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Topology of Configurations on Trees Configuration Spaces Discretized Model New Results Summary Topology of Configurations on Trees Safia Chettih Reed College WSU Vancouver Math and Statistics Seminar, Feb 22, 2017 Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Physical Intuition: Robots on Tracks A configuration is a snapshot at a moment in time. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Physical Intuition: Connectedness Is it always possible for the robots to move from one configuration to another without colliding? Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Physical Intuition: Connectedness It depends on the whole system! Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Topological Definitions A topological space is a set along with a notion of which points are ‘near’ each other (more formally, a definition of which subsets are ‘open’): R with unions of open intervals R with unions of half-open intervals [a; b) Any set W with W and ; This way lies point-set topology . Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Definition of Configuration Space The set of configurations of n ordered points in X is called Confn(X). Definition n Confn(X) := f(x1;:::; xn) 2 X j xi 6= xj if i 6= jg Confn(X) := Confn(X)=Σn If X is a topological space, then Confn(X) and Confn(X) are too. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Definition of Configuration Space Example Let I = [0; 1], then Conf2(I) is I × I with the diagonal removed Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Mathematical Connections Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Algebraic Topology Equivalences homotopy, homeomorphism, diffeomorphism, . Invariants homotopy groups, homology groups, characteristic classes, . Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Definition (Homotopy) Given two continuous functions f ; g : X ! Y , we say f ' g if there exists a continuous function H : X × I ! Y such that H(x; 0) = f (x) and H(x; 1) = g(x) Example f ; g : I ! R2, H : I × I ! R2 Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Example f ; g : S1 ! R2 − ∗ are not homotopic maps in R2 − ∗ Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Definition (Fundamental Group) Consider the set of based maps α : S1 ! X, which must send a basepoint that we’ve chosen in S1 to the basepoint of X. We define the fundamental group π1(X) as set of the homotopy equivalent classes of such maps, and we represent [α] 2 π1(X) with the loop α. Think of α :[0; 1] ! X as a path that starts and ends at the basepoint. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Structure of the Fundamental Group We can ‘add’ loops by concatenating. α ? β is a new path that follows α first, then β. But α ? β must be a map from S1 into X, so α ? β follows α twice as fast, then β twice as fast, so it can complete the entire circuit in time. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Structure of the Fundamental Group We ‘add’ loops by concatenating: [α ? β] The constant loop [c∗] is always in π1(X) c∗ ? α 6= α 6= α ? c∗ [c∗ ? α] = [α] = [α ? c∗] We ‘subtract’ loops by following the loop backwards: [−β] [β ? −β] = [c∗] (α ? β) ? γ 6= α ? (β ? γ) [(α ? β) ? γ] = [α ? (β ? γ)] Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Structure of the Fundamental Group We ‘add’ loops by (existence of a group concatenating: [α ? β] action) The constant loop [c∗] is (existence of a unit) always in π1(X) c∗ ? α 6= α 6= α ? c∗ [c∗ ? α] = [α] = [α ? c∗] (properties of a unit) We ‘subtract’ loops by (existence of an inverse) following the loop backwards: [−β] [β ? −β] = [c∗] (property of an inverse) (α ? β) ? γ 6= α ? (β ? γ) [(α ? β) ? γ] = [α ? (β ? γ)] (associativity of group ac- tion) Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Groups Therefore π1(X) has a specific mathematical structure. It is a group, with concatenation of paths as its group action. Z with + is a group Z with × is not a group n × n invertible matrices with matrix multiplication are a group Based maps S1 ! X with concatenation are not a group Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Definition (Homotopy Equivalence) We say X ' Y if there exist continuous functions f : X ! Y and g : Y ! X such that g ◦ f ' idX and f ◦ g ' idY Example (Deformation Retract) A deformation retract is a kind of homotopy equivalence D2 ' ∗ S1 × (0; 1) ' S1 Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary An Orange is not a Donut The fundamental group can tell the difference between the torus and the sphere. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Abelian Groups Definition A group is abelian if the order in which we ‘add’ elements doesn’t matter Z with + is an abelian group n × n invertible matrices with matrix multiplication are not an abelian group π1(X) is not, in general, an abelian group Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Fundamental Groups Example 1 1 Example π1(S _ S ) is the free group 1 ∼ on two generators, which is π1(S ) = Z It is generated by the loop not abelian. which goes once around the [α ? β] 6= [β ? α] circle. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary Let’s abelianize! Definition (First Homology Group) We define H1 as the abelianization of π1(X): π (X) H (X) = 1 1 [α ? β] ∼ [β ? α] H1(X) is an abelian group, and we can still think of elements as (homotopy classes of) loops in X. Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model Physical Intuition New Results Algebraic Topology Summary An Element of H1(Conf3(Γ)) S1 has dimension 1 =) only one point moves at a time, or two points move with a fixed distance between them Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary Definition If X is a space, n n Confn(X) = f(x1;:::; xn) 2 X jxi 6= xj if i 6= jg ⊆ X Example (Conf2(I)) Conf2(I) is I × I with the diagonal removed ' ∗ t ∗ Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary Confn(I) Confn(I) is just a bunch of points (n! of them, to be exact), so the ‘loops’ in Confn(I) are not very interesting. Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary Configuration spaces of graphs are not easy to visualize in general. Conf2(Y ) Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary Configuration spaces of graphs are not easy to visualize in general. Conf2(Q) Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary There is another way to think of a graph Definition A graph is a set of vertices V and a set of edges E where each edge attaches two (not necessarily distinct) vertices. Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary What if we required each point in a configuration to either rest at a vertex or move between unoccupied vertices? Example (Discretized Conf2(I)) Consider the unit interval I as a graph, with two vertices and a single edge connecting them. There are only two possible configurations, and neither point can move from either one. Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary Discretized Conf2(I) I = ' Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary This discretized version of configuration space Dn(Γ) has the structure of a cubical complex. Definition An i-cell is an i-dimensional cube. Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary Definition A cubical complex is a bunch of cells (possibly of different dimension) glued together along their faces.
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