Showing papers on "Equivariant map published in 1976"

A Geometric Approach to Homology Theory

Abstract: 1. Homotopy functors 2. Mock bundles 3. Coefficients 4. Geometric theories 5. Equivariant theories and operations 6. Sheaves 7. The geometry of CW complexes.

Linking forms and maps of odd prime order

Abstract: A differentiable orientation preserving map of odd prime period on a closed oriented differentiable manifold gives rise to two invariants taking values in a Witt group of bilinear forms. One is globally defined in terms of the rational cohomology of the manifold and the other is locally defined in terms of the fixed point set and its normal bundle. We show that these two invariants are, in fact, equal and apply this result to relate the structure of the manifold to that of the fixed point set and the quotient space. Let M4n be a closed oriented differentiable manifold and Tbe an orientation preserving diffeomorphism of M of odd prime period p. Using the representation T* of Zp in H 2n(M; Q), Conner and Raymond [81 defined a torsion element q(T, M) in the rational Witt ring W(Q). This is an invariant of the equivariant cobordism class of the action and vanishes if the action is fixed point free; hence it may be expressed in terms of the fixed point set and the action of Zp in its normal bundle. Conner and Raymond gave such an expression for p = 3. In [41 we announced a formula for all odd primes. The purpose of this paper is to give the details of the proofs along with some additional background and applications. An essential factor in the proof is the relationship between rational forms and forms on finite abelian groups. This approach has also been effective in dealing with other problems 111, [21. Here we use it to relate the peripheral invariant for a compact bounding (4n l)-dimensional manifold V, an element of W(Q) defined in [8], to the linking form on the torsion subgroup of H 2n(V; Z). ?1 contains the requisite algebraic material on bilinear forms and Witt groups. In the second section the peripheral invariant is defined and is shown to correspond to the negative of the linking form in the Witt group of finite forms. The formula expressing q(T, M) in terms of the fixed point data is established in ?3 and applied to prove the following: (3.3) If T* is the identity on H2n(M; Q), then (a) for p 3 (mod 4), sgn M sgn F (mod 4); Received by the editors May 5, 1975. AMS (MOS) subject classifications (1970). Primary 5SC35, 57D99; Secondary 1SA63. (1) This research was supported by NSF Grant GP-43775. Copyright

Borel Cross-Sections and Maximal Invariants

Abstract: A measurable cross-section for orbits of a sample space under a free (exact) transformation group is shown to exist under topological regularity conditions. This is used to represent the sample space as essentially the product of a maximal invariant and an equivariant part, which implies Stein's representation for the density of the maximal invariant.

Equivariant stable homotopy and framed bordism

Abstract: This paper gives an elementary proof of the result that equivariant stable homotopy is the same as equivariant framed bordism.

Nonimmersion of lens spaces with 2-torsion

Abstract: From a study of the equivariant unitary K-theory of the Stiefel manifold Vk 1+2(C), it is shown that the lens space Lk(n), with n a multi- ple of 22k-1-a(k-l) does not immerse in Euclidean space of dimension 4k -

Characteristic numbers for unoriented singular $G$-bordism

Abstract: We develop the notion of characteristic numbers for unoriented singular G-manifolds in a G-space, G being a finite group, and prove their invariance with respect to unoriented singular G-bordism. Thom [5] gave the notion of Stiefel Whitney numbers and Pontrjagin numbers of a manifold Mn and proved its invariance with respect to bordism. Chung N. Lee and Arthur Wasserman [4] developed these notions for Gmanifolds. In this note we have developed these notions for unoriented singular principal G-manifolds in a G-space, G being a finite group, and proved their invariance with regard to unoriented singular G-bordism. 1. Characteristic numbers. Let X be a finite CW-complex with free action of G, G being a finite group, and X/G be again a finite CW-complex. Let h* be an equivariant cohomology theory and h* be the associated equivariant homology theory [1]. Let h* = H* ? A and h* = HI* A, where A is a functor from the category of G-spaces and equivariant maps to the category of topological spaces and continuous maps, H* is the singular cohomology theory and H* is the associated singular homology theory. Let : h*(X; G) (H(pt.) h*(X; G) -* H*(pt.) be the Kronecker pairing. Let us assign to each compact G-manifold W, a class [W,aW] E h*(W,aW;G) such that (a) [WI U W2, aWI U aW2] = [WI,aWI] + [W2,aW2 (b) a*[W,aw] = [awl. Suppose [M',f; G] is an element of unoriented bordism group R,(X; G) [3] and x E h*(B(O,G) );G),B(O,G),, being the classifying space for G-vector bundles of dimension n. Then the x-characteristic number of the map f: Mn X associated with an element a'm E hm(X; G) is defined to be Received by the editors August 22, 1974. AMS (MOS) subject classifications (1970). Primary 57D85.

Homotopy Triangulations of Pseudo Lens Spaces and Actions on Products of Spheres

Abstract: In the last decade the classification of free actions of cyclic groups Zr on homotopy spheres has been studied extensively. Given a free Zr action on a homotopy sphere the orbit space is homotopy equivalent (but not necessarily simple homotopy equivalent) to a lens space [t13]. So the study of free Zr actions on homotopy spheres is equivalent to the problem of classifying manifolds of the homotopy type of lens spaces. Surgery theory gives a complete classification in the PL and topological categories. The case r = 2 was attacked by Lopez de Medrano [8] and Wall [16, § 14D]. The case r odd was settled in the work of R. Lee [7], Browder et al. [4, 5]. As a next step one can consider free Zr actions on manifolds E homotopy equivalent to the cartesian product of two spheres. One might then trie to replace the lens space by a manifold defined as follows: Let a, b complex representations of Zr of degree p + 1, q + 1 in V~, V b. Let S,, Sb the unit spheres in V,, Vb and assume that they are free Zr-manifolds. Our candidade is then M'=S,, x Sb/Z r. Such an orbit space we call a pseudo lens space. However in general it is not true that the orbit space E/Zr is homotopy equivalent to such a pseudo lens space. But we think it is worthwile to study those actions having this property. They can be obtained from actions whose orbit space is simple homotopy equivalent to M by operation of the Whiteheadgroup W(Z,). So we further restrict to so called simple actions, thus obtaining at least a lower bound for the set of all actions. The precise problem we study is the following: In the PL category we consider m-dimensional closed oriented manifolds E with free Zr-actions, an orientation preserving Zfequivariant homotopy equivalence e:E---,SoxSb inducing a simple homotopy equivalence between orbit spaces. Two such (E i, e~), i= 1, 2 are considered equivalent if there is an equivariant PLhomeomorphism c:Et~Ez with e2oc equivariantly homotopic to e t. The set 5~(M) of equivalence classes of simple actions is in 1 I correspondence to the set hT(M) of equivalence classes of homotopy triangulations [16, p. 102] on M. The sugery exact sequence [16, p. 107, 1 t l ]