2013/14, first term. Code: 34325. Credits: 7.
Teacher: Marco Grandis
Subject: Homology theories.
GOALS
Algebraic Topology studies topological problems by reducing them to simpler algebraic problems. Its main tools are homology theories, which assign to a space X a sequence of abelian groups
Hn(X),
and homotopy theory, which assigns to a pointed space X the sequence of its homotopy groups
πn(X) (starting with the fundamental group).
These groups can detect the presence of n-dimensional 'cavities' in the given space, and allow one to prove important results, for instance the invariance of topological dimension.
The procedures used in the construction and study of homology theories gave rise to Homological Algebra and the Theory of Abelian Categories.
The course can be given in English.
PROGRAM
1. Singular Homology. Cubes, faces and degeneracies. The singular cubical set of a space and the complex of normalised cubical chains. Chain complexes of abelian groups and their homology. Definition of singular homology (by cubical chains) and functorial properties. Elementary computations of singular homology. The Homotopy Invariance Theorem.
2. Computing singular homology. Exact sequences. The exact homology sequence of a short exact sequence of chain comlexes. Subdivision Theorem. The Mayer-Vietoris exact sequence. Computing the homology of spheres. Applications: invariance of topological dimension, problems on retracts, Brouwer Fixed Point Theorem, vector fields on spheres, extensions of the Intermediate value theorem. Computing the homology of various compact surfaces and other topological spaces.
3. Relative singular homology and homology theories. Relative singular homology. Eilenberg-Steenrod's axioms for homology theories. Homotopy Invariance, exactness and excision for relative singular homology.
4. Tensor products. Modules, abelian groups and vector spaces. The Hom functor. Tensor products of modules: definition, fundamental properties, computations. The torsion product of abelian groups; its exactness properties (without proof).
5. Relative singular homology with coefficients in a group. Definition. Subdivision Theorem. Eilenberg-Steenrod's axioms. The Mayer-Vietoris exact sequence. Computations and applications. The Universal Coefficient theorem (without proof).
6. Relative singular cohomology with coefficients in a group. Properties of the Hom functor. Cochain complexes. Definition of singular cohomology. Eilenberg-Steenrod's axioms and the Mayer-Vietoris exact sequence. The Universal Coefficient theorem (without proof).
7. Complements. A review of the fundamental group of a pointed topological space. Higher homotopy groups, their definition and commutativity. Hurewicz theorem (without proof). The free group generated by a set; free direct product and pushout of groups. The theorem of van Kampen (without proof). The fundamental group of the figure-eight space. Hints at de Rham cohomology of open euclidean spaces and its relationship with singular cohomology with real coefficients.
REFERENCES
Lecture Notes (a brief summary): in pdf (20 p.)
(a) Algebraic Topology
J. Vick, Homology Theory, Academic Press 1973.
W. Massey, Singular Homology Theory, Springer 1980.
W. Massey, Algebraic Topology, an Introduction, Harcourt 1967.
S. Eilenberg - N. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press 1952.
A. Dold, Lectures on algebraic topology, Springer 1972.
E. Spanier, Algebraic topology, McGraw-Hill 1966.
R. Brown, Topology, Ellis Horwood 1988.
S.T. Hu, Homotopy theory, Academic Press 1959.
A. Hatcher, Algebraic Topology, 2002. http://www.math.cornell.edu/~hatcher/
(b) Homological Algebra
H. Cartan - S. Eilenberg, Homological algebra, Princeton Univ. Press 1956.
S. Mac Lane, Homology, Springer 1963.
C.A. Weibel, An introduction to homological algebra, Cambridge Univ. Press 1994.
(c) Category Theory
S. Mac Lane, Categories for the working mathematician, Springer 1971.
J. Adámek - H. Herrlich - G. Strecker, Abstract and concrete categories, Wiley Interscience Publ., 1990.
Office Hours. On Mondays, 15-18.