**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

http://ncatlab.org/nlab/show/cohomology

http://www.map.him.uni-bonn.de/Bordism

http://www.map.him.uni-bonn.de/Complex_bordism

Obvious Wikipedia pages including http://en.wikipedia.org/wiki/List_of_cohomology_theories

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)