Topics in Geometry
- Professors:
- Cornalba Maurizio, Ghigi Alessandro
- Year:
- 2015/2016
- Course code:
- 504161
- ECTS:
- 6
- SSD:
- MAT/02
- DM:
- 270/04
- Lessons:
- 56
- Period:
- I semester
- Language:
- Italian
Objectives
An introduction to homotopy and homology.
Teaching methods
Lectures and problem sessions
Examination
Oral exam
Prerequisites
The basic notions of group theory, vector space theory and general topology.
Syllabus
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-Poincarè characteristic, orientation, the classification of surfaces.
Basic notions of homological algebra. Singular homology and its homotopic properties, relative homology, axiomatic homology theory, the Künneth formula.
Other homology theories. Simplicial complexes, CW-complexes. cohomology and Poincarè duality.
Covering spaces. Fundamental group and covering spaces. Higher homotopy groups, maps between spheres, degree,the Jordan curve theorem, invariance of domain. Triangulations, Euler-Poincarè characteristic, orientation, the classification of surfaces.
Basic notions of homological algebra. Singular homology and its homotopic properties, relative homology, axiomatic homology theory, the Künneth formula.
Other homology theories. Simplicial complexes, CW-complexes. cohomology and Poincarè duality.
Bibliography
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".
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".
Modules
- Professor:
- Cornalba Maurizio
- Lessons:
- 28
- ECTS:
- 3
- SSD:
- MAT/03
- Professor:
- Ghigi Alessandro
- Lessons:
- 28
- ECTS:
- 3
- SSD:
- MAT/03
Dipartimento di Matematica ''F. Casorati''
Università degli Studi di Pavia -
Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602