Constructing Manifolds Lecture 3 - February 3, 2009-1-2 PM Outline Sets of gluing data The cocycle condition Parametric pseudo-manifolds (PPM s) Conclusions 2 Let n and k be integers such that n 1 and k 1 (or k = ). A set of gluing data is a triple G = ( ( ) i I, (j ) (i,j) I I, (ϕ ji ) (i,j) K ), where I and K are countable sets and I is non-empty, satisfying the following three properties: 3
(1) For every i I, the set is a non-empty open subset of called parametrization domain, for short, p- domain, and the are pairwise disjoint (i.e., Ω j = for all i j). Ω 2 Ω 3... 4 (2) For every pair (i, j) I I, the set j is an open subset of. Furthermore, i =, and Ω ji if and only if j. Each non-empty j (with i j) is called gluing domain. Ω 21 Ω 2 Ω 3 2 Ω 31... 3 5 (3) If we let then K = {(i, j) I I j }, ϕ ji :j Ω ji is a C k bijection for every (i, j) K, called a transition function or gluing function. 6
The transition functions tell us how to glue the p- domains. Ω 21 Ω 2 Ω 3 2 ϕ 21 ϕ 12 ϕ 31 Ω 31... ϕ 13 3 7 The transition functions must satisfy the following conditions: (a) ϕ ii = id Ωi, for all i I, ϕ ii = id Ωi 8 (b) ϕ ij = ji, for all (i, j) K, and ϕ ij (p) p ji (p) Ω j 9
(c) for all i, j, and k, if Ω ji Ω jk then ji (Ω ji Ω jk ) k and ϕ ki (x) =ϕ kj ϕ ji (x), for all x ji (Ω ji Ω jk ). Ω ji ji k Ω j Ω jk Ω ji Ω jk ji (Ω ji Ω jk ) 10 The evil cocycle condition ϕ ki (x) =ϕ kj ϕ ji (x), for all x ji (Ω ji Ω jk ). Ω ji j Ω j ϕ ji Ω jk ϕ kj Ω k x ϕ ki = ϕ kj ϕ ji k Ω ki Ω kj 11 The cocycle condition implies conditions (a) and (b): (a) ϕ ii = id Ωi, for all i I, and (b) ϕ ij = ji, for all (i, j) K. 12
The statement if Ω ji Ω jk then ji (Ω ji Ω jk ) k is necessary! Ω ji ji k Ω j Ω jk Ω ji Ω jk 13 ji (Ω ji Ω jk ) Things can go wrong if the condition is false... Consider the p-domains (i.e., open line intervals) =(0, 3), Ω 2 =(4, 5), and Ω 3 =(6, 9). Ω 2 Ω 3 0 1 2 3 4 5 6 7 8 9 R 14 Consider the gluing domains 2 =(0, 1) and 3 =(2, 3), Ω 21 =Ω 23 =(4, 5), and Ω 32 =(8, 9) and Ω 31 =(6, 7). 2 3 Ω 21 =Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 R 15
Consider the transition functions: ϕ 21 (x) =x +4, ϕ 32 (x) =x +4, and ϕ 31 (x) =x +4. 2 3 Ω 21 =Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 R ϕ 21 ϕ 31 ϕ 32 16 Obviously, ϕ 32 ϕ 21 (x) =x +8, for all x 2. 2 3 Ω 21 =Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 R ϕ 21 ϕ 32 Note that but Ω 21 Ω 23 =Ω 2 =(4, 5), 21 (Ω 21 Ω 23 )=(0, 1) (2, 3) = 3. 17 So, the statement is false! if Ω 21 Ω 23 then 21 (Ω 21 Ω 23 ) 3 2 3 Ω 21 =Ω 23 Ω 31 Ω 32 0 1 2 3 4 5 6 7 8 9 ϕ 31 R It turns out that ϕ 31 is undefined in 21 (Ω 21 Ω 23 ). 18
The question now becomes: Given a set of gluing data, G, can we build a manifold from it? Indeed, such a manifold is built by a quotient construction. We form the disjoint union of the and we identify j with Ω ji using ϕ ji, an equivalence relation,. We form the quotient ( ) M G = /,. i 19 Theorem 1 [Gallier, Siqueira, and Xu, 2008] For every set of gluing data, G = ( ( ) i I, (j ) (i,j) I I, (ϕ ji ) (i,j) K ), there is a n-dimensional C k manifold, M G, whose transition functions are the ϕ ji s. 20 REMARK: A condition on the gluing data is needed to make sure that M G is Hausdorff: (4) For every pair (i, j) K, with i j, for every x (j ) and every y (Ω ji ) Ω j, there are open balls, V x and V y centered at x and y, so that no point of V y Ω ji is the image of any point of V x j by ϕ ji. 21
Theorem 1 is very nice, but... Our proof is not constructive; M G is an abstract entity, which may not even be compact, orientable, etc. So, the question that remains is how to build a concrete manifold. Let us first formalize our notion of concreteness. 22 R m Big Picture θ 1 θ 2 ϕ 12 2 Ω 21 Ω 2 ϕ 21 23 Let n, m, and k be integers, with m>n 1 and k 1 or k =. A parametric C k pseudo-manifold of dimension n in R m is a pair M =(G, (θ i ) i I ), such that G = ( ( ) i I, (j ) (i,j) I I, (ϕ ji ) (i,j) K ) is a set of gluing data, for some finite I, and each θ i is a C k function, θ i : R m, called a parametrization, such that the following holds: 24
(C) For all (i, j) K, we have θ i = θ j ϕ ji. R m θ 1 θ 2 θ i (p) θ j ϕ 21 (p) ϕ 12 p 2 Ω 21 Ω 2 ϕ 21 25 The subset M = i I θ i ( ) of R m is called the image of the parametric pseudomanifold. 26 M R m θ 1 θ 2 ϕ 12 2 Ω 21 Ω 2 ϕ 21 27
When m =3and n =2, we say that M is a parametric pseudo-surface. Under certain conditions (which we shall see in the next slide), the image of a parametric pseudo-surface is a surface in R 3. 28 We proved that M can be given a manifold structure if we require the θ i s to be bijective and to satisfy the following conditions: (C ) For all (i, j) K, θ i ( ) θ j (Ω j )=θ i (j )=θ j (Ω ji ). (C ) For all (i, j) K, θ i ( ) θ j (Ω j )=. 29 Conclusions We can build a parametric pseudo-manifold (PPM) from a set of gluing data and, under certain conditions, the image of a PPM can be given the structure of a manifold. In the last lecture, we will describe a new constructive approach to define a set of gluing data from a triangle mesh. We also describe how to build a parametric C pseudosurface from the set of gluing data. The image of this parametric pseudo-surface approximates the vertices of the mesh. 30
Suggested Reading Gallier, J.; Chapter 3 - Construction of Manifolds from Gluing Data, Notes on Differential Geometry and Lie Groups. Download a PDF from the course web page: http://w3.impa.br/ lvelho/ppm09 31