A complement to the theory of equivariant finiteness obstructions

Transcription

1 F U N D A M E N T A MATHEMATICAE 151 (1996) A complement to the theory of equivariant finiteness obstructions by Paweł A n d r z e j e w s k i (Szczecin) Abstract. It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family w H α (X) of elements of the groups K 0 (Z[π 0 (W H(X)) α]). We prove that every family {w H α } of elements of the groups K 0 (Z[π 0 (W H(X)) α]) can be realized as the family of equivariant finiteness obstructions w H α (X) of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall s obstruction ([1], [2]). Introduction. The purpose of this paper is a clarification of the theory of equivariant finiteness obstructions. At present there are four different approaches to this subject. Two of them are equivariant generalizations of Wall s and Ferry s ideas (see [1] [3] and [4] respectively). In 1985 W. Lück [5] suggested a purely geometric construction of the finiteness obstruction and then he developed the global algebraic approach to the equivariant finiteness obstruction [6] which covers all the constructions mentioned above. In [7], Theorem F, C. T. C. Wall proved that if Y is a finite CW-complex then each element of the group K 0 (Z[π 1 (Y )]) can be realized as the finiteness obstruction of a finitely dominated CW-complex. We shall establish among other things a similar theorem for equivariant finiteness obstructions proving in Section 2 that if Y is a finite G-complex then every family {w H α } of elements of the groups K 0 (Z[π 0 (W H(Y )) α]) can be realized as the family of equivariant finiteness obstructions w H α (X) of an appropriate finitely dominated G-complex X. This result, in turn, will be used in Section 3 to show the existence of a natural equivalence between the geometric finiteness obstruction introduced by Lück [5] and the obstructions w H α (X). Throughout the paper G denotes a compact Lie group Mathematics Subject Classification: Primary 57S10, 55S91; Secondary 19J05. [97]

2 98 P. Andrzejewski 1. A short review of the equivariant finiteness obstruction. In this introductory section we recall a construction of the equivariant finiteness obstruction based on the ideas of C. T. C. Wall [7] and described by the author in [1] and [2]. As a result of this construction one gets a family of invariants which decide whether a finitely G-dominated G-complex is G-homotopy finite. Roughly speaking, the family of obstructions we want to introduce is defined for each component Xα H by means of the invariants w G (X, A) (see [1], 1, or [2], 2). Precisely, let H denote a closed subgroup of G and let Xα H be a connected component of X H. We define an equivalence relation in the set of such components Xα H by setting Xα H Xβ H iff there exists an element n G such that nhn 1 = K and n(xα H ) = Xβ H. We denote the set of equivalence classes of this relation by CI(X). Note that this definition is functorial, i.e. a G-map f : X Y induces a map CI(f) : CI(X) CI(Y ). If X is finitely G-dominated by a complex K and Xα H denotes a component of X H which represents an element of the set CI(X) then the group (W H) α acts on the pairs (Xα H, X α >H ) and (Kβ H, K>H β ) in such a way that (Xα H, X α >H ) is relatively free and (Kβ H, K>H β ) is relatively free and relatively finite. By the relative version of Proposition 1.3 in [1] we see that the pair (Kβ H, K>H β ) (W H) α -dominates the pair (Xα H, X α >H ). Definition ([1], [2]). We define a Wall-type invariant w H α (X) to be w H α (X) = w (W H)α (X H α, X >H α ) = w(c ( X H α, X>H α )) K 0 (Z[π 0 (W H) α]). The elements wα H (X) are invariants of the equivariant homotopy type and they vanish for finite G-complexes. Moreover, the invariant wα H (X) does not depend (up to canonical isomorphism) on the choice of the representative Xα H from the equivalence class [Xα H ] in CI(X) (see [1]). The fundamental property of the invariants wα H (X) is that they are actually obstructions to homotopy finiteness of X: Theorem 1.1 ([1] [3]). Let a G-complex X be G-dominated by a finite G- complex K. Then there exist a finite G-complex Y and a G-homotopy equivalence h : Y X iff all the invariants w H α (X) vanish. Moreover, if the complex X contains a finite G-subcomplex B and dim K = n then Y and h can be chosen in such a manner that B Y, dim Y = max(3, n) and h B = id B. 2. The realization theorems for the equivariant finiteness obstruction. As in the proof of Theorem 1.1 (see [1] or [2]) we begin with the case of a relatively free action which will serve as an inductive step in the proof of the main result.

3 Equivariant finiteness obstructions 99 Proposition 2.1. Let (Y, A) be a relatively free, relatively finite G-CW - pair and w 0 K 0 (Z[π 0 (G(Y ) )]) be an arbitrary element. Then there exist relatively free G-CW -pairs (X, A) and (K, A) and a G-retraction r : X Y inducing the isomorphism of fundamental groups such that Y X, Y K, (K, A) is a relatively finite G-CW -pair and G-dominates (X, A) and the equality r (w G (X, A)) = w 0 holds where r denotes the isomorphism induced by r on K 0. R e m a r k. Here w G (X, A) denotes the algebraic Wall finiteness obstruction of a finitely dominated chain complex C ( X, Ã) of free Z[π 0(G(Y ) )]- modules (see [1], p. 12, or [2], 2). P r o o f. Let P and Q be finitely generated, projective Z[π 0 (G(Y ) )]- modules with P Q = B a free module. Let w 0 = ( 1) n [P ] = ( 1) n+1 [Q] where n > 2. Let p : B P and q : B Q denote projections and C be the chain complex of the form... B q B p B q B with C k = 0 for k < n. We shall construct a relatively free G-CW -pair (X, Y ) such that C = C ( X, Ỹ ). Suppose rank(b) = m and let Y 1 be a G-complex obtained from Y by attaching m free G-n-cells via trivial G-maps φ i : G S n 1 Y, φ i (g, x) = g y 0, where y 0 Y is fixed. We shall show inductively that for each k 0 there exists a relatively free G-CW -pair (X k, Y ) and a G-map r k : X k Y 1 such that C = C ( X k, Ỹ ) for n+k 1 and that P (respectively Q) is a direct summand in π n+k (r k ) for odd (resp. even) k. We start with the inclusion r 0 : Y = X 0 Y 1. Since the attaching maps of free G-n-cells in Y 1 are equivariantly trivial there exists an exact sequence... π n (Y 1 ) π n (r 0 ) π n 1 (Y ) π n 1 (Y 1 )... with π n (r 0 ) = B and = 0. Let ξ j (j = 1,..., m) denote free generators of the module B and a j = q(ξ j ) B = π n (r 0 ). If r 1 : X 1 Y 1 is obtained from r 0 by attaching m free G-n-cells to Y = X 0 via a j π n (r 0 ) then one has the split exact sequence... π n+1 (r 0 ) π n+1 (r 1 ) P 0 and P is a direct summand in π n+1 (r 1 ). Since = 0, the attaching maps of G-n-cells in X 1 are equivariantly trivial. Hence there is a G-homotopy equivalence k 1 : Y 1 X 1.

4 100 P. Andrzejewski Let further b j = p(ξ j ) P π n+1 (r 1 ) and let r 2 : X 2 Y 1 be obtained from r 1 by attaching free G-(n + 1)-cells via b j. We have the split exact sequence... π n+2 (r 1 ) π n+2 (r 2 ) Q 0 and Q is a direct summand in π n+2 (r 2 ). It follows from the construction that C ( X 1, Ỹ ) = C for n and C ( X 2, Ỹ ) = C for n + 1. The inductive step goes alternately. Set X = k 0 X k and r : X Y 1 by r Xk = r k. Then for K = X 1 we see that the pair (K, A) G-dominates the pair (X, A) with the section given by the composition Finally, we have by definition (X, A) r (Y 1, A) k 1 (K, A). r (w G (X, A)) = ( 1) n+1 [C n+1 ( X, Ỹ )/B n+1( X, Ỹ )] = ( 1) n+1 [C n+1 / im n+2 ] = ( 1) n+1 [B/P ] = ( 1) n+1 [Q] = w 0. We will also need the following technical result concerning the glueing equivariant domination maps. Lemma 2.2. Let A X be a G-cofibration, Y a G-space and r : Y A a G-domination map with a section s : A Y. Then in the commutative diagram X id A A id id X s A Y the map r extends to a G-domination map R : X s Y X id A = X. Now we can formulate the realization theorem. Theorem 2.3. Let Y be a finite G-complex and {wα H } be a family of elements indexed by the set CI(Y ), with wα H K 0 (Z[π 0 (W H(Y ) )]). Then there exist a G-complex X and a G-retraction r : X Y inducing bijections and isomorphisms r : π 0 (X H ) π 0 (Y H ) r : π 1 (X H α ) π 1 (Y H α ) such that Y X, X is finitely G-dominated and r (w H α (X)) = w H α. r s

5 Equivariant finiteness obstructions 101 P r o o f. Note that the set CI(Y ) consists of one connected component from each W H-component (W H)Yα H. One can assume, in view of Proposition 2.14 in [6], that H runs through a complete set of representatives for all the isotropy types (H) occurring in X. We may suppose, in view of Proposition 2.12 in [6], that the set CI(Y ) is finite. Let Y H p α q, with 1 p r, 1 q s p, denote the representatives of W H p -components in the set CI(Y ). Order the set of pairs {(p, q) : 1 p r, 1 q s p } lexicographically. For each pair (p, q) we shall construct inductively a G-complex X p,q with the following properties: (1) Y X p,q and there exists a G-retraction r p,q : X p,q Y inducing bijections on the π 0 -level and isomorphisms of fundamental groups of appropriate fixed point set components. (2) If (p, q) (m, n) then X p,q X m,n. (3) The complex X p,q is G-dominated by the finite G-complex K p,q. (4) w H α (X p,q ) = w H α for (H) = (H i ), 1 i < p and for any α. (5) w H p α j (X p,q ) = w H p α j for 1 j q. (6) w H p α j (X p,q ) = 0 for j > q. (7) w H α (X p,q ) = 0 for (H) = (H i ), i > p and for any α. Then the complex X r,sr obtained as a result of the final inductive step satisfies the assertion of the theorem. Let X 0,0 = Y and suppose that X p,q has been constructed. There are two cases to consider. C a s e I: q < s p. Simplify the notation by setting H = H p and α = α q+1. Then ((X p,q ) H α, (X p,q ) >H α ) is a relatively free and relatively finite (W H) α -CW -pair (by property (6) and Theorem 1.1). Since π 1 (Yα H ) = π 1 ((X p,q ) H α ) we can assume that CI(Y ) = CI(X p,q ) and wα H K 0 (Z[π 0 (W H(X p,q )) α]). By Proposition 2.1 there exists a relatively free (W H) α -CW -pair (Z, (X p,q ) >H α ) such that (a) (Z, (X p,q ) >H α ) is (W H) α -dominated by a relatively free, relatively finite (W H) α -CW -pair (K, (X p,q ) >H α ), (b) (X p,q ) H α Z and there exists a (W H) α -retraction r : Z (X p,q ) H α, and (c) r (w (W H)α (Z, (X p,q ) >H α )) = wα H. Let d : (K, (X p,q ) >H α ) (Z, (X p,q ) >H α ) denote a (W H) α -domination map with a section s : (Z, (X p,q ) >H α ) (K, (X p,q ) >H α ).

6 102 P. Andrzejewski One can treat the pair (Z, (X p,q ) >H α ) as an (NH) α -pair and then the inclusion (b) extends to the inclusion of G-pairs (G (NH)α (X p,q ) H α, G (NH)α (X p,q ) >H α ) (G (NH)α Z, G (NH)α (X p,q ) >H α ) and the retraction r : Z (X p,q ) H α to the G-retraction r : G (NH)α Z G (NH)α (X p,q ) H α. If Z 1 = (G (NH)α Z) q G(X p,q ) >H α then by Lemma 2.2 we have the inclusion (X p,q ) (H) α Z 1 and the G-retraction r 1 : Z 1 (X p,q ) (H) α. By the inductive assumption (conditions (6), (7) and Theorem 1.1) the pair (X p,q, (X p,q ) (H) α ) is relatively finite and taking Z 2 = X p,q Z 1 one can extend the inclusion (X p,q ) (H) α Z 1 to the inclusion X p,q Z 2 and the retraction r 1 : Z 1 (X p,q ) (H) α to a G-retraction r 2 : Z 2 X p,q such that the G-pair (Z 2, Z 1 ) is relatively finite. If K 1 = (G (NH)α K) q G(X p,q ) >H α, then we can extend the domination d to the G-domination map d 1 : (K 1, G(X p,q ) >H α ) (Z 1, G(X p,q ) >H α ) such that the pair (K 1, G(X p,q ) >H α assumption (property (3)) G(X p,q ) >H α G(K p,q ) >H α = K. Let φ : K G(X p,q ) >H α denote this domination and s 1 : G(X p,q ) >H α K its section. Applying Lemma 2.2 to the diagram ) is relatively finite. By the inductive is G-dominated by a finite G-complex K 1 G(X p,q ) >H id α G(X p,q ) >H α id we get the G-domination map K 1 G(X p,q ) >H s 1 id α K φ 1 : K 1 s1 K K 1 where K 1 = K 1 s1 K is a finite G-complex. Then the composition K 1 φ 1 d 1 K1 Z1 s 1

7 Equivariant finiteness obstructions 103 is a finite domination over Z 1. Invoking Lemma 2.2 again we get a G-domination map d 2 : K 2 Z 2 with K 2 a finite G-complex. Hence X p,q+1 = Z 2 is G-dominated by a finite G-complex K p,q+1 = K 2 and the composition r X 2 r p,q p,q+1 Xp,q Y defines a G-retraction r p,q+1 : X p,q+1 Y. Finally, it follows from the construction that X p,q+1 properties (4) (7). C a s e II: q = s p. This is similar to Case I. has the desired 3. The geometric finiteness obstruction of W. Lück and the invariants w H α (X). Let X be a G-complex G-dominated by a finite one. W. Lück [5], [6] defined geometrically a group W a G (X) and an element w G (X) W a G (X) that decides when the G-complex X has the G-homotopy type of a finite G-complex. The aim of this section is to connect Lück s obstruction with the invariants w H α (X). This theorem along with results of [1], 4, completes the proof of the equivalence of three out of four definitions of the equivariant obstruction to finiteness. We start by recalling the construction from [5], [6]. Let X be an arbitrary G-complex. Consider the set of G-maps f : Y X, where Y ranges through finitely G-dominated G-complexes. We define an equivalence relation as follows: f 0 : Y 0 X and f 4 : Y 4 X are equivalent iff there exists a commutative diagram i 0 j 1 P PPPP A AAf1 P PPPP A f AA f 2 0 P PP Y 0 Y 1 Y 2 Y 3 Y 4 X j 2 i 3 f 3 } }} n nn } }} n nnnn f 4 n nnnn such that j 1 and j 2 are G-homotopy equivalences and i 0, i 3 are inclusions such that the G-CW -pairs (Y 1, Y 0 ) and (Y 3, Y 4 ) are relatively finite. Let W a G (X) denote the set of equivalence classes. The disjoint union induces an addition on W a G (X) and the inclusion of the empty space defines a neutral element. One can show that this addition gives W a G (X) the structure of an abelian group ([5], p. 370, or [6], p. 51). Definition. Let X be a finitely G-dominated G-complex. We define its geometric obstruction to finiteness as w G (X) = [id : X X] W a G (X).

8 104 P. Andrzejewski Then we have the following result. Theorem 3.1. ([5], Theorem 1.1, or [6], 3). Let X be finitely G-dominated. Then (a) W a G : G-CW Ab is a covariant functor from the category of equivariant CW -complexes to the category of abelian groups. (b) w G (X) is an invariant of the G-homotopy type. (c) A G-complex X is G-homotopy equivalent to a finite G-complex iff w G (X) = 0. Let X be a G-complex. We define a homomorphism F : W a G (X) K 0 (Z[π 0 (W H(X)) α]) CI(X) by the formula F ([f : Y X]) = f (w H α (Y )) where f : K 0 (Z[π 0 (W H(Y )) α]) K 0 (Z[π 0 (W H(X)) α]) denotes the homomorphism induced on K 0 by f. The following result gives the precise relation between Lück s obstruction w G (X) and Wall-type invariants w H α (X). Theorem 3.2. Suppose X is a G-complex such that (1) X has finitely many orbit types, (2) π 0 (X H ) is finite for any subgroup H of G occurring on X as an isotropy subgroup, (3) π 1 (X H α, x) is finitely presented for any representative X H α from the class [X H α ] CI(X) and for any x X H α. Then the natural homomorphism F : W a G (X) CI(X) K 0 (Z[π 0 (W H(X)) α]) is an isomorphism. If the G-complex X is finitely G-dominated then F (w G (X)) = w H α (X). R e m a r k. Observe that any finitely G-dominated G-complex satisfies conditions (1) (3) of Theorem 3.2. Before presenting a proof of the theorem let us recall one technical lemma from [6] which will be used in the proof. Lemma 3.3 ([6], Lemma 14.7). Let f : Y X be a G-map between G-complexes. Suppose the sets Iso(X) and Iso(Y ) of orbit types on X and Y, respectively, are finite. Suppose that for any H Iso(X) Iso(Y ) the sets π 0 (X H ) and π 0 (Y H ) are finite and the fundamental groups π 1 (Yα H, y) and π 1 (Xβ H, x) are finitely presented for any y Y α H, x Xβ H. Then one

9 Equivariant finiteness obstructions 105 can extend the map f to a G-map g : Z X such that for any subgroup H of G, g : π 0 (Z H ) π 0 (X H ) is bijective and g : π 1 (Z H α, z) π 1 (X H α, g(z)) is an isomorphism for any component Z H α and any point z Z H α. P r o o f o f T h e o r e m 3.2. Suppose an element [f : Y X] belongs to the kernel of F. The assumptions on X and Lemma 3.3 imply that there exists a G-complex Z obtained from Y by attaching finitely many G-cells and an extension g : Z X of the map f such that (1) g : π 0 (Z H ) π 0 (X H ) is a bijection and (2) g : π 1 (Z H α ) π 1 (X H α ) is an isomorphism. Note that [f] = [g] in the group W a G (X). Hence Since (1), (2) and F ([f]) = F ([g]) = g (w H α (Z)) = 0. g : K 0 (Z[π 0 (W H(Z)) α]) K 0 (Z[π 0 (W H(X)) α]) are bijections we have CI(Z) = CI(X) and wα H (Z) = 0 for any component Zα H which represents an element of the set CI(Z). It follows from Theorem 1.1 that there exists a finite G-complex Z 1 and a G-homotopy equivalence h : Z 1 Z. Then the diagram id O g O OO g Y Z Z Z 1 X h n nn g h } }} n nnnn } }} n nnnn shows that [f] = 0 in the group W a G (X). Thus F is a monomorphism. Similarly, the assumptions on the space X and Lemma 3.3 applied to the map X imply that we can find a finite G-complex K and a G-map g : K X such that g : π 0 (K H ) π 0 (X H ) is a bijection and g : π 1 (K H α ) π 1 (X H α ) is an isomorphism. Then CI(K) = CI(X) and g induces an isomorphism g : K 0 (Z[π 0 (W H(K)) α]) K 0 (Z[π 0 (W H(X)) α]).

10 106 P. Andrzejewski By the commutativity of the diagram W a G (K) g W a G (X) F 1 F CI(K) K 0 (Z[π 0 (W H(K)) α]) = CI(X) K 0 (Z[π 0 (W H(X)) α]) it suffices to show that F 1 : W a G (K) CI(K) K 0 (Z[π 0 (W H(K)) α]) is an epimorphism. Let wα H K 0 (Z[π 0 (W H(K)) α]) be an arbitrary element. By Theorem 2.3 there exists a G-complex L, G-dominated by a finite G- complex and a G-retraction r : L K such that r (wα H (L)) = wα H. Then [r : L K] W a G (K) and F 1 ([r]) = r (w H α (L)) = w H α. References [1] P. Andrzejewski, The equivariant Wall finiteness obstruction and Whitehead torsion, in: Transformation Groups, Poznań 1985, Lecture Notes in Math. 1217, Springer, 1986, [2], Equivariant finiteness obstruction and its geometric applications a survey, in: Algebraic Topology, Poznań 1989, Lecture Notes in Math. 1474, Springer, 1991, [3] K. Iizuka, Finiteness conditions for G-CW -complexes, Japan. J. Math. 10 (1984), [4] S. Kwasik, On equivariant finiteness, Compositio Math. 48 (1983), [5] W. Lück, The geometric finiteness obstruction, Proc. London Math. Soc. 54 (1987), [6], Transformation Groups and Algebraic K-Theory, Lecture Notes in Math. 1408, Springer, [7] C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), Department of Mathematics University of Szczecin Wielkopolska Szczecin 3, Poland pawelan@uoo.univ.szczecin.pl pawelan@euler.mat.univ.szczecin.pl Received 27 February 1995