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) := {(x1,..., xn) ∈ X | xi 6= xj if i 6= j}

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] ∈ 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) = {(x1,..., xn) ∈ X |xi 6= xj if i 6= j} ⊆ 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.

Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary

The cubical structure of Dn(Γ) looks like: 0-cell = each point fixed at a different vertex of the graph 1-cell = one point moving along an edge, disjoint from all other points 2-cell = two points moving along edges, disjoint from all other points . .

Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary

Example (D2(Q)) Consider the graph Q, which has three vertices (0,1,2) and three edges. One edge connects 0 to 1, the other two edges connect 1 to 2. There are 6 possible configurations, but it is impossible to move from a configuration where the first point is at 0 to a configuration where the second point is at 0.

2 Q D (Q) 6' Conf2(Q)

Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary

We have to subdivide the graph to get a good model of Confn(Γ)! Theorem (A. Abrams, 2000) For any graph Γ and any n > 1, there exists a subdivision Γ0 of Γ such that 1 each path between distinct vertices of valence not equal to two passes through at least n − 1 edges, and 2 each path from a vertex to itself that cannot be shrunk to a point in Γ passes through at least n + 1 edges n 0 Then D (Γ ) is a deformation retract of Confn(Γ).

n 0 ∼ In particular, this means that H1(D (Γ )) = H1(Confn(Γ)).

Safia Chettih Topology of Configurations on Trees Configuration Spaces Configuration Spaces Discretized Model A Better Way New Results A True Model Summary

Example (The graph H)

2 D (H) ' Conf2(H)

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

We will focus on configurations of two ordered points in graphs that have no loops, which are called trees.

Remember that Conf2(T ) = {(x1, x2)|x1, x2 ∈ T , x1 6= x2}

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Homology Classes of Conf2(T )

There are three basic non-trivial paths that we will encounter:

γABC ∈ H1(Conf2(T ))

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Homology Classes of Conf2(T )

µABCD ∈ H1(Conf2(T )) Notice that µCDAB = µABCD = −µADCB

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Homology Classes of Conf2(T )

0 µACBD ∈ H1(Conf2(T )) µACBD ∈ H1(Conf2(T ))

0 µACBD = σ(µACBD) where σ switches the labels of x1 and x2

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Homology Classes of Conf2(T )

τAB,DE ∈ H1(Conf2(T )) τAB,ED ∈ H1(Conf2(T ))

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Some Homology Relations for Conf2(T )

γCDE + τAB,ED = τAB,DE

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Some Homology Relations for Conf2(T )

τAB,GH = τAB,DE + τCE,GH

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Embeddings of the trees Y , X and H into more complicated trees induce inclusions of homology classes. Theorem (S. Chettih)

This gives a complete description of H1(Conf2(T )) in terms of these representatives.

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Theorem (S. Chettih) k H1(Conf2(T )) ' Z where X k = −1 + (µ(v) − 1)(µ(v) − 2) v vertex s.t. µ(v)>1 and where µ(v) is the valence of v

Hint of proof: Apply discrete Morse theory . . .

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

We can define Conf2(T ) → Conf3(T ) by adding a third point to any configuration at one of the extremal vertices of T , after first ’pushing in’ any points in the configuration on that edge. Example 61 H1(Conf3(X)) ' Z but the images of γ and µ, along with their relations, give a total rank of 59.

The maps on homology induced by ‘pushing in’ a new point from the extremities are not surjective!

Safia Chettih Topology of Configurations on Trees Configuration Spaces Homology Classes Discretized Model Conf (T ) New Results 3 Conf (T ) Summary n

Theorem (S. Chettih and D. Lütgehetmann) If T is a tree, consider the subgraphs of T which are either star graphs or homeomorphic to the H graph. Then Hq(Confn(T ); Z) is generated by ‘products’ of classes where points are constrained to these subgraphs of T

Theorem (S. Chettih and D. Lütgehetmann)

Hq(Confn(Γ); Z) is torsion-free for any q ≥ 0

Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model New Results Summary Summary

A loop in X represents a class in π1(X). If we forget the order in which we concatenate, a loop can also represent a class in H1(X). Configuration spaces of graphs can be modeled by a cubical complex.

We can build H1(Conf2(T )) out of homology classes on a particular set of subgraphs of T .

We can build Hq(Confn(T )) out of products of homology classes on a particular set of subgraphs of T .

Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model New Results Summary Further Directions

Confn(Γ) We have a basis for Hq(Confn(T )). What does a basis for Hq(Confn(Γ)) look like? Non-k-equal configurations What can we say about Confn,k (Γ)? Graph Theory What properties of graphs are these configuration spaces ‘measuring’?

Safia Chettih Topology of Configurations on Trees Configuration Spaces Discretized Model New Results Summary

Thank you! A. Abrams and R. Ghrist. Finding topology in a factory: configuration space Amer. Math. Monthly, v109, pg140-150, 2002. S. Chettih and D. Lütgehetmann. The Homology of Configuration Spaces of Graphs http://arxiv.org/abs/1612.08290, 2017 D. Farley. Presentations for the Cohomology Rings of Tree Braid Groups http://arxiv.org/abs/math/0610424, 2006.

Safia Chettih Topology of Configurations on Trees