Special course at Faculty MIM UW
Stefan
Jackowski Office hours: Tuesday 15:00-16:00,
room 4590, or on appointment by e-mail
Location and time. Faculty MIM UW, ul. Banacha 2.
Lecture and problem session: Friday,
10:15-14:00 , room 1770
Many interesting geometric
objects carry a structure of a differentiable manifold, e.g. sets of solutions
of the nonsingular equations. It turns out that the differential structure and
related geometric structures (Riemann metric) are very helpful in studying
topological properties of such spaces.
The most spectacular achievements of differential topology include:
classification of closed manifolds up to the bordism relation (Rene Thom) and proof of the
Poincare conjecture, first in dimensions >4 (Stephen Smale)
and then 4 (Michael Freedman), and 3 (Georgi Prelelman). A method
of decomposition of a smooth manifold to cells, known as the Morse theory led
to the first proof of the periodicity theorem for homotopy groups of the linear
group (Raoul Bott).
Development of the surgery of manifolds
(William Browder, Dennis Sullivan) led to deep understanding of the structure of
manifolds. Today it is known that there
are many topological manifolds which do not posses any smooth
structure - their analysis is often
based on methods developed originally for the smooth manifolds. There are also
many topological manifolds which admit several non-diffeomorphic smooth structures
(John Milnor, Simon Donaldson).
Differential topology uses tools from methods of differential geometry
and algebraic topology. The distinguished role of
Differential Topology in XXth century mathematics is
illustrated by several Fields medals awarded for achievements in this field.
Below we list several mathematicians who contributed fundamental results in
Differential Topology (in parentheses country and information about the Fields
medal):
Lev
Pontrjagin (RUS), Charles
Ehresmann (F), Hassler Whitney (USA), George
De Rham (CH), Rene
Thom (F, Fields medal 1958),
Friedrich
Hirzebruch (D), Raoul Bott
(USA), John
Milnor (USA, Fields medal 1962), Michele Kervaire
(CH), Dennis
Sullivan (USA), Stephen
Smale (USA, Fields medal 1966), William
Browder (USA), Sergei
P. Novikov (RUS, Fields medal 1970), Simon
Donaldson (UK, Fields medal1986),
Grigori Y. Perelman (RUS, Fields medal 2006).
Bibliography
& links
·
R.Bott,
L.Tu Differential Forms in Algebraic Topology.. Springer-Verlag
·
G. Bredon,
Topology and Geometry. Springer-Verlag, New
York, 1993
·
Th. Bröcker, K. Jänich Introduction
to differential topology.
·
Th. Bröcker, T.
tom Dieck Kobordismentheorie
Lecture Notes in Mathematics 178.
Springer 1970
·
J. Dieudonne A History of Algebraic and
Differential Topology 1900-1960. Birkhaeuser 1989
·
B.A.Dubrovin,
A.T.Fomenko, S.P.Novikov Modern Geometry - Methods and Applications:
Part I. Springer 1991 (Russian original 1979)
·
B.I.Dundas Differential Topology.
2002--2013
·
M. Gualtieri
Geometry
and Topology, 1300Y , Lecture 2012 , Lecture
2009
·
M.W. Hirsch, Differential
Topology. Springer 1976
·
V. Guillemin, A. Pollack, Differential
Topology.
·
S. Jackowski Topologia Rozmaitości. Notatki do wykładu 2003
·
A. A. Kosinski
Differential
Manifolds Academic Press 1993
·
S. Lang Differential
manifolds. 1972 (nowe wydanie
2002)
·
J. W. Milnor, Topology
from the differentiable viewpoint. The University Press of Virginia
·
----------------
Singular points of complex surfaces. Princeton UP 1968
·
----------------
Lectures
on the h-cobordism theorem. Princeton UP 1965
·
----------------- Morse
Theory. Princeton UP 1963
·
---------------- Lectures on
Differential Topology (video, 1965) Lecture 1, Lecture 2, Lecture 3.
·
J.W. Robbin
D. A. Salamon Introduction to Differential Topology.
·
V.A. Rokhlin,
D.B.Fuks
Beginner's Course in Topology: Geometric Chapters. Springer
2004 (Russian original 1977)
·
J. T. Schwartz
Differential Geometry and Topology. New York, 1968
·
A.H. Wallace
Differential Topology. First Steps. W.A.Benjamin
1968
·
F.W. Warner Foundations of
Differentiable Manifolds and Lie Groups. Springer 1983
·
S. Weinberger Differential Topology.
Course at the University of Chicago
Last update: 15.05. 2014