallings
``A topological proof of Stallings' theorem on lower central series of groups'' T.D. Cochran