Coarse Geometry 1 1 University of Connecticut Fall 2014 - S.i.g.m.a. Seminar
Outline 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Outline Basic Problem 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Introduction Basic Problem The idea behind coarse geometry is to look at spaces only at large scale, to neglect all local, innitesimal structure (topological or geometric) and consider only the geometric properties that are global. If (X,d) is a metrix space, then d (x,y) = min{1,d(x,y)} induces same topology. In Coarse geometry, induces same structure d (x,y) = max{1,d(x,y)}
Introduction Basic Problem The idea behind coarse geometry is to look at spaces only at large scale, to neglect all local, innitesimal structure (topological or geometric) and consider only the geometric properties that are global. If (X,d) is a metrix space, then d (x,y) = min{1,d(x,y)} induces same topology. In Coarse geometry, induces same structure d (x,y) = max{1,d(x,y)}
Introduction Basic Problem Coarse equivalence is the analogous form of a homeomorphism, where coarse properties are invariant. Why? Utility in work towards Baum-Connes Conjecture and Novikov conjecture.
Introduction Basic Problem Coarse equivalence is the analogous form of a homeomorphism, where coarse properties are invariant. Why? Utility in work towards Baum-Connes Conjecture and Novikov conjecture.
Introduction Basic Problem Coarse equivalence is the analogous form of a homeomorphism, where coarse properties are invariant. Why? Utility in work towards Baum-Connes Conjecture and Novikov conjecture.
Goal Basic Problem We rst describe some coarse invariant properties. We will show that the topological property of separability is not invariant under coarse equivalence. We will show that a certain nonseparable space is coarse equivalent to the separable space L p if and only if p = 1.
Goal Basic Problem We rst describe some coarse invariant properties. We will show that the topological property of separability is not invariant under coarse equivalence. We will show that a certain nonseparable space is coarse equivalent to the separable space L p if and only if p = 1.
Goal Basic Problem We rst describe some coarse invariant properties. We will show that the topological property of separability is not invariant under coarse equivalence. We will show that a certain nonseparable space is coarse equivalent to the separable space L p if and only if p = 1.
Outline 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Example Intuitively, X is coarse equivalent to Y if you zoom out far enough on a space X, there will be a point at which X starts to look similar to Y. The space(r, ) is coarse equivalent to (Z, ).
Denition of a Coarse Function Denitions Let X and Y be metric spaces, and let f : X Y be any map. (a) The map f is (metrically) proper if the inverse image, under f, of each bounded subset of Y, is a bounded subset of X. (b) The map f is (uniformly) bornologous if for every R > 0 there is S > 0 such that d X (x,y) < R d Y (f (x),f (y)) < S (c) The map f is coarse if it is proper and bornologous.
Example of a Coarse Function Example The function f : R + {0} R + {0} dened by f (x) = x is coarse.
Example of a function that is not coarse Example The function f : R\{0} R dened by f (x) = 1 x is not coarse.
Denition of a Denitions (a) Two maps f,f from a set X into a metric space Y is close if d (f (x),f (x)) is bounded uniformly in X. (b) We say metric spaces X and Y are coarse equivalent if there exists coarse maps f : X Y and g : Y X such that f g and g f are close to the identity maps on Y and on X, respectively.
Examples We show (R, ) is coarse equivalent to (Z, ). Dene F : R Z by F (x) = x and dene G : Z R by G(x) = x. Its easy to see that F,G are coarse. Note that F G (x) x = x x = x x = 0, and G F (x) x = x x 1 and so (R,d) and (Z,d) are coarse equivalent.
Examples Theorem The space Z n is coarse equivalent to R n.
Examples Example Let (X,d X ) be a metric space where the space is bounded. Let Y = ([0,M], ) where M = sup d X (x,y). x,y X Then X is coarse equivalent to Y. Example Let X be a bounded metric space and Y a nite space. Then X and Y are coarse equivalent. Example All nite spaces are coarse equivalent.
Examples Example Let (X,d X ) be a metric space where the space is bounded. Let Y = ([0,M], ) where M = sup d X (x,y). x,y X Then X is coarse equivalent to Y. Example Let X be a bounded metric space and Y a nite space. Then X and Y are coarse equivalent. Example All nite spaces are coarse equivalent.
Examples Example Let (X,d X ) be a metric space where the space is bounded. Let Y = ([0,M], ) where M = sup d X (x,y). x,y X Then X is coarse equivalent to Y. Example Let X be a bounded metric space and Y a nite space. Then X and Y are coarse equivalent. Example All nite spaces are coarse equivalent.
Theorem (Yu) Let G be a nitely generated group whose classifying space BG has the homotopy type of a nite CW-complex. If G has nite asymptotic dimension as a metric space with a word length metric, then the Novikov conjecture holds for G. Theorem (Yu )Let Γ be a discrete metric space with bounded geometry. If Γ admits a uniform (coarse) embedding into the Hilbert Space, then the Baum-Connes conjecture holds for Γ.
Examples: Groups as Metric Spaces Interesting examples involve dening metrics on nitely generated groups. Let G be a group and suppose Γ generates G ; that is every g G can be written as a word of members of Γ and their inverses; that is g = γ 1 γ 2 γ n and we let the smallest number n of generators that can be used to form g to be called the word length (relative to Γ), and write it as g.
Examples: Groups as Metric Spaces D n = r,s r n = 1,s 2 = 1,sr = r 1 s has Γ = { s,r,r 1}. It turns we can write the whole list as D n = { 1,r,r 2,...,r n 1,s,sr,...,sr n 1}. So the elements of D n have either word lengths of 1 to n. The group Z with Γ = { 1,1}. Note that the length of a word in Z is simply the absolute value. It is not too hard to show that d(g,h) = g 1 h denes a metric on G, known as the word metric. d(g,h) is the shortest length of a word w of G such that w = gh 1.
Examples: Groups as Metric Spaces Theorem Let Γ and Γ be two generating sets for the same group G ; and let d and d be the associated word metrics. Then the identity map (G,d) (G,d ) is a coarse equivalence. Thus any nitely generated group carries an intrinsic coarse geometry.
Invariant Coarse Geometric Properties Bounded Geometry: R > 0, N(r) > 0 such that #(B(x,r)) < N(r) for all x. Asymptotic dimension: Similar to Covering Dimension. Amenability: Property A
Invariant Coarse Geometric Properties Bounded Geometry: R > 0, N(r) > 0 such that #(B(x,r)) < N(r) for all x. Asymptotic dimension: Similar to Covering Dimension. Amenability: Property A
Invariant Coarse Geometric Properties Bounded Geometry: R > 0, N(r) > 0 such that #(B(x,r)) < N(r) for all x. Asymptotic dimension: Similar to Covering Dimension. Amenability: Property A
Invariant Coarse Geometric Properties Bounded Geometry: R > 0, N(r) > 0 such that #(B(x,r)) < N(r) for all x. Asymptotic dimension: Similar to Covering Dimension. Amenability: Property A
Invariant Coarse Geometric Properties Bounded Geometry: R > 0, N(r) > 0 such that #(B(x,r)) < N(r) for all x. Asymptotic dimension: Similar to Covering Dimension. Amenability: Property A
Outline 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Coarse version of the Lebesgue Covering Dimension of Topology Denition (Gromov) The asymptotic dimension of a metric space X is the smallest integer n such that for every R > 0, there exists a uniformly bounded cover U = {U i } i I of X so that (1) The diameters of U i s are bounded, (2) Each R ball intersects at most n + 1 members of the covering We write asdim X = n.
Calculating asdim Z To calculate asdim Z we let R > 0 and consider collection Taking R = 1 we have U = {[2Rn,2(n + 1)R)} n Z. U = {...,[ 4, 2),[ 2,0),[0,2),[2,4)}. Note that these sets are disjoint, with length 2R and hence uniformly bounded. Also note that a ball of radius R centered at any point x Z can intersect at most 2 elements of U. This shows asdim Z 1. We can also show that asdim Z 1 (harder), hence asdim Z = 1. Since Z is coarse equivalent to R, then asdim R = 1.
Z n Theorem If X and Y are coarse equivalent then asdimx = asdimy. One can prove that asdim Z n = n. By the theorem, the groups Z n and Z m are coarse equivalent i n = m.
Properties of asdim Let X,Y be metric space. If X Y then asdimx asdimy. Also, asdim X Y asdimx + asdimy. If X = X 1 X 2 then asdimx = max{asdimx 1,asdimX 2 }. this contrasts with classical dimension where dimx dimx 1 + dimx 2 + 1.
Finite asdim Calculating asdim can be hard. We are often concerned in knowing if a space has nite asymptotic dimension. There are several theorems that help us check if a given group as nite asdim.
Finite asdim Theorem Let G be a nitely generated group and N a normal nitely generated subgroup. Then asdimg (asdimn + 1)(asdimG/N + 1) 1. There are other theorems that relates conditions of Lipschitz maps to nite dimension.
Hyperbolic groups Hyperbolic groups have nite dimension. This gives us a several examples of spaces with asdimx <. Finitely generated free groups Hyperbolic plane. The fundamental groups of surfaces with genus two. (e.g. double torus) Z 2 is not hyperbolic.
Innite Dimension Let Z = i=1 Z equipped with metric d (x,y) = i=1 where x,y Z. Then asdimz =. Dene f n : Z n Z by i x i y i, f (x 1,x 2...,x n ) = (x 1,x 2...,x n,0,0...). The map is a quasi-isometry (think homomorphism). Its image is coarse equivalent to Z n. But note that by a previous property asdimz asdimz n = n, for all n.
Outline 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Motivation Theorem (Yu )Let Γ be a discrete metric space with bounded geometry. If Γ admits a uniform (coarse) embedding into the Hilbert Space, then the Baum-Connes conjecture holds for Γ.
Denition Let X,Y be metric spaces. A function f : X Y is a coarse embedding if there exists non-decreasing functions ρ 1,ρ 2 : [0, ) [0, ) satisfying (1) ρ 1 (d X (x,y)) d Y (f (x),f (y)) ρ 2 (d X (x,y)) for all x,y X, (2) lim t ρ 1 (t) = +.
Theorem The spaces X and Y are coarse equivalent if and only if there exists a coarse embedding f : X Y such that for each y Y there exists x X such that d Y (f (x),y) C for some C > 0 independent of y and x. Note that the second condition is a coarse version of being surjective. Sometimes this is an easier condition to work with when proving coarse equivalence directly.
Theorem Coarse equivalence is an equivalence relation. Proof. Transitivity is the only nontrivial property. Suppose X coarse Y then let f,ρ 1,ρ 2 be as in the denition. Suppose Y coarse Z then let g,φ 1,φ 2 be as in the denition. Not too hard to see that φ 1 (ρ 1 (d X (x,y))) d Z (g (f (x)),g (f (y))) φ 2 (ρ 2 (d X (x,y))) for all x,y X. Then h : X Z dened by h = g f is a coarse embedding with φ 1 ρ 1 and φ 2 ρ 2 as its associated non-decreasing functions.
Proof. Now show the coarse version of surjective. Take z Z, there exists y z such that d Y (g (y z ),z) C 1. There exists x X such that d (f (x),y z ) C 2. Then d Z (g f (x),z) d Z (g f (x),g (y z )) + d Z (g (y z ),z) φ 2 (d Y (f (x),y z )) + C 1 φ 2 (C 2 ) + C 1.
Outline The partition space P ([a,b]). Preliminaries Main Result 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Goal The partition space P ([a,b]). Preliminaries Main Result Using function spaces over [a, b], we show Separability is not Coarse Invariant. We rst construct the Nonseparable space P ([a, b]).
Denitions The partition space P ([a,b]). Preliminaries Main Result Denition A partition P of [a,b] is an ordered set P = {a = x 0,x 1,...x n 1,x n = b} such that a = x 0 x 1 x n 1 x n = b Denition A partition function f P of an interval [a,b], is a function f P : P R where P is a partition of [a,b]. We set the Partition Space P ([a,b]) to be the set of all partition functions f P : P R on [a,b].
Denitions The partition space P ([a,b]). Preliminaries Main Result Denition A partition P of [a,b] is an ordered set P = {a = x 0,x 1,...x n 1,x n = b} such that a = x 0 x 1 x n 1 x n = b Denition A partition function f P of an interval [a,b], is a function f P : P R where P is a partition of [a,b]. We set the Partition Space P ([a,b]) to be the set of all partition functions f P : P R on [a,b].
Denitions The partition space P ([a,b]). Preliminaries Main Result Denition A partition P of [a,b] is an ordered set P = {a = x 0,x 1,...x n 1,x n = b} such that a = x 0 x 1 x n 1 x n = b Denition A partition function f P of an interval [a,b], is a function f P : P R where P is a partition of [a,b]. We set the Partition Space P ([a,b]) to be the set of all partition functions f P : P R on [a,b].
The function Lin The partition space P ([a,b]). Preliminaries Main Result We dene the the linear interpolating operator. Denition Let f P be a partition function on [a,b]. Dene Lin : P ([a,b]) R 1 by { fp (x i ),x = x i Lin(f P (x)) := f P (x i ) f P (x i 1 ) x i x i 1 (x x i 1 ) + f P (x i 1 ),x (x i 1,x i ) Theorem The function Lin (f P ) is coarse.
A Partition function The partition space P ([a,b]). Preliminaries Main Result The partition function f P
The Lin function The partition space P ([a,b]). Preliminaries Main Result The function Lin(f P ).
The metric space P ([a,b]) The partition space P ([a,b]). Preliminaries Main Result Theorem Dene d P : P ([a,b]) P ([a,b]) R for f P1,g P2 P ([a,b]) as d P (f P1,g P2 ) = { b a Lin (f P 1 (x)) Lin (g P2 (x)) dx,p 1 = P 2 b a Lin (f P 1 (x)) Lin (g P2 (x)) dx + 1,P 1 P 2 then (P ([a,b]),d P ) is a metric space.
Separability The partition space P ([a,b]). Preliminaries Main Result Denition A metric space X is said to be separable if it contains a countable dense subset. Theorem P ([a,b]) is not separable. Dene the partition function g P (x) = 1 where P is a partition, and set A = {g P P is a partition of [a,b]}. Let D be a dense subset. For each g P A the ball of radius 1 centered at g 2 P intersects D\{g P }. Since each g P is a distance of 1 apart, this implies that D is uncountable. P ([a,b]) does not have the discrete topology.
Outline The partition space P ([a,b]). Preliminaries Main Result 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
The partition space P ([a,b]). Preliminaries Main Result We dene R 1 ([a,b]) to be the set of Riemann Integrable functions on [a, b]. We recall from analysis that for a partition P of [a,b] and function f : [a,b] R we have Theorem U(f,P) = n i=1 M i x i and L(f,P) = n i=0 m i x i. where M i,m i are the sup's and inf's of respective intervals. R 1 ([a,b]) is separable.
Outline The partition space P ([a,b]). Preliminaries Main Result 1 Motivation 2 3 The partition space P ([a, b]). Preliminaries Main Result 4
Main Result The partition space P ([a,b]). Preliminaries Main Result We prove that there exists a nonseparable space coarse equivalent to a separable space. Theorem (P ([a,b]),d P ) is coarse equivalent to ( R 1 ([a,b]), 1 )
Overview of Proof The partition space P ([a,b]). Preliminaries Main Result The two coarse functions F,G we use that are needed to show a coarse equivalences are the following: Dene F : P ([a,b]) R 1 ([a,b]) as F (f P ) = Lin(f P ). F is coarse by a previous theorem. Let ε > 0. Dene G : R 1 ([a,b]) P ([a,b]) as G(f ) = f (x), x P N where P n = { a + b a n i} n and i=0 N = min{n N U(f,P n ) L(f,P n ) < ε}. G is well dened since N is well ordered and by Riemann Criterion.
Proof The partition space P ([a,b]). Preliminaries Main Result We see that F G (f ) is close to f R 1.
Denition of L p A generalization of R ([a,b]). We set L p (X ) = {f : X C f measurable f L p < } where f L p = ( X ) 1/p f p dm where m is the lebesgue measure and X R.
Results Theorem L p ([a,b]) is separable for 1 p <. Theorem R p ([a,b]) is coarse equivalent to L p ([a,b]). Theorem P ([a,b]) is coarse equivalent to L p ([a,b]) if and only if p = 1.
Results Theorem L p ([a,b]) is separable for 1 p <. Theorem R p ([a,b]) is coarse equivalent to L p ([a,b]). Theorem P ([a,b]) is coarse equivalent to L p ([a,b]) if and only if p = 1.
Results Theorem L p ([a,b]) is separable for 1 p <. Theorem R p ([a,b]) is coarse equivalent to L p ([a,b]). Theorem P ([a,b]) is coarse equivalent to L p ([a,b]) if and only if p = 1.
Summary Basic Intro into Coarse Geometry and some of its properties. We showed the existence of non-separable space that is coarse equivalent to a separable space. Hence the topological property of separability is not coarse invariant. Outlook Necessary and sucient conditions that allows a space X to be coarse equivalent to L p (R)?
Novikov Conjecture Let M be a closed oriented n dimensional smooth manifold with a map f : M BG for some discrete group G and let α H n 4 (BG;Q) be a rational cohomology class. The higher signature of M dened by (f,α) is the rational number σ α (M,f ) = L M f α,[m] Q where L M H 4 (M;Q) is the Hirzebruch L-class of M. Let h : N M be a homotopy equivalence of closed oriented smooth manifolds. The Novikov conjecture states σ α (N,f h) = σ α (M,f ) for all G,f,α and for all homotopy equivalences h : N M.
Baum-Connes Conjecture The Baum-Connes conjecture implies the Novikov conjecture. Let Γ be a second countable locally compact group. One can dene a morphism µ Γ i : RK Γ i (EΓ) K i (C λ (Γ)), called the assembly map, from the equivariant K -homology with Γ-compact support of the classifying space of proper actions E Γ to the K-theory of the reduced C -algebra of Γ. The idex can be 0 or 1. Then the conjecture says that the assemply map µ Γ i is an isomorphism.
Boo!!
References P. Mariano., Rose-Hulman Undergraduate Math Journal Vol. 14, No. 2 (2013). P. Nowak. of Metric Spaces into Banach Spaces Proceedings of the American Mathematical Society, Vol. 133, No. 9 (Sep., 2005), pp. 2589-2596. J. Roe. Lectures on Coarse Geometry American Mathematical Society, 2003. P Nowak and G. Yu. Large Scale Geometry Lecture notes, 2011.