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The Euler Characteristic THE EULER CHARACTERISTIC LIVIU I. NICOLAESCU Abstract. I will describe a few basic properties of the Euler characteristic and then I use them to prove special case of a cute formula due to Bernstein-Khovanskii-Koushnirenko. 1. Basic properties of the Euler characteristic The Euler characteristic is a function  which associates to each reasonable1 topological space X an integer Â(X). For us a reasonable space would be a space which admits a ¯nite simplicial decomposition (a.k.a. triangulation.) For example, all algebraic varieties are reasonable. In the sequel we will tacitly assume that all spaces are reasonable and so we will drop this attribute from our discourse. More explicitly, the Euler characteristic of X is de¯ned as the alternating sum X k k Â(X) = (¡1) dimR Hc (X; R); k¸0 ² where Hc (X; R) denotes the cohomology with compact supports and real coe±cients of the space X. The Euler characteristic is uniquely determined by the following properties. ² Normalization. Â(fpointg) = 1: ² Topological invariance. Â(X) = Â(Y ) if X is homeomorphic to Y: ² Proper homotopy invariance Â(X) = Â(Y ) for any homotopic compact spaces X and Y: ² Excision. Â(X) = Â(C) + Â(X n C); for every closed subset C ½ X: The excision property has a dual form Â(X) = Â(U) + Â(X n U) for every open subset U ½ X: ² Multiplicativity. Â(X £ Y ) = Â(X)Â(Y ): The excision property is frequently used under the guise of the inclusion-exclusion formula. More precisely, if X is a union of two closed sets X = S1 [ S2 then Â(X) = Â(S1) + Â(S2) ¡ Â(S1 \ S2): Date: September, 2006. Notes for the "Math Club" talk, Sept 21, 2006. 1For example, every space de¯ned by polynomial equalities and inequalities is reasonable. The technical term would be subanalytic. 1 2 LIVIU I. NICOLAESCU Indeed Â(X) = Â(S1) + Â(X n S1) = Â(S1) + Â(S2 n S1 \ S2) = Â(S1) + Â(S2) ¡ Â(S1 \ S2): We have a similar formula with closed replaced by locally closed 2. Let us compute the Euler characteristic of a few reasonable spaces. Note ¯rst that the Euler characteristic of a ¯nite set (equipped with the discrete topology) is equal to the cardinality of that set. Denote by ¢n the closed n-dimensional simplex. Thus ¢0 is a point, ¢1 is a segment, ¢2 is a triangle, ¢3 is a tetrahedron etc. Every simplex ¢n is homotopic to a point and thus ∆ ∆ ∆ 1 2 0 ∆3 Figure 1. Simplices. Â(¢n) = 1; 8n ¸ 0: n¡1 0 Observe that @¢n is homeomorphic to the (n¡1)-sphere S . Since S is a union of two points we deduce Â(S0) = 2. In general, the n-dimensional sphere is a union of two closed hemispheres intersecting along the Equator which is a (n ¡ 1) sphere. Hence, n n¡1 n¡1 Â(S ) = 2Â(¢n) ¡ Â(S ) = 2 ¡ Â(S ): We deduce inductively 2 = Â(Sn) + Â(Sn¡1) = Â(Sn¡1) + Â(Sn¡2) = ¢ ¢ ¢ = Â(S1) + Â(S0) so that ½ 2 if n 2 2Z Â(Sn) = 1 + (¡1)n = : 0 if n 2 2Z + 1 n Now observe that the interior of ¢n is homeomorphic to R so that n n¡1 n¡1 n Â(R ) = Â(¢n) ¡ Â(@¢n) = 1 ¡ Â(S ) = 1 ¡ (1 + (¡1) ) = (¡1) : The excision property implies the following useful formula. Suppose ; ½ X(0) ½ X(1) ½ ¢ ¢ ¢ ½ X(N) = X is an increasing ¯ltration of X by closed subsets. Then Â(X) = Â(X(0)) + Â(X(1) n X(0)) + ¢ ¢ ¢ + Â(X(N) n X(N¡1)): Let's apply this in the case when X is a simplicial complex. We denote by X(k) the union of the simplices of dimension · k. Then X(k) nX(k¡1) is the union of the interiors of the k-dimensional simplices. We denote by fk(X) the number of such simplices. Each of them is homeomorphic 2A subset of a topological space is called locally closed if is the intersection of an open subset with a closed subset THE EULER CHARACTERISTIC 3 k k (k) (k¡1) k to R and thus its Euler characteristic is equal to (¡1) . Hence Â(X n X ) = (¡1) fk(X) so that X k Â(X) = (¡1) fk(X): k¸0 Suppose ¼ : X ! Y is a d : 1 covering map. Then we can ¯nd an increasing ¯ltration of Y by closed subsets Y (0) ½ ¢ ¢ ¢ ½ Y (k) ½ Y (k+1) ½ ¢ ¢ ¢ ½ Y such that over Y (k) n Y (k¡1) the projection ¼ is a trivial covering map. If we set X(k) = ¼¡1(Y (k)) then X(k) n X(k¡1) is homeomorphic to a product F £ (Y (k) n Y (k¡1)), where F is a ¯nite set of cardinality d. Hence Â(X(k) n X(k¡1)) = dÂ(Y (k) n Y (k¡1)) and we deduce Â(X) = dÂ(Y ): More generally, suppose f : X ! Y is a reasonable3, proper, continuous map with ¯nite ¯bers. Denote by Y (k) the subset of Y consisting of points y such that the ¯ber f has cardinality k. We set ¡ ¢ X(k) := f ¡1 Y (k) : The map f : X(k) ! Y (k) is a k : 1 cover and we deduce Â(X(k)) = k ¢ Â(Y (k)): Summing over k we deduce the slicing formula X X ¡ ¢ Â(X) = kÂ(Y (k)) = k ¢  fy 2 Y ; jf ¡1(y)j = kg : (1.1) k¸0 k¸0 2. ABezout¶ type formula To put things into perspective we start with a classic elementary fact. Suppose we are given a polynomial with complex coe±cients in one complex variable k n A(z) = akz + ¢ ¢ ¢ + anz ; k < n; ak ¢ an 6= 0: Then for generic choices of coe±cients the number of nonzero roots of P is equal to n ¡ k. We want to rephrase this in a more sophisticated way. For a polynomial P in º complex variables ~z = (z1; ¢ ¢ ¢ ; zº) ¤ ¤ º ZP := f~z 2 (C ) ; P (~z) = 0g: ¤ In our special case the number of nonzero roots of A is the Euler characteristic of ZA. ® ®1 ®º To every nonzero monomial a®~z = a®z1 ¢ ¢ ¢ znº of P we associate the point º º ® = (®1; ¢ ¢ ¢ ; ®º) 2 Z ½ R : We obtain in this fashion a ¯nite set of lattice points denoted by supp P ½ Rº. The Newton polytope of P is by de¯nition the convex hull of supp P . We denote it by ¢(P ). For example, supp A(z) = fk; k + 1; ¢ ¢ ¢ ; ng ½ R1; and ¢(A) is the line segment connecting the points k; n 2 R. The fact that a generic polynomial A with Newton polytope [k; n] has (n ¡ k) nonzero roots can be rewritten in the following sophisticated fashion ¤ ¡ ¢ Â(ZA) = vol1 ¢(A) : 3A reasonable map would be for example a real analytic map. 4 LIVIU I. NICOLAESCU Above, vol1 denotes the 1-dimensional volume of the segment [k; n], i.e. its length. The above equality is a special case of the following general result. Theorem 2.1 (Bernstein-Khovanskii-Koushnirenko). Fix a convex polytope in Rº with vertices º in (Z¸0) then for a generic polynomial P such that ¢(P ) = ¢ we have ¤ º¡1 Â(ZP ) = (¡1) º! ¢ volº(¢); º where volº denotes the º-dimensional Euclidean volume in R . ut Let us prove this theorem in a special case when º = 2 and P = ax4 + bx3y4 + cy5 + d: The Newton polytope of P is depicted in Figure 2. y 5 C B 4 A O 3 4 x Figure 2. The Newton polygon of ax4 + bx3y4 + cy5 + d. To compute the area of this polygon we decompose it into two triangles OAB and OBC and we have ¯ ¯ ¯ ¯ ¯ 4 0 ¯ ¯ 3 0 ¯ 2Area (OABC) = 2Area (OAB) + 2Area (OBC) = ¯ ¯ + ¯ ¯ = 16 + 15 = 31: ¯ 3 0 ¯ ¯ 4 5 ¯ Consider the curve 2 ZP = f(x; y) 2 C ; P (x; y) = 0; y 6= 0g: 0 5 and set ZP = f(0; y) 2 ZP g = f(0; y); cy + d = 0g. For generic (c; d) we have Â(ZP ) = 5 so that ¤ 0 Â(ZP ) = Â(ZP ) ¡ Â(ZP ) = Â(ZP ) ¡ 5: To compute the Euler characteristic of ZP we use the slicing formula (1.1) applied to the map ¤ f : ZP ! C ; ZP 3 (x; y) 7! y: ¡1 Note that for every y0 2 C the ¯ber f (y0) is the intersection of the horizontal line y = y0 with the curve ZP . The points on this intersection are found by solving the polynomial equation in x 4 3 4 5 ax + bx y0 + cy0 + d = 0: For all but ¯nitely many y0's this equation has exactly 4 distinct solutions. The intersection contains less than 4 points precisely when the horizontal line y = y0 is tangent to the curve ZP , ¤ ¤ i.e. y0 is a critical value of the map ¼ (see Figure 3). Denote by S the set of y0 2 C such that ¤ ¤ the line y = y0 intersects the curve ZP transversally. The discriminant set is D = C n S . THE EULER CHARACTERISTIC 5 regular level set singular level set ¤ Figure 3. Slicing the curve ZP . For generic coe±cients a; b; c; d the curve ZP can be represented near a horizontal tangency point as the graph of the implicit function y = y(x). The horizontal tangency condition can be rewritten as dy = 0: dx Derivating the equation P = 0 with respect to x we deduce dy y3(4bx3 + 5cy) = ¡x2(4ax + 3by4); xy 6= 0: dx The points with horizontal tangents on ZP are obtained by solving the system ax4 + bx3y4 + cy5 + d = 0; x2(4ax + 3by4) = 0; xy 6= 0: (2.1) We distinguish two cases, x = 0 and x 6= 0.
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