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Course, academic year 2023/2024
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Introduction to Differential Topology - NMAT009
Title: Úvod do diferenciální topologie
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Oldřich Kowalski, DrSc.
Classification: Mathematics > Topology and Category
Is incompatible with: NMAG452
Is interchangeable with: NMAG452
Annotation -
Last update: T_MUUK (17.05.2001)
This lecture is based on a text of the world's leading topologist John Milnor, and it is an introduction to a very modern part of topology. In contrary to the general ("point-set") topology, where the basic notions are continuous mapping and homeomorphism, the basic notions in differential topology are smooth mapping and diffeomorphism. Instead of general topological spaces, more special objects, so-called smooth manifolds, are studied. But here a highly nontrivial fact can be shown that a diffeomorphism is a more subtle equivalence relation than a homeomorphism. The topics studied in the lecture are ,e.g., the integer-valued degree of a map and the index of a vector field at its isolated null point. Besides many interesting theorems one can solve, by the developed technics, various well-known mathematical puzzles, like the problem of "hair-styling of a sphere". The subject can be tought in English.
Aim of the course -
Last update: KOWALSKI/MFF.CUNI.CZ (28.03.2008)

The goal of this topic is to acquaint the students with a new branch of topology, which completes other courses of

topology taught at the Faculty.

Literature -
Last update: KOWALSKI/MFF.CUNI.CZ (28.03.2008)

J. Milnor, A.H. Wallace: Differencial topology, introductory course.

Teaching methods -
Last update: KOWALSKI/MFF.CUNI.CZ (28.03.2008)

The method of teaching is a standard lecture.

Syllabus -
Last update: T_MUUK (16.05.2003)

Smooth manifolds and maps. Regular and critical values, Theorems by Sard and Brown. Manifolds with boundary. Classification of 1-dimensional manifolds with boundary. Brouwer's fixed-point theorem in smooth case. The degree modulo 2 of a map. Smooth homotopies and isotopies, the proof of the unique existence of the degree modulo 2 in the non-oriented case. Oriented manifolds. Brouwer's integer degree of a map. Applications: Problem of existence of smooth nonzero vector fields on the spheres. Index of a vector field at an isolated zero. Poincare-Hopf theorem on the sum of indexes of a vector filed (with isolated zeros).

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