An introduction to homotopy and homology.

Lectures and problem sessions

Oral exam

The basic notions of group theory, vector space theory and general topology.

The fundamental group. Free groups. The theorems of Van Kampen. Other methods for computing fundamental groups.
Covering spaces. Fundamental group and covering spaces. Higher homotopy groups, maps between spheres, degree,the Jordan curve theorem, invariance of domain. Triangulations, Euler-Poincaré characteristic, orientation, the classification of surfaces.
Basic notions of homological algebra. Singular homology and its homotopic properties, relative homology, axiomatic homology theory.
Other homology theories. Simplicial complexes, CW-complexes. cohomology and Poincaré duality.

A. Hatcher: "Algebraic Topology", Cambridge University Press (freely
available online)
M. Greenberg, J. Harper: "Algebraic Topology".
W. Massey: "A Basic Course in Algebraic Topology", Springer-Verlag.
E. Spanier: "Algebraic Topology".

Further references:
M. Greenberg: "Lectures on Algebraic Topology".
C. Kosniowski: "Introduzione alla topologia algebrica".
M. Massey: "Algebraic Topology, an Introduction".
M. Manetti, "Topologia", second edition, Springer, Milan 2014
E. Sernesi: "Geometria 2".
P. Hilton: "Introduction to Homotopy Theory".
S. Hu: "Homotopy Theory".
J. Milnor, Spivak: "Morse Theory".
W. Massey: "Singular Homology Theory".
S. Hu: "Homology Theory".
C. Maunder: "Algebraic Topology".
G. Bredon: "Topology and geometry".