MM811 - 2° semestre 2015

 (Co)homology products, characteristic classes, and Morse field theory
a.k.a. Homology II

1. basic notions of homology
2. general point set topology (compactness, connectedness, etc).
   
3. elementary techniques of smooth manifolds
   

(Co)homology products, characteristic classes, and Morse field theory

The lecture course has three parts. The first two parts cover products and operations in classical (co)homology theories and could be named Homology II whereas in part three our aim is to recover these notions using Morse theory. From now on and throughout we shall work with coefficients in the field Z/2Z which simplifies the theory considerably: Mod two (co)homology groups of a topological space are vector spaces (as opposed to modules). Our main text will be the (freely accessible) recent book by Hausmann.

    In Part I, which is a natural continuation of our course “Introdução à homologia” given in 2014-1, we start off with a quick recap of simplicial and singular (co)homology. Then we introduce the cup product in cohomology, thereby turning the cohomology vector space into a ring. The ring structure is very powerful: It allows to distinguish some topological spaces whose cohomologies are isomorphic as vector spaces. Further topics are cap product, cross product, and the Künneth formula. Applications of the latter include Euler characteristics of product spaces and the Thom isomorphism theorem.

  In Part II, turning to manifolds and vector bundles, we shall explain Poincaré duality, the Gysin homomorphism, as well as intersection and linking numbers. Furthermore, we discuss cohomology operations, such as Steenrod squares, and characteristic classes, such as Stiefel-Whitney classes. If time permits we will touch Thom's celebrated cobordism theory.

    In Part III we aim to recover (as many as possible of) the above structures in terms of the Morse-Witten complex, thereby understanding and providing the easy to grasp toy model for advanced topological Quantum Field Theories such as Floer homology.

Syllabus/Ementa

• Recap of homology and cohomology

• Cross, cup, and cap products, duality, Steenrod operations, Stiefel-Whitney classes

• Pontryagin-Thom construction and “umkehr maps”

• Recap of the Morse-Witten complex, functoriality

• Morse field theory represents them all

Literature

(Co)homology

• Hausmann, J.-C. “Mod two homology and cohomology”, Universitext. Springer, Cham, 2014.   downloadable from IMECC computers
• Hatcher  “Algebraic Topology”
• Munkres, J. R. “Elements of algebraic topology”, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
  
• Bredon “Topology and geometry”, GTM 139, Springer 1993.

• Vick “Homology theory. An introduction to algebraic topology.”, GTM 145, Springer 1994.
• tom Dieck, T. “Algebraic topology”, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.

• Steenrod, N. E. “Cohomology operations”, Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50. Princeton University Press, Princeton, N.J., 1962.
  

Characteristic classes

• Milnor, J. W., and Stasheff, J. D. “Characteristic classes”, Princeton University Press, Princeton, N. J., 1974. Annals of Mathematics Studies, No. 76.
  

Topological Quantum Field theory

• Atiyah, M. “Topological quantum field theories”,  Inst. Hautes E ́tudes Sci. Publ. Math., 68 (1988), 175–186.
• Cohen, R., and Norbury, P. “Morse field theory”, Asian J. Math. 16, 4 (2012), 661–711.

Joa Weber
IMECC UNICAMP