The Eilenberg-Steenrod axioms

Goal: understand the axiomatic formulation of (generalized) homology and cohomology theories and illustrate it for the case of singular homology and cohomology.

Study Team: Olimpiu Anton, Paul Tudorache, Narcisa Haiducu, ⁨Özgür Ionuț-Emre⁩, Cezar Valentin Ionescu, Octavian Ianc.

Past Members: Denisa Asan

Supervisor: Calin Lazaroiu

Scheme:

A. Basic theory of topological spaces, homeomorphisms and (strong)
homotopy equivalence. (slides 1)

1) Topological space
2) Continuous map
3) Homeomorphism
4) Homotopy between two continuous maps
5) Homotopy equivalence of topological spaces
6) Fundamental group of a topological space (first homotopy group)
7) Covering spaces
8) Covering maps
9) Higher homotopy groups
10) Weak homotopy equivalence of topological spaces
11) The category of good topological spaces and its homotopy category
12) The notion of model category (bonus)

Related topics:
a) Simplicial complexes
b) CW complexes

B. Basic theory of Abelian groups and modules over a commutative ring.

C. Basic homological algebra of (co)chain complexes of modules over a
commutative ring.

D. Construction of singular homology and cohomology theory with
integer coefficients and with coefficients in an Abelian group.

E. The Eilenberg-Steenrod axioms.

F. Proof that singular homology theory with integer coefficients
satisfies the E-S axioms.

Historical: 50 min presentation on the history of the theory and the main people who contributed to it.

References:

Bourbaki, Elements de Mathematique

Vick, "Homology Theory: An Introduction To Algebraic Topology"

Hatcher, "Algebraic topology"

Rotman, "An Introduction To Homological Algebra"

Bredon, "Topology and Geometry"

Eilenberg & Steenrod, "Foundations of Algebraic Topology"

Tom Dieck, "Algebraic Topology"

Weibel, "An Introduction to Homological Algebra"

Gelfand & Manin, "Homological Algebra"

Mac Lane: "Categories for the working mathematician"

More references here and here.

Summer projects

I. Fundamentals of homological algebra

Responsible:  Narcisa Haiducu

Scheme:

a. The category of modules over a commutative ring

b. Abelian categories and exact sequences

c. (Co)chain complexes and their cohomology

d. Basic lemmas of homological algebra; diagram chases

e. Exact functors and related results; naturality.

Main refs:

Bourbaki, Algebra I, Chaps 1-3
Freyd, Abelian Categories
Weibel, Homological algebra

II. Fundamentals of homotopy theory

Responsible: Paul Damian Tudorache

Scheme:

a. The first homotopy group and covering space theory.

b. The higher homotopy groups of a topological space and their basic properties (Kunneth theorem etc.)

c. Strong and weak equivalence of topological spaces

d. Fibrations and cofibrations

e. The category of good topological spaces and its homotopy category. 

 Main refs:

Munkres, Topology
Dwyer and Spalinski, Homotopy theory and model categories
May, More Concise Algebraic Topology
Riehl, Notes on Categorical Homotopy Theory