2017/18, first term.                             Credits: 3 (24 hours).

Teacher: Marco Grandis

Contents: singular homology with integral coefficients.

Goals:   like any homology theory, singular homology assigns to a space a sequence of abelian groups, transforming topological properties into simpler algebraic properties; this allows us to prove important results, for instance the invariance of topological dimension.
    The course can be given in English.

PROGRAM

1. Main definitions. 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. The Subdivision Theorem (without proof). The Mayer-Vietoris exact sequence. Computing the homology of spheres, of various compact surfaces and other topological spaces.

3. Applications. Invariance of topological dimension. Problems on retracts. The Brouwer Fixed Point Theorem. The degree of an endomap of spheres. Vector fields on spheres. Extensions of the Intermediate Value theorem.

4. Complements. A review of the fundamental group. Hints at higher homotopy groups and the Hurewicz theorem. Hints at singular homology with coefficients in an abelian group.

REFERENCES

Lecture Notes (a brief summary): in pdf (20 p.); covering a wider program, without proofs.

(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, 10-12.