Showing papers on "Equivariant map published in 1974"

Equivariant homology and duality

Abstract: This note is concerned with stable G-equivariant homology and cohomology theories (G a compact Lie group). In important cases, when H-equivariant theories are defined naturally for all closed subgroups H of G, we show that the G-(co)homology groups of G xH X are isomorphic with H-(co)homology groups of X. We introduce the concept of orientability of G-vector bundles and manifolds with respect to an equivariant cohomology theory and prove a duality theorem which implies an equivariant analogue of Poincare-Lefschetz duality.

Equivariant function spaces and stable homotopy theory I

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Bordismengruppen unitärer Torusmannigfaltigkeiten

Abstract: Let G be a k-dimensional torus. Let U * G , denote the homotopical unitary bordism theory. We show that restriction to the fixed point set determines an element of U * G . This implies that the bordism class of a unitary G-manifold is determined by its characteristic numbers in equivariant K-theory.

Line Bundles over Framed Manifolds

Abstract: where IKP(oo) denotes the oo dimensional (left) IK projective space, d=dim~aIK, and of, r( ) denotes the framed bordism homology theory. In this note we develop some elementary properties of S~:, and show how the Adams-Toda complex e invariant er 2]~Q/7Z can be computed in terms of 2 $ M. We then apply these results to finite cyclic coverings of framed manifolds coming from principal circle actions. This is followed by applications to homogeneous spaces of compact Lie groups. After indicating the modifications needed for the quaternionic case we close the paper with some results indicative of the more general theory of equivariant framed manifolds which we develop in a sequel. It is a pleasure to thank the Mathematische Institut at Oberwolfach for their kind hospitality during the topology conference in September 1973 when this research was begun.

Almost free actions on manifolds

Abstract: Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and(2) the (even) number of fixed points.Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

Free ¹ actions and the group of diffeomorphisms

Abstract: Let S1 act linearly on S2px D2q and D2p x S2q-1 and let f: S2p-1 x S2q-I _ S2px S2q-1 be an equivariant diffeomorphism. Then there is a well-defined S1 action on S2p-I x D2q UD2p x S2q_I. An S1 action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres. 0. Introduction. Let 51 act linearly on Cm+1 by (g, (uo,., um))= (gu0,.* * *, gum) for g ES 1 and (uo, . . ., um) ecm+l. Let p = [(m + 1)/2] and q = m + 1 p. It is clear that 52p_1 x D 2q, D2p x S2q1 and S2P-l X S2q-I are invariant subspaces. Let A denote the induced actions. Let / be an equivariant diffeomorphism of (S2P-1 x S2q-1, A). We can define an action A(f) on 1(f) where 52p-1 x D2q U D2p x S2q-I so that A(/) I S2pxD2= A and A(/) I D2p x S2qI-I A. A free S5 action (12m+1, F) on a homotopy sphere 12m+1 is decomposable if there is an equivariant diffeomorphism / of (S2p-l x S2q-l, A) such that (12m+l, F) is equivalent to (Z(/), A(/)). It is clear that if f is equivariantly pseudo-isotopic to g, then (E(f), A(1/)) is equivalent to (E(g), A(g)). Hence the study of decomposable actions is reduced to the study of the group of equivariant pseudo-isotopy classes of equivariant diffeomorphisms of (S2p-1 x S2q-l, A) or, equivalently, the group of diffeomorphisms of S2P-1 x S2q-I/A. Let (E(f), A(/)) and (:(g), A(g)) be two decomposable actions. We define (E(f), A(/)) * (X(g), A(g)) = (Y(f . g), A(f . g)). We will show this is well defined and makes the set of decomposable free 51 actions a group such that the splitting invariants are homomorphisms. Furthermore, we are able to calculate its rank and determine its torsion elements. As applications we have Received by the editors March 5, 1973 and, in revised form, July 19, 1973. AMS (MOS) subject classifications (1970). Primary 57E30.

Equivariant homotopy theory

Abstract: In this note we announce an obstruction theory for extending (continuous) equivariant maps defined on a certain class of G-spaces, where G is a compact Lie group. The details of this work will be published elsewhere. Our results barely touch upon the attendant problem of providing techniques that would serve in practice for the computation of the obstruction groups. In general this last problem presents considerably greater difficulties than in the case of a finite group G, which has been treated fairly exhaustively in [1]. The author expresses his deep gratitude to Professor Glen E. Bredon in consultation with whom these results were obtained. Let G be a compact Lie group. If H is a closed subgroup of G, a closed Gstem of type (H) and equivariant dimension n is defined to be a Gspace which is equivariantly homeomorphic to B x G/H, where B is the standard «-cell, GjH is the homogeneous G-space consisting of the left cosets of H in G, and the action of G is the product of the trivial action on B and the usual action on G/H. A Hausdorff G-space K is said to be a G-complex if it is filtered by an ascending sequence of closed invariant subspaces K, whose union is K, such that Kr=0 and, for each n9 K is obtained from K** by attaching any number of «-stems by equivariant maps defined on the boundaries S~~xG/H of the standard «-stems B X GjH. A G-complex K is also required to have the topology coherent with the sequence of subspaces K. The least integer n such that K=K is called the equivariant dimension of K and is denoted by dim# K. The class of G-complexes is the analogue in the equiyariant category of CW-complexes in the topological category. (See [3] for the definition and some of the properties of a CW-complex.) When G is a finite group G-complexes have been defined by Bredon in [1]. Our notion is derived from his and we extend his techniques to the more general case. Matumoto has defined in [2] what he calls a G-CW-complex. His definition is equivalent to that of a G-complex K whose orbit space K\\G is a locally finite CW-complex. He has also indicated in [2] a proof of the important result that a differentiable G-manifold is a G-CW-complex

The image of unitary bordism in unoriented bordism—the equivariant case

Abstract: It is shown that the image of unitary G-bordism in unoriented G-bordism is the set of squares in unoriented G=bordism if G is Z2 and properly contains the squares if G is an odd order finite group. 1. Statement of results. In [3], Milnor shows that the image of the reduction homomorphism from unitary to unoriented bordism is the same as the image of the squaring homomorphism from unoriented bordism into itself. We consider an equivariant version of this result. Specifically, if U*G and N*G denote, respectively, the unitary and unoriented equivariant bordism of smooth G actions on closed manifolds (with unrestricted isotropy subgroups), p: UGNG denotes the reduction homomorphism which "'forgets" the stably almost complex structure, and o: N G -+NG is defined by o(x) = x2, then Theorem 1. If G = Z2 then the image of p equals the image of o. Theorem 2. If the order of G is odd, then the image of a is properly contained in the image of p. 2. Proof of Theorem 1. Denote by U*(X) and N*(X), respectively, the unitary and unoriented bordism of the space X. Assigning to an involution on a manifold, the normal bundle to its fixed set gives rise to fixed point homomorphisms z 2Z F: N2 DN*(BO(k)) and Fu: U2* 2 +U*(BU(k))There are exact sequences: (1) 0 o U * 2 U* -2 ffl u* 1(BoU()) -* 0 k=~O Received by the editors October 18, 1973. AMS (MOS) subject classifications (1970). Primary 57D85.

Classifying spaces for equivariant k-theory

Abstract: In this paper the methods of M. Karoubi (MR 41 #6205) are generalized to the case of equivariant K-theory. The sets of Fredholm operators in certain (Hilbert) spaces of representations of finite groups G are described which are classifying spaces for equivariant K-functors. The results were announced in the paper MR 46 #2702. Bibliography: 16 items.

Equivariant method for periodic maps

Abstract: The notion of coherency with submanifolds for a Morse function on a manifold is introduced and discussed in a general way. A Morse inequality for a given periodic transformation which compares the invariants called qth Euler numbers on fixed point set and the invariants called qth Lefschetz numbers of the transformations is thus obtained. This gives a fixed point theorem in terms of qth Lefschetz number for arbitrary q. Letf be a periodic transformation of a closed m-dimensional manifold M with fixed point set N. We develop in this note an equivariant approach using Morse theory. We introduce in ?2 the notion of coherency with a submanifold S of M for a Morse function and show that such S-coherent Morse functions are dense in C (M). Furthermore, in this approximation f-invariance will be preserved (?3). The coherency with the fixed point set N of f makes it possible to compare the difference of qth Euler number of N and qth Lefschetz number of f. More precisely, let /8q(N) and Aq(f) be respectively the qth Betti numbers of N and the trace of f* on the qth homology group Hq(M) with real coefficients. Let Bq(N) and Aq(f) be their alternative sums respectively, i.e., 13q(N) = Aq(N) 13q_j(N) + _.. + (_1)qpo3(N), Aq(f) = Xq(fq) -X l (q) + *_* * + (-l)qXo(f), where 0 < q < m. We establish in ?5 an inequality for arbitrary q that I 1q(N) Aq(f A) is no greater than the qth Morse difference of an arbitrary finvariant N-coherent Morse function. We obtain as corollaries a fixed point theorem in terms of arbitrary Aq (when q = m, this is the Lefschetz fixed point theorem) and a more geometric proof of the fact that P. (N) = An(f), i.e., the Euler number of N is equal to the Lefschetz number of f. The Lemma 1 (? 1) which states that a smooth function can be approximated by a Morse function with prescribed "boundary value" is essential to the construction of the approximations. 1. A Morse extension. For a real-valued smooth function F on M, let C(F) denote the set of all critical points of F. F is called a Morse function if for any p E C(F), the determinant of the Hessian at p does not vanish. We assume without loss of generality that M is a riemannian manifold with a metric g. Let g,j be the metric tensor of g with respect to a local coordinate (xi) Received by the editors November 12, 1971 and, in revised form, June 24, 1972. AMS (MOS) subject classifications (1970). Primary 57D70, 53C99.

Equivariant immersion and imbedding up to cobordism

Abstract: This paper determines the smallest possible integer r so that a given manifold with involution is equivariantly bordant to an immersed or imbedded submanifold of some Rx Rr with involution I x (-1)