топология / Tom Dieck T. Algebraic topology (EMS, 2008)

308 Chapter 12. Cellular Homology

in terms of the basis .xt j t 2 S.p 1// can have a non-zero coefficient only for the xs.i/. The coefficient of xs.i/ is seen from the commutative diagram

structed so far an isomorphism of C .K/ with the cellular chain complex C .jKj/ of jKj. Let Pp W Hp.C .K// ! Hp.C .jKj// be the induced isomorphism. Let Qp W Hp.C .jKj// ! Hp.jKj/ be the isomorphism in the proof of (12.2.1). Tracing through the definitions one verifies Rp D QpPp. Hence Rp is the composition of two isomorphisms.

An interesting consequence of (12.3.1) is that W C .K/ ! S .jKj/ is a chain equivalence. Thus, for a finite complex K, the singular complex of jKj is chain equivalent to a chain complex of finitely generated free abelian groups, zero above the dimension of K.

(12.3.2) Example. A circle S1 can be triangulated by a regular n-gon with ver-

Problems

1. Let K be the tetrahedral simplicial complex; it consists of E D f0; 1; 2; 3g, and all subsets are simplices. Verify Hi .K/ D 0 for n > 0. Generalize to an n-simplex.

12.4 The Euler Characteristic

Let X be a finite C W -complex and fi .X/ the number of its i-cells. The combinatorial Euler characteristic of X is the alternating sum

P

.X/ D i 0. 1/i fi .X/:

The fundamental and surprising property of this number is its topological invariance, in fact its homotopy invariance – it does not depend on the cellular decomposition of the space. The origin is the famous result of Euler which says that in the case X D S2 the value .X/ always equals 2 ([61], [62], [64]).

We prepare the investigation of the Euler characteristic by an algebraic result about chain complexes. Let M be a category of R-modules. An additive invariant

for M with values in the abelian group A assigns to each module M in M an element

.M / 2 A such that for each exact sequence 0 ! M0 ! M1 ! M2 ! 0 in M the additivity

.1/ .M0/ .M1/ C .M2/ D 0

holds. For the zero-module M we have .M / D 0 since there exists an exact sequence 0 ! M ! M ! M ! 0. We consider only categories which contain with a module also its submodules and its quotient modules as well as all exact sequences between its objects. Let

be a chain complex in this category. Then its homology groups Hi .C / are also contained in this category.

Zi D Ker @i . For k D 1 there exist, by definition of homology groups, exact sequences

0 ! B0 ! C0 ! H0 ! 0; 0 ! H1 ! C1 ! B0 ! 0:

We apply the additivity (1) to both sequences and thereby obtain .H0/ .H1/ D

.C0/ .C1/. For the induction step we consider the sequences

C 0 W 0 ! Ck 1 ! ! C0 ! 0;

0 ! Hk ! Ck ! Bk 1 ! 0; 0 ! Bk 1 ! Zk 1 ! Hk 1 ! 0I

the last two are exact and the first one is a chain complex. The homology groups of the chain complex are, for k 2,

Hi .C 0 / D Hi .C /; 0 i k 2; Hk 1.C 0 / D Zk 1:

310 Chapter 12. Cellular Homology

The relation of the combinatorial Euler characteristic to homology groups goes back to Henri Poincaré [150], [152]. The i-th Betti number, named after

Enrico Betti [20], bi .X/ of X is the rank of Hi .XI Z/, i.e., the cardinality of a basis of its free abelian part, or equivalently, the dimension of the Q-vector space Hi .XI Z/ ˝ Q Š Hi .XI Q/. The result of Poincaré says:

(12.4.2) Theorem. For each finite CW-complex X the combinatorial Euler characteristic equals the homological Euler characteristic Pi 0. 1/i bi .X/.

Proof. For finitely generated abelian groups A 7!rank A is an additive invariant. We apply (12.4.1) to the cellular chain complex C.X/ of X and observe that rank Ci .X/ D fi .X/.

If is an additive invariant for M and C a chain complex of finite length in

Proof. Apply (12.4.1) to the given exact sequence, considered as chain complex, and order the terms according to H , H 0, and H 00.

One can define the Euler characteristic by homological methods for spaces which are not necessarily finite CW-complexes. There are several possibilities depending on the homology theory being used.

Let R be a principal ideal domain. We call .X; A/ of finite R-type if the groups

Hi .X; AI R/ are finitely generated R-modules and only finitely many of them are non-zero. In that case we have the associated homological Euler characteristic

equality .X; AI Z/ D .X; AI R/ holds.

Proof. If .X; A/ is of finite Z-type, then the singular complex S .X; A/ is chain equivalent to a chain complex D of finitely generated free abelian groups with only finitely many of the Dn non-zero (see (11.6.4)). Therefore

Proposition (12.4.3) has the following consequence. Suppose two of the spaces A, X, .X; A/ are of finite R-type. Then the third is of finite R-type and the additivity relation

.AI R/ C .X; AI R/ D .XI R/

holds. Let A0; A1 be subspaces of X with MV-sequence, then

.A0I R/ C .A1I R/ D .A0 [ A1I R/ C .A0 \ A1I R/

provided the spaces involved are of finite R-type. Similarly in the relative case. Let

.X; A/ and .Y; B/ be of finite R-type. Then the Künneth formula is used to show that the product is of finite R-type and the product formula

.X; AI R/ .Y; BI R/ D ..X; A/ .Y; B/I R/

holds. These relations should be clear for finite CW-complex and the combinatorial Euler characteristic by counting cells.

For the more general case of Lefschetz invariants and fixed point indices see [51], [52], [109], [116].

12.5 Euler Characteristic of Surfaces

We report about the classical classification of surfaces and relate this to the Euler characteristic. For details of the combinatorial or differentiable classification see e.g., [167], [80], [123]. See also the chapter about manifolds.

Let F1 and F2 be connected surfaces. The connected sum F1#F2 of these surfaces is obtained as follows. Let Dj Fj be homeomorphic to the disk D2 with boundary Sj . In the topological sum F1 X D1ı C F2 X D2ı we identify x 2 S1 with '.x/ 2 S2 via a homeomorphism ' W S1 ! S2. The additivity of the Euler characteristic is used to show

.F1/ 1 C .F2/ 1 D .F1#F2/;

i.e., the assignment F 7! .F / 2 is additive with respect to the connected sum. Let mF denote the m-fold connected sum of F with itself. We have the standard surfaces sphere S2, torus T , and projective plane P . The Euler characteristics are

.S2/ D 2; .mT / D 2 2m; .nP / D 2 n:

If F is a compact surface with k boundary components, then we can attach k disks D2 along the components in order to obtain a closed surface F . By additivity.F / D .F / C k. Connected surfaces Fj with the same number of boundary components are homeomorphic if and only if the associated surfaces Fj are homeomorphic.

312 Chapter 12. Cellular Homology

(12.5.1) Theorem. A closed connected surface is homeomorphic to exactly one of the surfaces S2, mT with m 1, nP with n 1. The nP are the non-orientable surfaces.

The homeomorphism type of a closed orientable surface is determined by the orientation behaviour and the Euler characteristic. The homeomorphism type of a compact connected surface with boundary is determined by the orientation be-

12.5.2 Platonic solids. A convex polyhedron is called regular if each vertex is the end point of the same number of edges, say m, and each 2-dimensional face has the same number of boundary edges, say n. If E is the number of vertices, K the number of edges and F the number of 2-faces, then mE D 2K and nF D 2K. We insert this into the Euler relation E C F D K C 2, divide by 2K, and obtain

m1 C n1 D K1 C 21 :

We have m 3, n 3. The equation has only the solutions which are displayed in the next table.

12.5.3 Lines in the projective plane. Let G1; : : : ; Gn be lines in the projective plane P . We consider the resulting cells decomposition of P . Let tr be the number of points which are incident with r lines. We have the Euler characteristic relation f0 f1 C f2 D 1 where fi is the number of i-cells. Thus f0 D t2 C t3 C . From an r-fold intersection point there start 2r edges. The sum over the vertices yields f1 D 2t2 C 3t3 C 4t4 C . Let pn denote the numbers of n-gons, then f2 D P ps ; 2f1 D P sps . We insert these relations into the Euler characteristic relation and obtain

Pr 2.3 r/tr C Ps 2.3 s/ps D 3f0 f1 C 3f2 2f1 D 3:

We now assume that not all lines are incident with a single point; then we do not have 2-gons. From 2f1 3f2 and then f1 3.f0 1/ we conclude

t2 3 C Pr 4.r 3/tr :

314 Chapter 12. Cellular Homology

Proof. For each y 2 Y the set p 1.y/ X is closed and hence compact. The pre-images p 1.y/ in a ramified covering are always discrete, hence finite. The set V is also discrete and hence finite. The map p is, as a continuous map between compact Hausdorff spaces, closed. Thus we have shown that the map in question is proper. Now we use (12.5.4).

(12.5.6) Proposition (Riemann–Hurwitz). Let p W X ! Y be a ramified covering between compact connected surfaces. Let P1; : : : ; Pr 2 X be the ramification points with ramification index v.Pj /. Let n be the cardinality of the general fibre. Then for the Euler characteristics the relation

Proof. Let Q1; : : : ; Qs be the images of the ramification points. Choose pairwise disjoint neighbourhoods D1; : : : ; Ds Y where Dj is homeomorphic to a disk.

Then p W X0 D X X SsD p 1.Dı/ ! Y X S D Dı D Y0

j 1 j j 1 j

is an n-fold covering (see (12.5.4)). We use the relation .X0/ D n .Y0/ for n-fold coverings. If C is a finite set in a surface X, then .X X C / D .X/ jC j. Thus we see

Moreover

An interesting application of the Riemann–Hurwitz formula concerns actions of finite groups on surfaces. Let F be a compact connected orientable surface and G F ! F an effective orientation preserving action. We assume that this action has the following properties:

(1)The isotropy group Gx of each point x 2 F is cyclic.

(2)There exist about each point x a centered chart ' W U ! R2 such that U

is Gx -invariant and ' transforms the Gx -action on U into a representation on R2, i.e., a suitable generator of Gx acts on R2 as rotation about an angle

2 =jGx j.

In this case the orbit map p W F ! F=G is a ramified covering, F=G is orientable, and the ramification points are the points with non-trivial isotropy group. One can show that each orientation preserving action has the properties (1) and (2). Examples are actions of a finite group G SO.3/ on S2 by matrix multiplication and of a finite group G GL2.Z/ on the torus T D R2=Z2 by matrix multiplication.

The ramified coverings which arise as orbit maps from an action are of a more special type. If x 2 F is a ramification point, then so is each point in p 1px, and these points have the same ramification index, since points in the same orbit have conjugate isotropy groups. Let C1; : : : ; Cr be the orbits with non-trivial isotropy group and let nj denote the order of the isotropy group of x 2 Ci ; hence jCi jni D jGj. The Riemann–Hurwitz formula yields in this case:

that .F=G/ 0 is not compatible with 12.5.7, hence .F=G/ D 2 and the orbit space is again a sphere. We also see that r 3 and r D 0; 1 are not possible. For r D 2 we have 2=jGj D 1=n1 C 1=n2, 2 D jC1 C jC2. Hence there are two fixed points (example: rotation about an axis). For r D 3 one verifies that

has the solutions (for n1 n2 n3) displayed in the next table.

Examples are the standard actions of subgroups of SO.3/, namely D2n (dihedral), A4 (tetrahedral), S4 (octahedral), A5 (icosahedral). Up to homeomorphism there are no other actions. Þ

12.5.9 Action on the torus. Let F D T D S1 S1 be the torus, .F / D 0. The Riemann–Hurwitz formula shows that for r 1 we must have .F=G/ D 2. The cases r 5 and r D 1; 2 are impossible. For r D 4 we must have n1 D n2 D n3 D n4 D 2 and G D Z=2. For r D 3 the solutions of 12.5.7 are displayed in a table.

The cyclic groups generated by A; A2; A3 realize cases 2 and 1 of the table and the case r D 4 above. The matrix B realizes case 3 of the table. Þ

Problems

1.Let G act effectively on a closed orientable surface F of genus 2 preserving the orientation. Then jGj divides 48 or 10. There exist groups of orders 48 and 10 which act on a surface of genus 2. The group of order 48 has a central subgroup C of order 2 and G=C is the octahedral group S4 acting on the sphere F=C . Study the solutions of 12.5.7 and determine the groups which can act on F . Use covering space theory and work towards a topological classification of the actions.

2.The nicest models of surfaces are of course Riemann surfaces. Here we assume known the construction of a compact Riemann surface from a polynomial equation in two variables.

The equation y2 D f .x/ with 2gC2 branch points defines a surface of genus g. Such curves are called hyper-elliptic (g 2). It is known that all surfaces of genus 2 are hyper-elliptic. A hyper-elliptic surface always has the hyper-elliptic involution I.x; y/ D .x; y/. Here

are some examples. Let us write e.a/ D exp.2 ia/.

(1)y2 D x.x2 1/.x2 4/ has a Z=4-action generated by A.x; y/ D . x; e.1=4/y/. Note A2 D I .

(2)y2 D .x3 1/=.x3 8/ has a Z=3-action generated by B.x; y/ D .e.1=3/x; y/. Since B commutes with I , we obtain an action of Z=6.

(3)y2 D .x3 1/=.x3 C 1/ has a Z=6-action generated by C.x; y/ D .e.1=6/x; 1=y/. Since C commutes with I , we obtain an action of Z=6 ˚ Z=2. It has an action of Z=4 generated by D.x; y/ D .1=x; e.1=4/y/. Note D2 D I . The actions C and D do not commute, in fact CD D DC 5. Thus we obtain an action of a group F which is an extension

1 ! Z=2 ! F ! D12 ! 1

where Z=2 is generated by I and D12 denotes the dihedral group of order 12.

(4) y2 D x.x4 1/ has the following automorphisms (see also [121, p. 94])

p

K.x; y/ D . .x i/=.x C i/; 2 2e.1=8/y=.x C i//; K3 D I:

The elements G; H; K generate a group of order 48. If we quotient out the central hyperelliptic involution we obtain the octahedral group of order 24 acting on the sphere. Thus there also exists an action of a group of order 24 such that the quotient by the hyper-elliptic involution is the tetrahedral group (and not the dihedral group D12, as in (3)).

(5) y2 D x5 1 has an action of Z=5 generated by J.x; y/ D .e.1=5/x; y/. It commutes with I and gives an action of Z=10.

3. By an analysis of 12.5.7 one can show that, for an effective action of G on a closed orientable surface of genus g 2, the inequality jGj 84.g 1/ holds. There exists a group of order 168 which acts on a surface of genus 3 [63, p. 242].