Euler Characteristic

Transcription

1 Euler Characteristic Beifang Chen September 2, Euler Number There are two rules to follow when one counts the number of objects of finite sets. Given two finite sets A, B, we have (1) Addition Principle A B = A + B A B ; (2) Multiplication Principle A B = A B. These two rules are necessarily true if A, B are not finite sets. For instance, let A be a set of disjoint union of two disks and a zero-shape of R 2, and B be a set of disjoint union of a disk and the same zero-shape of R 2, see Figure 1. Notice that A = 3, B = 2, A B = 4, A B = 2. If the addition principle is true, then 4 = = 3 is a contradiction. However, if we think of the contribution of the zero-shape is zero to the counting of A B (the zero-shape like the symbol 0), then the addition principle is still valid. Indeed, there is a generalized counting measure χ, which is the counting for finite sets and the Euler number for topological spaces (precisely the Euler characteristic with compact support), and is a unified counting for both discrete and continuous sets. A Figure 1: Three disks and a zero-shape. Given a collection S of sets containing the empty set, closed under intersection and union, that is, if A, B S then A B, A B S. A set function ν on S with values in an abelian group is said to be a finitely additive measure if ν( ) = 0, ν(a B) = ν(a) + ν(b) ν(a B), A, B S. Since we do not consider measures in the sense of analysis (requiring countable additivity), we call finitely additive measures simply as valuations. To construct a valuation we usually begin with a class of sets, closed under intersection, known as intersectional class. 1 B

2 Let I be an intersectional class with the empty set. A set function ν on I with values in an abelian group is said to be a valuation if ν( ) = 0 and whenever A, B, A B I, ν(a B) = ν(a) + ν(b) ν(a B). In the following we consider a special case of intersectional class of coordinate parallelotopes. A closed interval is a set I = [a, b], where a, b are real numbers and a b. A coordinate box (or paralleotope of R n is a set P = n i=1 I i, where each I i is a closed interval. We denote by Par(n) the class of all coordinate parallelotopes of R n. Let B(Par(n)) denote the class of sets which are obtained by taking unions, intersections, and relative complement finitely many times, called the relative Boolean algebra generated by Par(n). Consider a translation invariant valuation ν : B(Par(1)) R[t] defined for closed intervals I = [a, b] by µ(i) = (b a)t + 1, where t is a real variable. There exists a unique valuation ν : B(Par(n)) R[t] such that n ( n ) ν(i n ) = µ(i) µ(i) = (b a) i t i. }{{} i n i=0 Writing ν(i n ) = n µ i (I n )t i, where µ i : B(Par(n)) R are translation invariant valuations defined by ( n ) µ i (I n ) = (b a) i. i i=0 To exclude possible pathological cases that we have no interest, we require the translation invariant valuation ν on B(Par(1)) to be continuous. A valuation ν on B(Par(n)) is said to be continuous if lim k ν(i k ) = ν(i) for a sequence I k = [a k, b k ] of closed intervals convergent to a closed interval I = [a, b], i.e., lim k a k = a and lim k b k = b. Note that each object P of B(Par(1)) is a disjoint union of finitely many singletons and open intervals, that is, P = k {a i } i=1 Then χ : B(Par(1)) Z, defined by l (b j, c j ) (disjoint union) j=1 χ(p) = k l, is a translation invariant valuation. Another translation invariant valuation on B(Par(1)) is the length function. Any linear combination of the two valuations is a valuation. The valuation ν, defined on B(Par(n)), can be extended to the class U(K n ) of finite union of compact convex sets of R n. In fact, if K is a compact convex set and B is the unit ball of R n then ν i can be extended to U(K n ) such that n vol n (K + rb) = ν i (K)ω n i r n i, (1) i=0 where K + rb = {x + ry : x K, y B}, r 0, and ω i is the volume of the i-dimensional unit ball. The identity (1) is called the Steiner formula. 2

3 Exercise 1. (a) Show that χ is a translation invariant valuation on B(Par(1)). (b) Show that every member P of B(Par(n)) is a disjoint union of finitely many relatively open coordinate parallelotopes i, b i ) i I(a {c j }, j J where I J = and I J = {1,..., n}. A subset S of R n is said to be convex if tx + (1 t)y S for x, y S and 0 t 1. Let K be a compact convex set of R n, called convex body. We denote by K the affine linear space that K spans. The interior of K in K is called the relative interior of K, denoted K ; the boundary of K is the set K := K K. A polytope of R n is a convex body that can be written as the intersection of finitely many half-spaces. A half-space of R n is a subset of the form H(φ a) = {x R n : φ(x) a}, where φ is a linear functio on R n and a R; its interior is called an open half-space, denoted H(φ < a). The intersection of finitely many half-spaces is called a convex polyhedron; its interior in its affine span is called a relatively open convex polyhedron. A bounded convex polyhedron is called a polytope, its int. It is well known that each polytope is the convex hull of some finite number of points (often adopted as definition of polytope). The classes of convex bodies and polytopes in R n are denoted by K n and P n respectively. We shall see that the counting measure χ (Euler characteristic) can be defined on the relative Boolean algebra B(K n ) generated by convex bodies of R n. Exercise 2. Show that a subset P of R n is a polytope if and only if P is the convex hull of finite number of points. Fix two real numbers p, q such that p+q = 1. For each real-valued function f on R with bounded support, continuous everywhere except discontinuous at finite number of places, we define the formal integral χ(f) = f(x)dχ(x) = [ f(x) pf(x ) qf(x + ) ], (2) x R called the Euler-Schanuel integral of f, where f(x ) and f(x + ) denote the left and right limits of f at x. The class of Euler-Schanuel integrable functions forms a real vector space. Each delta function δ a, defined by δ a (x) = 1 if x = a and δ a (x) = 0 otherwise, is Euler-Schanuel integrable and χ(δ a ) = 1. The indicator function 1 (a,b) of an open interval (a, b) is Euler- Schanuel integrable and χ(1 (a,b) ) = 1. Since each step function is a linear combination of some delta functions and indicator functions of some open intervals, all step functions are Euler-Schanuel integrable. More generally, for f a real-valued function on R n with bounded support, we define the the Euler-Schanuel integral of f as the following iterated integral χ(f) = f(x)dχ(x) = f(x 1,...,x n )dχ(x 1 ) dχ(x n ), (3) which is well-defined provided that the resulted integrand from each iterated integral is continuous, except at finite number of places and has left and right limits at those discontinuous places. Again the class of Euler-Schanuel integrable multivariable functions is a real vector space. 3

4 Let U be a relatively open convex set, the relative interior of a convex body. The image π(u) under the orthogonal projection π : R R n 1 is also a relatively open convex set. The inverse image π 1 (y) for each y π(u) is either a singleton or an open interval. If π 1 (y) for one y π(u) is a singleton, then dimπ(u) = dim U. If π 1 (y) for one y π(u) is an open interval, then dimπ(u) = dim U 1. For each y π(u), we have f(y, x n )dχ(x n ) = ( 1) dim U dim π(u). Hence fdχ(x n ) = ( 1) dim U dim π(u) 1 π(u). By induction on dim U, we obtain the following proposition. Proposition 1.1. Let U be a relative open convex set of R n. Then χ(1 U ) = ( 1) dim U. (4) A convex polyhedral set of R n is an intersection of finitely many half-spaces. A polyhedron of R n is a subset which can be obtained from half-spaces by taking union, intersection, and relative complement finitely many times. Every bounded polyhedron P can be written as a disjoint union of relatively open convex polytopes P i so that 1 P = i 1 Pi. Applying χ to both sides, we obtain the following theorem. Theorem 1.2. Let P be a bounded polyhedron decomposed into disjoint relatively open convex polytopes, and denote by α i the number of i-dimensional relatively open polytopes. Then χ(p) = i ( 1) i α i. (5) Example 1.3. The Euler number of an oriented surface Σ g of genus g in R 3 is χ(σ g ) = 2 2g. Assume that Σ g is standardly sitting in R 3 and its orthogonal projection to R 2 is the figure in Figure 2 with g = 3. Taking the Euler-Schanuel integral of the indicator function 1 Σ3 of Σ 3 along the last coordinate x 3, the result is a function f(x 1, x 2 ) on R 2, whose level set is indicated in Figure 2. Taking the Euler-Schanuel integral of f along the coordinate x 2, we Figure 2: Projection of Σ 3 to R 2. obtain a function h(x 1 ) on R whose graph is given in Figure 3. Taking the Euler-Schanuel integration for h, we see that the Euler number of Σ 3 is χ(1 Σ3 ) = 2 6 = 4. The general case of χ(σ g ) = 2 2g is analogous. 4

5 1 2 Minkowski Algebra Figure 3: Projection of Σ 3 to R. Let P d denote the class of polytopes of R d, and Q d the class of all closed convex polyhedral sets of R d. We denote by U(P d ) the class of all possible finite unions P i with P i P d. Let B(P d ) denote the relative Boolean algebra generated by P d, i.e., the members of B(P d ) are obtained from the members of P d by taking union, intersection, and relative complement finitely many times. Let F(P d ) denote the vector space of all positive finite linear combinations a i I Pi, where a i R, P i P d, and I Pi is the indicator function of polytope P i. It is well-known that there is a linear functional χ : F(P d ) R, called the Euler characteristic, such that χ(i X ) = χ(x) for each X B(P d ). Notice that χ(p) = 1 and χ( P) = ( 1) dim P for each polytope P. Let C d denote the class of all convex closed polyhedral cones of R d, and let F(C d ) denote the class of functions that are linear combinations of indicator functions of relatively open convex cones. There is a linear involution N : F(C d ) F(C d ), N(f)(u) := χ(f I H u ), (6) where H u = {v Rd : u, v 0} is a half-space of R d if u 0 and H u = Rd if u = 0. For each relatively open convex cone C C(V ), its normal cone is C = {u R d : u, v 0 for all v C}. Lemma 2.1. For each relatively open convex cone C P d, N(I C ) = ( 1) dim C I C. Proof. The map N is well-defined and linear since χ is a linear functional. Given a relatively open convex cone C. For each vector u C, it is clear that C H u, thus N(I C )(u) = χ(c H u ) = χ(c) = ( 1)dim C. For u C, we have either C H u = or C H u ; the set C H u in latter case is a half-closed and half-open convex cone; hence N(I C )(u) = χ(c H u ) = 0. Let σ be a relatively open convex polyhedral set of a vector space V. For each face ρ of σ, the tangent cone of σ at ρ is the relatively open convex cone cone(σ, ρ) = {v V : p ρ, ε > 0 s.t. p + tv σ, 0 < t < ε}, and the tangent cone indicator of σ at ρ is the function T ρ (σ) = ( 1) dim ρ I cone(σ, ρ). The tangent cone of σ at is the relatively open convex cone cone(σ, ) = { v V : v 0, p σ s.t. p + tv σ, t > 0}, 5

6 and the tangent cone indicator of σ at is the function T (σ) = ( 1) dim σ 1 I cone(σ, ). If σ is bounded, it is clear that the tangent cone cone(σ, ) is empty so that T (σ) = 0. Let P be a polyhedron of a Euclidean space E d, possibly non-convex and unbounded. A face system of P is a decomposition of its closure P into a finite polytope cell complex F(P), that is, each member of F(P) is a relatively open convex polyhedron (allowed to be unbounded), ρ σ with ρ, σ F(P) implies ρ σ (written ρ σ), and P = σ (disjoint). The tangent cone of P at a face ρ is the tangent indicator of P at ρ is σ F(P) cone(p, ρ) := and the tangent indicator of P at is σ F(P) T ρ (P) := σ F(P) T (P) := σ F(P) cone(σ, ρ), T ρ (σ), T (σ). Let f be a polyhedron function on E d. A face system of f is a face system of supp f. The tangent indicator of f at its face ρ with respect to a face system F(f) is T ρ (f) := f(σ)t ρ (σ), σ F(f) where f(σ) = f(x) for any x σ, and the tangent indicator of f at is T (f) := σ X f(σ)t (σ). One can define tangent cone and tangent indicator in a more intrinsic way for a polyhedron function on E d. For each vector v R d, we associate a function Let f, g F(P d ). It is clear that for x E d and v R d, f v (x) = lim f(x + tv). (7) t 0 + (af + bg) v (x) = af v (x) + bg v (x). It is easy to see that f v is also a polyhedron function on E d. The tangent cone of f at x may be defined as cone(f, x) := {v R d f v (x) 0}. The tangent indicator of f at x is the function T x (f) : R d R defined by T x (f)(v) := f v (x). 6

7 So we have the tangent function T(f) : E R R, (x, v) f v (x), and the tangent functor T : F(C d ) E d F(C d ), f T(f). The tangent indicator of f at is defined by T (f)(v) := lim f(x tv) t + if v 0 and the half-line x + R( v) is contained in supp f, otherwise T (f)(v) = 0. Lemma 2.2. The map T(f) : E d F(C d ), x T x (f), is piecewise constant. So T(f) is integrable with respect to the Euler measure χ. Proof. Let F(f) be a face system of f. Then T ρ (f) = T x (f) for each ρ F(f) and all x ρ. In fact, T σ (f)(v) = χ(i σ f v ) for all v R d and T ρ (I σ ) = ( 1) dim ρ I cone(σ, ρ) for ρ, σ F(f) with ρ σ. Theorem 2.3. For each polyhedron function f on E d, E d T x (f)dχ(x) = χ(f)i {0}, where I {0} is the indicator function of the singleton {0} of the origin. In particular, if F(f) is a face system of f, then ρ X T ρ (f) = T (f) + d T i (f, X) = χ(f)i {0}. i=0 Proof. First Proof: For each nonzero vector v R d, we compute the integral of T x (f)(v) with respect to the Euler measure χ. Notice that T x (f)(v) = f v (x) and T x (f)(v)dχ(x) = f v (x)dχ(x). E d E d By definition of f v (x), the support of f v on p + Rv is a disjoint union of finitely many half-closed and half-open intervals, and f v is constant on each of such intervals. In order to compute the above integral with respect to χ, choose frame (v 1,...,v d ) of E d so that v 1 = v. We see that f v (x 1 v x d v d )dχ(x 1 ) = 0. It follows that f v (x)dχ(x) = E d f v (x 1 v x d v d )dχ(x 1 ) dχ(x d ) = 0. If v = 0, then f 0 (x) = f(x). Thus f 0 (x)dχ(x) = χ(f). The proof is finished. Second Proof: Since f = f(σ)i σ, it suffices to prove the case of I σ for each σ F(f). In fact, (I σ ) v (x) = 1 if and only if v cone(σ, x), that is, σ (x + Rv) is an open segment. Thus if v 0, the support of (I σ ) v is the union of σ and a cell of dimension dim σ 1; so χ((i σ ) v ) = 0. If v = 0, then (I σ ) v = I σ ; hence χ((i σ ) 0 ) = χ(i σ ). Therefore f v (x)dχ(x) = f(σ) (I σ ) v (x)dχ(x) E d σ E d = σ f(σ)χ(i σ )I {0} = χ(f)i {0}. 7

8 Let f F(Q ) be a polyhedron function with a face system F(f). The normal indicator of f at a point x R d is K x (f) := N(T x (f)), (8) or alternatively for each v R d, K x (f)(v) := lim χ ( ) f I r 0 x+ B d (r) H (v), where x E d ; K (f)(v) := lim lim χ ( ) f I t r 0 x+tv+ Bd (r), where x + Rv supp f. The normal indicator of f at a face ρ F(f) { } is K ρ (f) := N(T ρ (f)). Theorem 2.4. For each polyhedron function f, K x (f)dχ(x) = χ(f)i R d. If F(f) is a face system of f, then ρ F(f) K ρ (f) = χ(f)i R d. Proof. It is trivial when v = 0. Assume v 0. Let π : R d L and π v : R d L be the orthogonal projections, where L = Rv. Let X be decomposed into frame cells σ i with respect to a frame, where v is a member of the frame. For the cell σ i, we may further assume that σ i π 1 (a i+1 ) is a closed frame cell; otherwise we may divide σ i πv 1(a i+1) so that. Thus K x (σ i )(v) = χ(σ i ) if x πv 1(a i+1) and K x (σ i )(v) = 0 otherwise. Then K x (σ i )(v)dχ(x) = χ(σ i ). Hence K x (X)(v)dχ(x) = K x (σ i )(v)dχ(x) = χ(σ i ) = χ(x). π : R d L be the orthogonal projection, where L = Rv. Then R can be divided into finite singletons {a i } and open intervals (a i, a i+1 ) so that π 1 (a i, a i+1 ) (a i, a i+1 ) π 1 (a), where a (a i, a i+1 ) is a fixed number. Let π 1 (a) be decomposed into disjoint cells σ ij. Then K x (σ ij )(v) = 0 for x π 1 (a i, a i+1 ) and K x (σ ij )(v) = ( 1) dim σ ij for x σ ij π 1 (a i+1 ). Proposition 2.5. Let f be a function on R d. If χ(f, u) = χ(f, u σ ) for all permutations u σ = (u σ(1),...,u σ(d) ) of each frame u = (u 1,...,u d ), then χ(f, u) = χ(f, v) for all frames u, v. Proof. Fix a frame u = (u 1,..., u d ); every frame v can be obtained from u by a non-singular n-by-n matrix A, that is, v = ua. Note that A can be decomposed into a product of some (column) elementary matrices. Let E be an elementary matrix having the ith column multiplied by a nonzero constant a, that is, v = ue = (u 1,...,au i,..., u d ). It is obvious that χ(f, u) = χ(f, v). Consider the frame v = (u 1,...,u i,...,u j + au i,..., v d ) with a 0. Then g := fdχ(u 1 ) dχ(u i ) dχ(u j ) dχ(u d ) 8

9 is a well-defined function on span{u i, u j }. Note that g dχ(u i )dχ(u j ) = g dχ(u i )dχ(u j + au i ). It follows that χ(f, u) = χ(f, v). The following example shows that the iterated Euler-Schanuel integral depends on the frames (ordered bases) of a basis. Example 2.6. Consider the function f(x, y) = { y 2 if x 2 + y 2 < 4, 0 otherwise. Its two iterated Euler-Schanuel integrals with respect to the standard basis {e 1, e 2 } are f(x, y)dχ(x)dχ(y) = ( (r + s)y 2 )dχ(y) = 4, ( 2,2) f(x, y)dχ(y)dχ(x) = ( 2,2) ( (r + s)(4 x 2 ))dχ(x) = 0. Definition 2.7. We define frame cellular sets inductively as follows: Every open segment or a singleton in a one-dimensional subspace L is a frame cell with respect to each frame of L. Let u = (u 1,...,u d ) be a frame of R d. Let H i = span{u 1,..., û i,...,u d }, and π i : R d H i be the projection. A subset σ R d is called a frame cell with respect to u if π i (σ) is a frame cell with respect to the frame (v 1,..., ˆv i,...,v d ) and π 1 i (x) σ are simultaneously open intervals or simultaneously singletons for all x π i (σ). A subset X of R d is said to be frame cellular if X can be decomposed into a disjoint union of finitely many frame cells with respect to each frame of R d. Proposition 2.8. Each frame cellular set X is Euler-Schanuel integrable. Moreover, if {σ i } is a frame cellular decomposition of X, then χ(x) = I X dχ(x) = ( 1) dim σ i. i Proof. Let {σ i } be the cellular decomposition with respect to a frame u = (u 1,..., u d ). It is clear that I X = i I σ i and I σi dχ(u 1,...,u d ) = I σi dχ(u σ(1),..., u σ(d) ) = ( 1) dim σ i. The formula follows from Proposition 2.5 and the linearity of Euler-Schanuel integral. Let σ be a frame cell of R d. The tangent cone Let X be a nonempty closed subset of R d. A smooth stratification of X is a finite collection {X a a P } of subsets of R d, such that the following three conditions are satisfied. Each X a is a smooth submanifold of R d, called a pure stratum of X. 9

10 X = a P X a (disjoint). If X a X b then X a X b ; we say that X a is a face of X b if X a X b, written X a X b. So P becomes a poset, called the poset of strata. The space X together with a stratification {X a a P } is called a stratified space. Given a nonzero vector v R d ; let L = Rv be the subspace spanned by v and V = L the orthogonal complement of L. Let φ : R d L be the orthogonal projection. A nonempty closed subset X of R d is said to be cylindric along v if L can be divided into set function φ : L V Given a nonzero vector v R d ; we say that X is locally finite at a point x along v if the intersection X {x + tv : 0 t < ε} is either an empty set, or a singlton, or an half-closed and half-open segment, when ε is small enough. finite with respect to v if for each point x R d, the intersection X {x+tv : t R} is a finite union of singletons, or closed segments, or half-closed and half-open segments. finite along v if X {x+tv : t R} is a finite union of singletons, or closed segments, or half-closed and half-open segments. 3 Group Arrangements It is well-known that there exists a unique translation invariant measure vol on the σ-algebra generated by the Borel sets of R n such that vol ([0, 1] n ) = 1, called the volume of R n. 4 Grassmannian Let Gr n (R n+k ) be the set of all n-dimensional vector subspaces of R n+k, and Gr n (R ) the set of all n-dimensional vector subspaces of R. We shall make both Gr n (R n+k ) and Gr n (R ) into CW-complexes so that Gr n (R n+k ) is a subcomplex of Gr n (R ). Let B d denote the unit ball of R d, consisting of all vectors v with v = 1. The interior of B d is defined to be the subset B0 d consisting of all vectors v with v < 1. For the special case p = 0, both B d and B0 d consist of the single zero vector. Any space homeomorphic to B p is called a closed p-cell, and any space homeomorphic to B p 0 is called an open p-cell. For instance R p is an open p-cell. A CW complex is a Hausdorff space X, called the underlying space, together with a partition of X into a collection {e α } of disjoint subsets, such that the following four conditions are satisfied. CW1 Each e α is a topologically an open cell of dimension d(α) 0, that is, there exists a continuous map φ α : B d(α) X, called the characteristic map for the cell e α, such that φ α ( Bd(α) ) = e α and the restriction φ α : B d(α) e α is a homeomorphism. CW2 If e α ē β then ē α ē β ; we say that e α is a face of e β if ē α ē β. If {e α } contain only finite number of cells, we say that X satisfying CW1 and CW2 is a finite CW complex. CW3 Closure Finiteness. Each point of X is contained in a finite subcomplex. 10

11 CW4 Whitehead Topology. The space X is topologized as the direct limit of its finite subcomplexes, that is, a subset of X is closed if and only if its intersection with each finite subcomplex is closed in the finite subcomplex. 11