Lecture notes for Topology MMA100 J A S, S-11 1 Simplicial Complexes 1.1 Affine independence N A collection of points v0; v1; : : : ; vn in some Euclidean space R are affinely independent if the (affine hyper-) plane of least dimension containing them has dimension n. Synonymously, the points are in general or generic position. v3 This is equivalent to v1 − v0; v2 − v0; : : : ; vn − v0 being linearly independent 3 v vectors (or similarly with any vi replacing v0). Thus e.g. four points in R 2 can be in generic position (but not five). P v0 Another wayP to express genericity is to say that tivi = 0; where ti are v1 scalars with ti = 0; only if all ti = 0. Yet another criterion is to use the embedding RN ,! RN+1 given by adding R3 the scalar 1 as a first coordinate: x 7! (1; x). Then v ; v ; : : : ; v are in generic Figure 1: Four points in 0 1 n in general position position iff (1; v0); (1; v1);:::; (1; vn) are linearly independent vectors. Obviously, if v0; v1; : : : ; vn are linearly independent as vectors they are also affinely independent as points (considering vectors as position vectors for points). P P An point p is an affine combination of the points v0; v1; : : : ; vn if p = tivi; where ti = 1. If the points are in generic position, the affine coordinates ti are uniquely determined by p. The (n-dimensional) plane spanned by points v0; v1; : : : ; vn consists of all affine combinations of them. 3 E.g. the (two-dimensional) plane spanned by three points v0; v1; v2 in R consist of all points x = t0v0 + t1v1 + t2v2; with t0 + t1 + t2 = 1; or differently put: x = v0 + t1(v1 − v0) + t2(v2 − v0); with no condition on the scalars. This is a standard parametric equation for the plane. 1.2 Convex closure As you know a subset of RN is convex if whenever it contains two points p and q it contains the whole line segment between them. This is parametrized by (1 − s)p + sq; where s 2 [0; 1]: It follows directly that the intersection of any number of convex sets is again convex. Consequently, for any subset A ⊂ RN there is a smallest convex set containing it: the intersection of all convex sets containing it. This is the convex closure, co(A) of A. P RN 2 An affine combinationP tipi of points p0; p1; : : : ; pn in ; is a convex combination if all ti [0; 1] (in addition to ti = 1). An easy induction on n shows that if a0; a1; : : : ; an are points in A; and C is a convex set containing A; then any convex combination of the points is again in C. In particular the applies to C = co(A). It's equally easy to see that if p and q are convex combinations of points in A; then so is any (1 − s)p + sq; where s 2 [0; 1]. Thus, co(A) can be described intrinsically as the set of all convex combinations of points in A. 1 1.3 Simplexes N n N An n-simplex in R ; denoted σ = σ; is the convex closure of a set of n + 1 points v0; v1; : : : ; vn 2 R in generic position. It will be denoted by hv0; v1; : : : ; vni; a notation that will (hopefully) only be used under the proviso that the points are affinely independent. It's natural to ask if hv0; v1; : : : ; vni = hw0; w1; : : : ; wmi is possible without n = m and w0; w1; : : : wn being some reordering of v0; v1; : : : ; vn. The answer is no, as will be demonstrated later on. To wit, a simplex is the convex closure of exactly one set of affinely independent points. This observation makes it clear the vertexes can be used to define properties and derive concepts of simplexes, as is done bellow. P P Thus σ consist of all points p that can be (uniquely) expressed as p = tivi; where ti = 1 and all t1 2 [0; 1]. In this connection the scalars ti are called the barycentric coordinates of p and the vi:s are vertexes of σ. The dimension of σ is one less than the number of vertexes, and is thus the same as the dimension of the smallest (affine hyper) plane containing them. A simplex of dimension 0 is hence the same as a single vertex (or point). We allow the set of vertexes to be empty, so that the empty set, ;; is simplex of dimension −1. The barycenter of σ; or the center of gravity, is the point 1 1 1 1 (v + v + ··· + v ) = v + v + ··· + v 2 σ: n + 1 0 1 n n + 1 0 n + 1 1 n + 1 n Any (proper) subset of the vertexes spans a simplex τ which is a (proper) face of σ; written τ ≤ σ (or τ < σ). The boundary of a simplex σ; denotedP by @σ; is the union of all its proper faces. In terms of convex combinations it consists of all such tivi; where at least one coordinate is 0. Beware that this is generally not the topological boundary of σ as a subset of some RN . P ◦ The interior σ of a simplex σ is σ − @σ: This is the set of all tivi; where all ti > 0. This is generally not the same as the interior of σ in a containing Euclidean space. Lemma 1.1 A simplex of dimension > 0 is (homeomorphic to) the cone on its boundary. Proof Let σ = hv0; v1; : : : ; vni denote the simplex and b its barycenter. Define σ v3 f : @σ ×I ! σ by f(p; t) = (1−t)p+tb: The target is correct by the convexity of σ and f is continuous since it is the restriction of the continuous map t b v2 RN × ! RN I ; given by the same formula. Obviously, f maps the top of the q v0 cylinder on @σ to b. The usual compact-Hausdorff argument gives that, in p ∼ order to prove that f induces a homeomorphism C(@σ) = σ; we are left with v1 checking that f restricts to a bijection @σ × [0; 1) ! σ − fbg: Figure 2: Polar coordinates First note that if q =6 b is a point in σ and q = tb + (1 − t)p; where 0 ≤ t < 1 for q and p 2 @σ; then t=(n + 1) is the minimal barycentric coordinate of q; as some coordinate of p is 0: This shows the uniqueness of t and hence of p; so the restriction of f is injective. On the other hand with q as above, let t¯ < 1=(n + 1) be the minimal barycentric coordinate of q. Write t^= (n + 1)t¯ < 1: Then X X ti − t¯ q = (n + 1)tb¯ + (ti − t¯)vi = tb^ + (1 − t^) vi = tb^ + (1 − t^)p: 1 − t^ i i Notice that the sum of the barycentric coordinates of p is 1; and that at least one of them is 0; so p is a point in @σ: Hence the restriction of f is surjective. If q 2 σ is q = tp + (1 − t)b; with p 2 @σ and 0 ≤ t ≤ 1; then (t; p) are the polar coordinates for q. They are unique, with the exception that (0; p) corresponds to b for any p 2 @σ The standard n-simplex ∆n in Rn+1 is the convex closure of the standard basis n+1 of R ; which in this connection is enumerated e0; e1; : : : ; en. Explicitly, this e2 ∆2 e 2 1 e0 Figure 3: The standard 2- simplex sits in R3 means that X n ∆ = f(t0; t1; : : : ; tn) j ti = 1 and ti 2 [0; 1]; for all ig: Lemma 1.2 An n-simplex is homeomorphic to the standard n-simplex hence compact. P n Proof Define f : ∆ ! hv0; v1; : : : ; vni by f(t) = tivi. Then f is continuous. In fact, if v0; v1; : : : vn 2 Rm the same formula gives a continuous function Rn+1 ! Rm: The map f is obviously a bijection, ∆n compact and hv0; v1; : : : ; vni Hausdorff, so f is a homeomorphism. Lemma 1.3 An n-simplex (n ≥ 1) is homeomorphic to the cone on any of its n − 1-dimensional faces. Proof Let σ be an n+1-simplex with vertexes v0; v1; : : : ; vn and τ the face with the same vertexes except vn: Define a map f : τ × I ! σ by f(p; t) = (1 − t)p + tvn. It's injective, by uniqueness of barycentric coordinates, with the exception that f(p; 1) = en; for all p. All we need to check to see that f induces a homeomorphism on from the cone on τ to σ is the surjectivity of fj : τ × [0; 1) ! σ − fvng: P If q = tivi 2 σ and p =6 vn; i.e. tn =6 1; then X ti q = tnvn + (1 − tn) vi = tnvn + (1 − tn)p: 1 − tn i=6 n Notice that the (positive!) barycentric coordinates of p has sum 1 and at least one is 0: Thus p 2 τ and f(p; tn) = q: Lemma 1.4 An n-simplex is homeomorphic to an n-disk by a homeomorphism respecting the "bound- aries" of the two spaces. Proof It's enough to show this for the standard n-simplex ∆n: This simplex is the graph of the map n n n n f : T ! R; f(t0; : : : ; tn−1) = 1 − Σiti over the set T ⊂ R consisting of those t 2 R with each n n coordinate ti 2 [0; 1] and Σiti ≤ 1.