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