Topology: Basic ideas

Transcription

1 For I will give you a mouth and wisdom, which all your adversaries shall not be able to gainsay nor resist. Topology: Basic ideas Yuri Dabaghian Baylor College of Medicine & Rice University dabaghian at rice.edu

2 Historical background Topology appeared in late XIX early XX century H. Poincare (1895) F. Hausdorff (1914) XIX century geometry century N. Lobachevsky (1826) J. Boliai (1829) K. F. Gauss (1830s) Erlangen Program, F. Klein (1872)

3 What defines geometric properties? Geometric characteristics: angles, lengths, etc = 2, 3 = , 5 6, 5 7, etc. How to match figures in Euclidean geometry: use translations + rotations Other geometries other transformations, other geometric characteristics The larger the group of transformations, the fewer geometric properties What is the largest group of transformations?

4 Continuous transformations What is the largest group of transformations that does not violate the structure of the space itself? What is space? What is spatial continuity?

5 The fabric of space 1. Space consists of points 2. What turns a set of points into a space? 3. The notion of spatial proximity

6 The fabric of space, constructively Take a subset of points Tour de magie: define this set as a proximity neighborhood. From now on the red points are neighbors

7 The fabric of space, constructively Take another subset of points Tour de magie 2: define this set as a proximity neighborhood. From now on the blue points are also neighbors

8 The fabric of space, constructively Take another subset of points Tour de magie 3: define this set as another proximity neighborhood. From now on the green points are also neighbors

9 The fabric of space, constructively Take another subset of points Tour de magie 4: define this set as another proximity neighborhood. From now on the purple points are also neighbors

10 Topology, definition A set X with a system of neighborhoods, T, such that: 1. The whole set X and the empty set are neighborhoods, X T, 2. A finite intersection of neighborhoods is a neighborhood, 3. Any union of neighborhoods is a neighborhood i N i ii N i T n i1 T N i T

11 Continuous maps Continuous maps: maps that preserve topological structure Topology: area of mathematics concerned with spatial properties that are preserved under continuous deformations of spaces and geometric objects Intuitively: elastic stretches and deformations are allowed, cutting and gluing is not

12 Topological properties, examples What are the topological properties? How to define them? How to describe them? How to compute them?

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14 Topological properties, examples A space can have many pieces C 3 C 1 C 2

15 Topological properties, examples z γ y w γ 3 C 3 x γ 1 C 1 u γ 2 v C2 Path connectedness

16 Topological properties, examples x γ 1 C 1 z γ y w γ 3 C 3 Path: I u γ 2 0,1 ( t) v C 2 Path connectedness

17 Topological properties, examples γ z γ 3 C 3 y w Path: x γ 1 C 1 I u γ 2 0,1 ( t) v C 2 (0) y, 1 w...? Path connectedness

18 Topological properties, examples z [γ 3 ] γ y w C 3 x [γ 1 ] C 1 [γ] 1 C 1, [γ] 2 C 2, [γ] 3 C 3. [γ 2 ] u v C 2 Path connectedness: there are 3 types of paths

19 Paths Path γ x y I 0,1 ( t) Are there path types,[γ]?

20 Fundamental group X Path γ x γ I 0,1 ( t) 1. Use closed paths S 1 () t Equivalence class,[γ], is defined, m Topological index m of the path [γ] m

21 Fundamental group X 1. Use closed paths S 1 () t x γ 2. Paths combine: Equivalence classes combine: Topological index combines: 1 m 2 n 3 n m ( paths) ( indexes) (n,m) n+m Fundamental group, π 1 (X)= Z

22 Two holes ( index?) x γ

23 Two holes ( mn, ) x γ 1 ( m, m ) 2 ( n, n ) 3 ( m n, m n ) Fundamental group, π 1 = Z Z

24 3 holes ( m, n, k ) x γ Fundamental group, π 1 = Z Z Z

25 Fundamental group, examples Circle π 1 = Z Sphere π 1 = 0 Torus π 1 = Z Z

26 How useful are topological indexes? π 1 = 0 No loops π 1 = Z? 1 loop π 1 = Z Z? 2 loops

27 Topological equivalence and homotopies hollow

28 Topological equivalence and homotopies

29 Applied Topology: knots 2 3 1

30 Applied Topology: embedding aspect The fundamental group of the knot complements in R 3 are different knots are different π 1 (R 3 \ ) π 1 (R 3 \ )

31 Applied Topology: links The topological cycle of a replicon, EMBO reports (2004)

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33 Summary 1. Topology, the idea 2. The notion of continuity 3. Topological indexes 4. Fundamental group Next: Simplicial complexes and homologies Literature: 1. J. Munkres, Topology (2000) and Elements Of Algebraic Topology (1996) 2. A. Zomorodian, Topology for computing, (2009) 3. A. Hatcher, Algebraic Topology, (2002),