Teachers:
Agnieszka
Bojanowska e-mail:
aboj@mimuw.edu.pl tel.
5544463
Office hours:
room 4630, Monday 15:00-17:00 or on appointment.
Stefan
Jackowski e-mail:
sjack@mimuw.edu.pl tel.
5544459
Office hours:
room 4590, Tuesday 15:00-17:00 or on appointment.
Time and place:
Wydział MIM
UW, ul.Banacha 2 (wejście od ul. Pasteura),
Lecture
and problem session: Wednesday,
14:15-17:45, room 4060
Elements of category theory: categories, functors, natural transformations, adjoint functors, examples from algebra and topology.
Compact-open topology in mapping spaces.
Homotopy of maps, homotopy equivalence. Homotopy category of spaces and pointed spaces. Group and co-group objects.
Cofibrations and fibrations. Duality. Lecture Notes
D. Puppe exact sequences of homotopy classes.
CW-complexes. J.H.C. Whitehead theorem.
Homotopy groups. Action of the fundamental group on the higher homotopy groups. Homotopy exact sequence of the pair and of a fibration.
Calculations of homotopy groups. Elements of differential topology. Smooth approximation and cellular approximation.
H. Hopf classification of maps M^k ---> S^n for k\leq n.
Attaching cells. Spaces with prescribed homotopy groups.
The Eilenberg - MacLane spaces. The Postnikov tower.
Series 1. Categories and functors
Series 3. Cofibrations and fibrations.
Series 6. Spaces related to Linear Algebra and their Homotopy Groups
Series 7. Homotopy classification of maps
Algebraic Topology
as a separate filed of mathematics has not much more than 100 years.
Its aim is to study topological properties of spaces such that
manifolds, polyhedra etc. using algebraic methods. It uses also
analytic tools. Fundamental notions of algebraic topology - homotopy
and homology - were introduced by a great French mathematician Henri
Poincare (1854-1912). Poincare believed that those ideas will
play a fundamental role in many mathematical fields, also far away
from topology. His vision fulfilled - the methods of Algebraic
Topology influenced very strongly development of mathematics in XX
century. It is well seen if one looks at the list of mathematicians,
who contributed to development of Algebraic Topology and/or its
applications in other fields (in parenthesis countries in which they
worked and years of main contributions to Algebraic Topology; if
relevant, information about the
Fields medal):
|
James
Waddell Alexander (USA, 13-36),
|
Friedrich Hirzebruch (D,
53-56), |
To students interested in history of Algebraic Topology we recommend J. Dieudonne A History of Algebraic and Differential Topology 1900-1960 Birkhaeuser 1989; a collection of the classical papers Algebraic Topology - A Student's Guide collected by J. F. Adamsa (Cambridge Univ. Press 1972) and a survey paper by S.P. Novikov Topologia Algebraiczna (in Russian).
Texts
A.Bojanowska Notes on Algebraic Topology (in Polish)
A. Bojanowska, S. Jackowski Topology 2 (basic homotopy theory, fundamental group, covering spaces) (in Polish)
G. Bredon Topology and Geometry Springer-Verlag, New York, 1993
Tammo tom Dieck Algebraic Topolgy EMS Textbooks in Mathematics 2008
Allen Hatcher Algebraic Topology
J. W. Milnor, Topology from the differentiable viewpoint. (translated to Polish)
Jesper Michael Moller Algebraic Topology Notes - Addendum to Chapter 0 of A. Hatcher's book (above)
Neil Strickland Notes on Homotopy Theory
Other links of interest
M. F. Atiyah Mathematics in the 20th century
M.J. Hopkins Algebraic Topology and Modular Forms ICM 2002
J. Milnor The Poincare Conjecture 99 Years Later. A Progress Report. February 2003
Archives
December 26, 2009
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