Topology III - Homology

MATH-323

Media

This file is part of the content downloaded from Topology III - Homology.

General information

Lecture title: Topology III Homology 


  • Instructor: Leonid Monin
  • Main assistant: Archi Kaushik

  • Lectures:  Mondays 10-12 in MA B1 11
  • Exercises: Mondays  8-10 in MA B1 11


Course content:

Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand its properties and learn how to compute it. There will be many examples and applications. 

Concrete topics covered in the course are:  Simplicial and singular homology; exact sequences and excision; Mayer-Vietoris sequence; Eilenberg-Steenrod axioms; CW complexes and  cellular homology; cohomology and Poincaré duality.

Bibliography:

The main reference for the course are Chapters 2 and 3 of Allen Hatcher's book on algebraic topology. It is freely available here.

Additional references will be published in course materials.


Motivation for homology, definition of delta-complexes.

Simplicial homology, computation for surfaces, beginning of singular homology

Singular homology, homotopy invariance, beginning exact sequences

Proof of the homotopy invariance, Relative homology, long exact sequences of homology groups.

Formulation of Excision Theorem and using it to prove Theorem 2.13 in Hatcher. Begging of the proof of Excision theorem: Barycentric subdivision.

Proof of the Excision theorem.

I follow a slightly simplified proof of the excision theorem which you can find in the notes of Richard Schwartz uploaded below. 

Recap of the Excision and long exact sequence of a pair. Some remark and corollaries:
  • dimension theorem;
  • naturality of exact sequences;
  • excision for CW-pairs.
Equivalence of simplicial and singular homology.

Reminder of homotopy groups. Hurevicz theorem, the proof of the case of n=1 case.

  • Mayer-Vietoris sequence, Mapping tori and exact sequence of homology for them.
  • Homology with coefficients, Universal coefficients theorem
  • Axioms for homology.

Degree of maps of spheres and its properties. Vector fields on spheres and division algebras. Computation of degrees of maps via local degrees.

More on the degree maps. Definition of cellular homology.

For the differential geometric approach to degrees see Section 5 of Milnor's Topology from the differential viewpoint (you might need some previous sections for background).

Differential in the cellular chain complex via degrees. Homology of complex projective spaces and complex Grassmannians via cell decompositions. I am attaching notes of Jens Hemelaer onn Schubert cells of Grassmannians (see Section 2 there).

Euler characteristic, Lefschetz Fixed point theorem, Hopf Poincare theorem.

Cohomology: definition, universal coefficient theorem, cup product, Poincare duality.