Math 5311

Geometry and Topology II

Spring 2012

(Homology and Cohomology)

MWF 12 – 12:50 MSB 415 www.math.uconn.edu/~bridgeman/math5311s12/index.htm

Instructor: Jim Bridgeman bridgeman@math.uconn.edu, www.math.uconn.edu/~bridgeman/index.html

Hours: M/Th/F 10:00 – 11:30 W 3:00 – 4:30 F 1:00 – 2:00 or by appointment

Assignments (most recent on top)

For 4-27 Study pages 230 to 257 and 268 to 280 … sample some corresponding exercises (no need to hand in)

For 4-13 Study pages 206 to 228 and sample some exercises on 228 to 230 (no need to hand in, but ask if you have questions.)

For 4-2 Study pages 185 to 204 and sample some exercises (no need to hand in, but ask if you have questions) on 204 to 206.

For  3-19 work up solutions for the 10 problems you chose from 155-59.  Pick a topic to write your paper about and get ready to start working on it.  Study pages 160-165

For 2-29 Study pages 143-155; look over the exercises on 155-59 and pick out 10 to start working on for March 19 hand-in

For 2-15 Complete the exercises assigned last week.  Study the equivalence of singular and simplicial homology on pp. 128-31 (pay attention to the Five-Lemma, it will become an old friend.) Pick two exercises out of 10 thru 31 on pp. 131-33 that interest you and start trying to work them out (not to be handed in, but ask about them if you’re not sure about your solutions.)  Finally, study pages 134 to 143, the beginning of some real computational tools.

For 2-8 study the section on “exact sequences and excision” pp. 113 to 128.  Start writing up solutions to Exercises 1 thru 9 on page 131 (will be collected on 2-15).

For 2-1 read: the rest of the Appendix up to p. 525, reading for facts (proofs only as they interest you); introduction to Ch. 2 (pp. 97-101) and (now studying proofs, too) “simplicial homology”, “singular homology”, and “homotopy invariance” in Ch. 2.1 (pp. 104-113)

For 1-25 be sure to have read chapter 0, Δ-complexes in 2.1 (pp102-104), and Appendix up to but not including Prop.A.2 (pp519-521)

Text: Algebraic Topology, Allen Hatcher, Cambridge University Press 2001, primarily chapters 0, 2, 3 and Appendix

available online free at www.math.cornell.edu/~hatcher/AT/ATpage.html including useful supplemental material

Resources:

Elements of Algebraic Topology, James R. Munkres, Addison-Wesley 1984

A Concise Course in Algebraic Topology, J. P. May, University of Chicago Press 1999 online free at www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Algebraic Topology, E. H. Spanier, McGraw-Hill 1966

More Concise Algebraic Topology, Ponto-May, University of Chicago Press 2012

Stable Homotopy and Generalized Homology, J. F. Adams, University of Chicago Press 1974

Algebraic Topology from a Homotopical Viewpoint, Aguilar-Gitler-Prieto, Springer 2002

Rough syllabus:

CW-Complexes

Simplicial Complexes

Homology of Simplicial Complexes

Singular Homology

Properties of Singular Homology

Computations

Cellular Homology

Mayer-Vietoris Sequences

Coefficients

Axiomatics

Applications

Cohomology of Complexes

Universal Coefficient Sequence - Cohomolgy

Cohomology of Spaces

Products and Ring Structure

Künneth Formula - Cohomology

Some Examples and Applications

Orientation and Duality

Universal Coefficient Sequence – Homology

Künneth Formula – Homology

Sketch of the Homotopical Point of View

Additional Topics and Applications as time allows (suggestions welcome)

Work: come to class having read the assignment (posted above), 2 homework sets will be assigned (and posted above), plus one project/paper (pick a topic, learn about it, tell me about it in writing)