Some Generalizations of a Theorem of Dold

Cristian-Nicolae Costinescu

Abstract We obtain generalizations of a theorem of Dold to G-cohomology theories defined on some categories of G-spaces,where G is a compact Lie group.

Mathematics Subject Classification: 55N20 Key words: G-cohomology theory, weakly S-completeness.

Introduction

An old result of Dold says that if h^{⋆} is a generalized multiplicative cohomology theory defined on the category of CW -complexes and suppose h^{⋆}(X) is a projective module over h^{⋆}(point), then the product map

h^{⋆}(X) ⊗\overline{h} h^{⋆}(Y) → h^{⋆}(X × Y)

is an isomorphism for any Y, where \overline{h} denotes h^{⋆}(point).

The generalization of this theorem is not trivial. There is a well-known example of L. Hodgkin (see [4]): although h_{G}^{⋆}(X) is a free h_{G}^{⋆}(point)-module, this does not imply, in general, that the above product map is an isomorphism.

The beginning steps towards the generalization of Dold’s Theorem come from R. M. Seymour who considered in [7] G-cohomology theories which are ”complete” with regard to a family S of closed subgroups of G.

In this paper, we use one of the two notions of S-completeness defined by Seymour and we state a generalization of the Dold Theorem in terms of conditions concerning S and the associated H-cohomology theories for H ∈ S. Unfortunately we did not obtain satisfactory information regarding the category C where h_{G}^{⋆} is defined.

1

Let us recall some elements which we shall need. First, we shall use the words generalized cohomology theory to mean a family h^{⋆} = (h^{q})_{q∈Z} of contravariant ∂-functors in the sense of [2], defined on a category of pairs .

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 19-23 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

We shall assume h^{⋆} has the following properties:

(i) If f ≃ g, then h^{⋆}(f) = h^{⋆}(g).

(ii) If f is a relative homeomorphism, then h^{⋆}(f) is an isomorphism.

(iii)

h^{⋆}(∐_{α} X_{α}) →^{≅} ∏_{α} h^{⋆}(X_{α}),

for any family of spaces (X_{α}), where ∐ denotes the topological sum (for this definition see also [3]).

We consider now G-cohomology theories defined on the category of G-pairs in the sense of [5], section 5 (here G is a compact Lie group). In fact, we shall recall the generalized Borel construction to obtain a large class of examples: for any family S of closed subgroups of G and any G-space X, define a topological category C(X; S) whose objects are ∐_{H∈S} X^{H} and whose morphisms are

∐_{(H,K)∈S×S} Hom_{G}(G/H, G/K) × X^{K}.

Next we define the space X(S) to be the geometric realization without degeneracies of C(X; S), in the sense of [6], appendix A.

Clearly, if (X, A) is a G-pair, then A(S) is a closed subspace of X(S) and also, (., S) preserves all the above properties. Thus, if h^{⋆} is a generalized cohomology theory, we can define a G-cohomology theory on G-pairs by the formula

h_{G}^{⋆}(X, A; S) = h^{⋆}(X(S), A(S))

and we shall say that h_{G}^{⋆}(.; S) is the G-cohomology theory obtained from h^{⋆} by generalized Borel construction with regard to S.

Finally in this section, for a G-cohomology theory h_{G}^{⋆} and for a closed subgroup H of G, we define the associated H-cohomology theory h_{H}^{⋆} by the formula

h_{H}^{⋆}(X, A) = h_{G}^{⋆}(G ×_{H} X, G ×_{H} A)

where (X, A) is an H-pair and G ×_{H} X denotes the orbit space of G × X under the action of H given by

h · (g, x) = (gh^{−1}, hx).

Observe that,for a G-space X,then G ×_{H} X is naturally G-homeomorphic to G/H × X (here G acts on both factors). Thus the projection G/H × X → X induces a natural transformation

c_{H} : h_{G}^{⋆}(X) → h_{H}^{⋆}(X).

Because the Weyl group of H in G acts as a group of automorphisms of h_{H}^{⋆} restricted to G-spaces (see [7] section 3), the transformation c_{H} maps h_{G}^{⋆}(X) into the invariants of h_{H}^{⋆}(X) under this action. Again note that c_{H} is a ring homomorphism.

2

The following notions were introduced by R. M. Seymour (see [7]). Definiton 1. Let S be a family of closed subgroups of the compact group G. A G-map f : X → Y will be called a strong (G, S)- equivalence if f is an H-equivariant homotopy equivalence for each H ∈ S.

In view of Lemma 2.1 in [7], for the rest of this paper, we shall assume that the family S is closed with respect to conjugation and finite intersection.Such a family is called a complete family of subgroups. Remark. If G ∈ S, then a strong (G, S)-equivalence is just a G-homotopy equivalence. On the other hand, if S = {1}, then f is a strong (G, S)-equivalence if f is a G-map which is a homotopy equivalence. Generally G-homotopy equivalence implies strong (G, S)-equivalence . Definition 2. A G-cohomology theory h_{G}^{⋆} is weakly S-complete if the morphism f^{⋆} : h_{G}^{⋆}(Y) → h_{G}^{⋆}(X), induced by a strong (G, S)-equivalence f : X → Y is an isomorphism. Example. For a compact and connected Lie group G, we consider its maximal torus T and let K_{G}^{⋆} be the equivariant K(cf. [7], 5.1). Remark. The equivariant K-theory is not weakly (1)-complete. This is proved in [1], where an example of a G-map, homotopy equivalence f : X → Y is constructed, but which does not induce the isomorphism K_{G}^{⋆}(Y) → K_{G}^{⋆}(X).

To conclude this section, if h_{G}^{⋆} is a multiplicative G-cohomology theory, we are given a natural, associative pairing

μ_{G} : h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} h_{G}^{⋆}(Y) → h_{G}^{⋆}(X × Y),

where \overline{h}_{G} denotes the graded ring h_{G}^{⋆}(point). We assume \overline{h}_{G} has an identity and μ_{G} is graded commutative.

Again, if H is a closed subgroup of G and X, Y are H-spaces, then there are the pairings

μ_{H} : h_{H}^{⋆}(X) ⊗_{\overline{h}_{H}} h_{H}^{⋆}(Y) → h_{H}^{⋆}(X × Y).

Here we write \overline{h}_{H} for h_{H}^{⋆}(point).

3

Let C be some full subcategory of G-spaces. We shall recall a result of R.M.Seymour (6.1 in [7]): Proposition 1. For h_{G}^{⋆} a multiplicative G-cohomology theory we suppose h_{G}^{⋆}(X) is a projective \overline{h}_{G}-module and that

\overline{μ}_{G} : h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} \overline{h}_{H} → h_{H}^{⋆}(X)

is an isomorphism for each H ∈ S. Then the map μ_{G} from section 2 is an isomorphism for each Y in C.

For weakly S-complete theories we obtain the following result:

Proposition 2. Let S be a complete family and suppose h_{G}^{⋆} is a weakly Scomplete multiplicative theory. Let h_{G}^{⋆}(X) be a projective \overline{h}_{G}-module and suppose \overline{μ}_{G} is an isomorphism for each H ∈ S, X ∈ C. Then the map

μ_{G} : h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} h_{G}^{⋆}(Y) → h_{G}^{⋆}(X × Y)

is also an isomorphism for each Y ∈ C. Proof. We consider the natural transformation between G- cohomology theories

h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} h_{G}^{⋆}(.) → h_{G}^{⋆}(X × .).

Because h_{G}^{⋆}(G/H) = h_{H}^{⋆}(point) and h_{G}^{⋆}(X × G/H) ≅ h_{H}^{⋆}(X × point) = h_{H}^{⋆}(X), using one of the hypotheses, we obtain that the induced morphisms

h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} h_{H}^{⋆}(point) → h_{H}^{⋆}(X × point)

are isomorphisms for each H ∈ S and X ∈ C.

By a standard induction over the skeleton of the CW -complex L we conclude that

h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} h_{G}^{⋆}(L_{n}) → h_{G}^{⋆}(X × L_{n})

is an isomorphism for each n ≥ 0.

By the device of [7], Proposition 3.3, one can obtain that μ_{G} is an isomorphism for any CW -complex L.

Now define a topological category as follows:

Object space is ∐_{H∈S} G/H × Y^{H}

Morphism space is ∐_{(H,K)∈S×S} G/H × Hom_{G}(G/H, G/K) × Y^{K}.

Next observe that G acts on the objects and morphisms of this category by the left action of G on the factors G/H and all structural maps are equivariant regarding this action.Thus,we have an induced G-action on the geometric realization E(Y, S) of the above category.

Because S is a complete family,then E(Y, S) is strongly G-contractible with respect to S(see [7],Proposition 2.2).It follows that the induced morphism

p^{⋆} : h_{G}^{⋆}(X) → h_{G}^{⋆}(E(Y, S) × X)

is an isomorphism,where p : E(Y, S) × X → X is projection onto the second factor.

Finally we take E(Y, S) as the previous CW -complex L and that gives the result. Corollary 1. Let S be a complete family and h_{G}^{⋆} be a weakly S-complete multiplicative theory.We assume that h_{G}^{⋆}(X) is a free \overline{h}_{G}-module on basis B and that c_{H} maps B bijectively onto a basis for h_{H}^{⋆}(X) as a free \overline{h}_{H}-module for each H ∈ S.Then, if the morphism

μ_{H} : h_{H}^{⋆}(X) ⊗_{\overline{h}_{H}} h_{H}^{⋆}(Y) → h_{H}^{⋆}(X × Y)

is an isomorphism for each H ∈ S and X, Y ∈ C,it follows that μ_{G} is an isomorphism for X, Y objects of the category C. Proof. We denote by [B]R the free R-module on basis B.We have the following diagram of morphisms:

[B]\overline{h}_{G} ⊗_{\overline{h}_{G}} \overline{h}_{H}  →^{≅} h_{G}^{⋆}(X) ⊗_{\overline{h}_{G}} \overline{h}_{H} →^{μ} h_{H}^{⋆}(X)
c_{H} ⊗ 1↓  c_{H} ⊗ 1↓  ||

[c_{H}(B)]\overline{h}_{H} ⊗_{\overline{h}_{H}} \overline{h}_{H} →^{≅} h_{H}^{⋆}(X) ⊗_{\overline{h}_{H}} \overline{h}_{H} →^{≅} h_{H}^{⋆}(X)

The two vertical maps are isomorphisms by hypothesis and the first two horizontal isomorphisms are induced by the inclusions B ⊂ h_{G}^{⋆}(X) and

c_{H}(B) ⊂ h_{H}^{⋆}(X).Finally the second map from the lower line of diagram is a canonical isomorphism. It follows that μ is an isomorphism and applying the previous proposition we obtain the result.

In [7] R. M. Seymour obtained some conditions under which the hypotheses of above corollary hold (see proposition 6.3, [7]).

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

References

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[3] P.Hilton,General cohomology theory and K-theory, London Math. Soc. Lecture Notes 1, Cambridge 1971.

[4] L.Hodgkin, The equivariant Künneth formula in K-theory, Lectures Notes in Math. 496, Springer-Verlag, Berlin, 1975.

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Department of Mathematics Tehnical Univ. of Civil Engineering of Bucharest 72302 Bucharest, Romania