Homotopy theory in continuous world Homotopy theory in combinatorial world Homotopy theory in continuous and combinatorial world Jie Wu National University of Singapore 28 June, 2016 Zhejiang Normal University, Jinhua, China Homotopy theory in continuous world Homotopy theory in combinatorial world Outline Homotopy theory in continuous world Homotopy theory in combinatorial world Homotopy theory in continuous world Homotopy theory in combinatorial world Homotopy and Homotopy Groups Let f ; g : X ! Y be continuous map. A homotopy between f and g means a continuous map H : X × [0; 1] ! Y such that H(x; 0) = f (x) and H(x; 1) = g(x). n Homotopy Groups. πn(X) = [S ; X]. Note. π0(X) is the set of path-connected components of X. π1(X) is the fundamental group of X. πn(X) is abelian for n ≥ 2. Homotopy theory in continuous world Homotopy theory in combinatorial world Importance of Homotopy Groups–Core problem in homotopy theory Building blocks for cell-complexes: m n+1 2-cell complex: S [f D for some map f : Sn = @Dn+1 ! Sm. n+1 n n+1 In general Y = X [f D for some map f : S = @D ! X. Homotopy groups are far-unknown in general. Computation of homotopy groups is usually much harder than that of homology groups. Homotopy theory in continuous world Homotopy theory in combinatorial world Applications of Homotopy Theory In addition to traditional applications such as geometry, mathematical physics, From Wiki: The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. This is enabled by utilizing a homotopy-Maclaurin series to deal with the nonlinearities in the system. The HAM was first devised in 1992 by Liao Shijun of Shanghai Jiaotong University in his PhD dissertation.... Homotopy theory in continuous world Homotopy theory in combinatorial world There are various applications of topology in combinatorial or discrete world through combinatorial topology. The basic understanding could be that algebraic topology was developed from simplicial complex theory (triangulations of polyhedron). From such a view, I would not be surprised for some applications like persistent homology because one can make out many simplicial complex models. In continuous world, the homotopy and homology are algebraic invariants of geometric objects under continuous deformation ! In discrete/combinatorial world, there would be similar algebraic data which are invariants under certain rules. I will first display few references as applications of homotopy/homology in graph theory, and then, for illustration, I will talk an example how to get topology from combinatorics. Homotopy theory in continuous world Homotopy theory in combinatorial world Homology/homotopy of graphs and similar objects • H. Barcelo and R. Laubenbacher, Perspectives on A-homotopy theory and its applications, Discr. Math. 298 (2002), 39-61. • From their abstract: “This paper contains a survey of the A-theory of simplicial complexes and graphs, a combinatorial homotopy theory developed recently. The initial motivation arises from the use of simplicial complexes as models for a variety of complex systems and their dynamics. This theory diverges from classical homotopy theory in several crucial aspects. It is related to prior work in matroid theory, graph theory, and the work on subspace arrangements.” Homotopy theory in continuous world Homotopy theory in combinatorial world Homology/homotopy of graphs and similar objects • A. Cavicchioli and M. Meschiari, A homology theory for colored graphs, Discrete Math. 137 (1995), 99-136. • From their abstract:“We introduce a homology theory for colored graphs (G; CG) which is motivated by topological reasons. Indeed, it translates the usual one of polyhedron jGj canonnically associated to (G; CG). Then we obtain combinatorial analogs to exact homology sequences, cohomology groups, products, duality, etc. and prove some classical results of algebraic topology by graph-theoretic tools. Finally we study some combinatorial invariants of (G; CG) which are useful to describe the topological structure of jGj.” • B. Everitt and P. Turner, Homology of coloured posets: A generalisation of Khovanov’s cube construction, J. of Algebra, 322 (2009), 429-448. Homotopy theory in continuous world Homotopy theory in combinatorial world Coloring of a Graph Let V (G) be the vertex set of a graph G, and e(G) be the edge set. • a vertex coloring of G is a labeling of the graph’s vertices with colors such that no two vertices sharing the same edge have the same color. • A coloring using at most k colors is called a (proper) k-coloring. • A k-coloring is a function f : V (G) ! f1; 2;:::; kg such that f (a) 6= f (b) if (a; b) 2 e(G). Homotopy theory in continuous world Homotopy theory in combinatorial world Chromatic polynomial The chromatic polynomial of G, PG(t), is the number of t-colourings V (G) ! f1; :::; tg. It is known that PG(t) is a monic polynomial of degree jV (G)j. Example. PG(3) = 12 for the following graph G. http://upload.wikimedia.org/wikipedia/commons/b/b7/Graph_... 第1页 共1页 2/3/14 10:01 PM Homotopy theory in continuous world Homotopy theory in combinatorial world Motivation of Chromatic polynomial: Four color theorem The four color theorem, (or the four color map theorem), states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Homotopy theory in continuous world Homotopy theory in combinatorial world Chromatic polynomial and four color theorem The four color theorem says that PG(4) 6= 0 for any planner graph G. For having a graph, each region is considered as a vertex and draw an edge between them if two regions are adjacent. It is not true that PG(4) 6= 0 for any graph G. Homotopy theory in continuous world Homotopy theory in combinatorial world Topology from combinatorics: Chromatic Configuration Spaces Let G be a graph with vertex set V (G) labeled by f1; 2;:::; ng. Let M be a topological space. The chromatic configuration space n F(M; G) := f(x1;:::; xn) 2 M jj(i; j) 2 e(G) ) xi 6= xj g with subspace topology. A chromatic space is the space of n-coloring of G with labels in a topological space M. Homotopy theory in continuous world Homotopy theory in combinatorial world Example Let M = R2 = C. Let G be 3 Then F(M; G) = f(z1; z2; z3) 2 C j z1 6= z2 and z2 6= z3g. Homotopy theory in continuous world Homotopy theory in combinatorial world Euler Number For any polyhedron X, there is an Euler number (or Euler characteristic). The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula: χ = V ( number of vertices)−E( number of edges )+F( number of faces ): For any space X, there is an Euler number χ(X). Homotopy theory in continuous world Homotopy theory in combinatorial world Theorem–Jun Da Chen, Zhi Lu and W. Let PG(t) be the chromatic polynomial of simple graph G. Let M be a compact triangulated homology m-manifold with Euler number χ(M) > 0. Then χ(F(M; G)) = PG(χ(M)) Remark. Michael Eastwood and Stephen Huggett (Euler characteristics and chromatic polynomials,European Journal of Combinatorics, Volume 28, Issue 6, August (2007), 1553-1560.) show that the above formula is true for complex projective spaces. Homotopy theory in continuous world Homotopy theory in combinatorial world Configuration Spaces A graph G is called complete if G is a simple graph with the property that every pair of vertices is connected by a unique edge. Let G be a complete graph with jV (G)j = n. Then ×n F(M; G) = f(x1;:::; xn) 2 M j xi 6= xj for i 6= jg; which is called n-th ordered configuration space of M denoted by F(M; n). Homotopy theory in continuous world Homotopy theory in combinatorial world Braid groups Let the symmetric group Σn act on F(M; n) by permuting coordinates. The n-th unordered configuration space of M is defined by the orbit space B(M; n) = F(M; n)=Σn. The n-strand braid group over M: Bn(M) = π1(F(M; n)=Σn), the fundamental group of the n-th unordered configuration space. The n-strand pure braid group over M: Pn(M) = π1(F(M; n)). Homotopy theory in continuous world Homotopy theory in combinatorial world Artin Braid Group 2 The Artin braid group Bn = Bn(D ) with a presentation: generators σi , 1 ≤ i ≤ n − 1, and defining relations σi σj = σj σi for ji − jj ≥ 2 and σi σi+1σi = σi+1σi σi+1. 1 i-1 i i+1 n σi Homotopy theory in continuous world Homotopy theory in combinatorial world A robot moving between m obstacles gives an m+1 stranded braid Robot Motion Planning t i m e Homotopy theory in continuous world Homotopy theory in combinatorial world Braids and Links A link is the union of mutually disjoint simple closed (polygonal) curves in R3. Markov Theorem. ANY link can be obtained by closing up a braid. Homotopy theory in continuous world Homotopy theory in combinatorial world Have a fun with braids—Brunnian Braids A braid is called Brunnian if it becomes a trivial braid after removing ANY of its strands. -1 3 (σ1σ2 ) is Brunnian Homotopy theory in continuous world Homotopy theory in combinatorial world Brunnian braids and homotopy groups Berrick-Cohen-Wong-Wu: There is an exact sequence 2 2 2 2 1 ! Brunn+1(S ) ! Brunn(D ) ! Brunn(S ) ! πn−1(S ) ! 1 for n ≥ 5.