Topology from differentiable viewpoint  (Summer Semester 2013/14)

Special course at Faculty  MIM UW

 

Instructor, location and time

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

 

Lecture notes and exercises

 

Exercises 1.  Exercises 2.  Exercises 3.  Exercises 4.  Exercises 5.  Exercises 6. 

 

Problems for the oral exam.

 

Submanifolds (notes)

 

Syllabus

 

1.     Smooth maps on arbitrary subsets of the Euclidean space.  Tangent vectors and derivative.

2.     Topological  and smooth manifolds.  Smooth maps. Submanifolds  and other constructions on manifolds.

3.     Manifolds with boundary. Bordism ring.

4.     Local form of smooth maps -- constant rank theorem and Morse lemma.

5.     Vector bundles.  Riemann metrics. Tangent bundle and derivative.  Exponential map.

6.     Smooth approximation of continuous maps.

7.     Sard theorem.

8.     Transversality.

9.     Tubular neighborhood theorem.

10.  Differential interpretation of the Brouwer degree,  intersection number and linking number.

11.  The Pontriagin –Thom construction. Bordism and homotopy.

12.  Unoriented (co-)bordism as a generalized multiplicative  (co-)homology theory.

13.  The Thom class and the Euler class of a vector bundle.

14.  Formal groups.

15.  Cohomological operations in unoriented cobordism.

16.  Calculation of the bordism ring a la Boardman – Quillen.

 

About Differential Topology

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

 

Stefan Jackowski 

Last update: 15.05. 2014