MAT 542/540: Algebraic Topology

Stony Brook            Fall 2016

 

Announcements

Here is general information about the course. The MW lectures by Aleksey will take place 2:30-3:50pm in Physics P-128. The Friday discussions with Dennis/Chandrika will be held 12:30-2:30 in Math 5-127.

Possible Class Project

The second paragraph of this gives a quick definition of pseudocycle. It is then shown that the group of pseudocycle equivalences and the Z homology are isomorphic as graded abelian groups. Two natural questions arise.

  1. Suppose the pre-compactness condition is dropped. Does one get a group isomorphic to the Borel-Moore homology of a smooth manifold?
  2. The pseudocycle groups can be readily endowed with a product. Show that it is isomorphic to the usual homology intersection product, preferably in the context of non-compact manifolds and BM homology.
If you are interested in doing this project, please find someone else in the class to join you and discuss this with me. Depending on where you currently are in your graduate studies, this may be a great use of your time or not so great.

MAT 542 Course Instructor

Name: Aleksey Zinger     E-mail: azinger@math     Office: Math Tower 3-111
Office Hours: Tu 9-10 in Math P-143, 10-12 in Math 3-111 (or catch me at tea)

MAT 540 Course Instructor

Name: Dennis Sullivan     E-mail: dennis@math     Office: Math Tower 5-114
Office Hours: upon request or random encounter

 

Tentative Schedule

The MS reading assignments are from Characteristic Classes by Milnor-Stasheff.
All other reading assignements are from Munkres's Elements of Algebraic Topology.
You may also want to consult Hatcher's Algebraic Topology.

Date Topic Read Suggested Problems
08/29, MSimplicial complexesSections 1-4 HW1
note
08/31, WCW complexes and homologySection 38
09/02, FDiscussion with Dennis/Chandrika from Dennis
09/05, MLabor Day
09/07, WSimplicial homologySections 5,7,10 HW2
09/09, FDiscussion with Dennis/Chandrika from Dennis
09/12, MHomology of surfaces and conesSections 6,8 HW3
09/14, WRelative homology, Mayer-VietorisSections 9,23-25
09/16, FDiscussion with Dennis/Chandrika
09/19, MHomology pushforwards ISections 12,13,11 HW4
09/21, WChain equivalencesSections 13,46
09/23, FDiscussion with Dennis/Chandrika
09/26, MSimplicial subdivisionsSections 46,15,17 HW5
09/28, WTopological invarianceSections 17,18
09/30, FDiscussion with Dennis/Chandrika
10/03, MHomology pushforwards IISections 18,14 HW6
10/05, WSimplicial approximationsSections 16,19,20
10/07, FDiscussion with Dennis/Chandrika
10/10, MApplications and generalitiesSections 21,22,26-28 HW7
10/12, WSingular homologySections 29,30
10/14, FDiscussion with Dennis/Chandrika
10/17, MMayer-Vietoris and ExcisionSections 31-33 HW8
10/19, WSingular vs. simplicial homologySection 34
10/21, FDiscussion with Dennis/Chandrika
10/24, MSingular vs. CW homologySections 37-40 HW9
10/26, WInvariance of domainSection 36
10/28, FDiscussion with Dennis/Chandrika
10/31, MCohomology of chain complexesSections 41,45,46 HW10
11/02, WCohomology of topological spacesSections 44,42,43
11/04, FDiscussion with Dennis/Chandrika
11/07, MCohomology theorySections 44,47,48 HW11
11/09, WCup and cap productsSections 66,50,51,57
11/11, FDiscussion with Dennis/Chandrika
11/14, MKunneth formula for homologySections 54,58,55 HW12
11/16, WKunneth formula for cohomologySections 59,60
11/18, FDiscussion with Dennis/Chandrika
11/21, MHomological algebra wrapupSections 61,52,53 HW13
11/23, W Thanksgiving Break
11/25, F
11/28, MFundamental class of oriented manifoldMS pp270-275 HW14
11/30, WPoincare Duality for oriented manifolds MS pp276-279
12/02, FDiscussion with Dennis/Chandrika
12/05, MSome applicationsSection 65,68 HW15
12/07, WComputations on manifolds: an overview MS Sections 9-11
12/09, FDiscussion with Dennis/Chandrika
 
This page is maintained by Aleksey Zinger.
Last modified: December 6, 2016.