Equivariant Bundles Research Articles

Stable triples, equivariant bundles and dimensional reduction

This is a resubmission of preprint 9401008 , which has some TeXnical errors introduced by the reform procedure (designed to avoid precisely these problems!). The original can be formatted by editing out the messages %% following line cannot be broken... at line 1674. This problem has been corrected in the current version. The content of the paper has not been changed in any way. My aplogies for any inconvenience caused. SB

The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory

The geometry of the Lubin-Tate space of deformations of a formal group is studied via an étale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of the deformation space which, in turn, yields a formula for the dualizing complex in stable homotopy theory.

EQUIVARIANT BUNDLES ON TORAL VARIETIES

Equivariant bundles on toral varieties are described in terms of filtrations which arise canonically in the fiber over a fixed point. The cohomology groups and characteristic classes are computed in terms of these filtrations, and problems of linear algebra which arise from them are discussed. Bibliography: 20 titles.

Equivariant determinant bundles and supersymmetric closed strings

Abstract This work is to point out the relationship between the geometric approach to the string theories of Bowick and Rajeev, and Pilch and Warner and the recent research on the index of the Dirac operator in loop space. We consider character-valued Dirac index bundles instead of the vacuum bundle over some parameter space such as Diff S1/S1. Thus, some necessary conditions for the absence of the Virasoro anomaly can be derived from some formulas presented here.

Characteristic classes in borel cohomology

Let G be a topological group and let EC be a free contractible G-space. The Bore1 construction on a G-space X is the orbit space Xo = EC x G X. When asked what equivariant cohomology is, most people would answer Bore1 cohomology, namely H:(X) = H*(X,). This theory has the claim of priority and the merit of ready computability, and many very beautiful results have been proven with it. However, it suffers from the defects of its virtues. Precisely, it is ‘invariant’, in the sense that a G-map f: X + Y which is a nonequivariant homotopy equivalence induces an isomorphism on Bore1 cohomology. A quick way to see this is to observe that 1 x f: EC xX+ EC x Y is a map of principal G-bundles with base map lxof=fo:Xo-‘Y,, so that fc is a weak homotopy equivalence. As explained in [l], this invariance property is the crudest of a hierarchy of such properties that a theory might have. We shall show how to compute all characteristic classes in any invariant equivariant cohomology theory, the conclusion being that no such theory is powerful enough to support a very useful theory of characteristic classes. As explained in [2], a less crude invariance property can sometimes be exploited to obtain a calculation of equivariant characteristic classes in more powerful theories, such as equivariant K-theory. To establish context, consider a closed normal subgroup Z7 of a topological group r with quotient group G. If Y is a n-free r-space, we think of the orbit projection q : Y-* Y/Z7 as a particular kind of equivariant bundle. It is a principal U-bundle in the usual sense, its base space is a G-space and thus a Z7-trivial r-space, and its projection is a r-map. The classical case is r= G x Z7. Here q is called a principal (G, Ill)-bundle, and there is a theory of associated (G, I;T)-bundles exactly as in the nonequivariant case. For example, if G is a compact Lie group acting smoothly on a differentiable manifold M”, then the tangent n-plane bundle of M is a (G, O(n))bundle and is determined by its associated principal (G, O(n))-bundle. See e.g. [4; 5; 7, V $11 for background. There is a universal bundle E(Z7;r) -+ B(I7;I’) = E(Zl7; r)/Ii’ of the sort just specified. Its total space E(L7;T) is a Z7-free T-CW complex such that the /l-fixed point space E(Z7;r)” is contractible for any closed subgroup A of r such that

Equivariant Bundles and Cohomology

Let $G$ be a topological group, $A$ an abelian topological group on which $G$ acts continuously and $X$ a $G$-space. We define "equivariant cohomology groups" of $X$ with coefficients in $A$, $H_G^i(X;A)$, for $i \geq 0$ which generalize Graeme Segal’s continuous cohomology of the topological group $G$ with coefficients in $A$. In particular we have $H_G^1(X;A) \simeq$ equivalence classes of principal $(G,A)$-bundles over $X$. We show that when $G$ is a compact Lie group and $A$ an abelian Lie group we have for $i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)$ where $\tau A$ is the sheaf of germs of sections of the bundle $(X \times EG \times A)/G \to (X \times EG)/G$. For $i = 1$ and the trivial action of $G$ on $A$ this is a theorem of Lashof, May and Segal.

Equivariant bundles and cohomology

Let G G be a topological group, A A an abelian topological group on which G G acts continuously and X X a G G -space. We define "equivariant cohomology groups" of X X with coefficients in A A , H G i ( X ; A ) H_G^i(X;A) , for i ≥ 0 i \geq 0 which generalize Graeme Segal’s continuous cohomology of the topological group G G with coefficients in A A . In particular we have H G 1 ( X ; A ) ≃ H_G^1(X;A) \simeq equivalence classes of principal ( G , A ) (G,A) -bundles over X X . We show that when G G is a compact Lie group and A A an abelian Lie group we have for i > 1 H G i ( X ; A ) ≃ H i ( E G × G X ; τ A ) i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A) where τ A \tau A is the sheaf of germs of sections of the bundle ( X × E G × A ) / G → ( X × E G ) / G (X \times EG \times A)/G \to (X \times EG)/G . For i = 1 i = 1 and the trivial action of G G on A A this is a theorem of Lashof, May and Segal.

Equivariant bundles over a single orbit type

Equivariant bundles over a single orbit type

Subjective geometry and geometric psychology

“Subjective geometry” is a term coined by Weintraub and Krantz to describe the distortion imposed upon geometric patterns by the visual system itself—so-called optical illusions. The latter are widely regarded as being generated by misplaced “constancy” effects, i.e., they are regarded as stemming from the invariance of an object's appearance under wide variations in viewing conditions, such as obliquity, rotations, etc. The invariances represented by these constancies—shape constancy, size constancy, etc.—are spatiotemporal invariants of certain Lie subgroups of P 4 ( R ) ⊕ CO (1, 3) ⊕ GL (4, R ) that govern Euclidean and non-Euclidean geometry. Tha Euclidean subgroups describe a Cyclopean visual world; the non-Euclidean, a binocular (bipolar) world of hyperbolic nature, according to the work of Luneburg, Blank, Indow, and others. The visual field of view is itself a geometric object involvingnot only “figure” and “ground” but also visual contours (orbits of the Lie groups involved), linear perspective, interposition, and contact and symplectic structures. The retina and “cortical retina” are both covered by a family of “circular-surround” cellular response fields (of a “Mexican hat” nature) which constitute an atlas for the visual manifold S . Upon this manifold are defined certain equivariant vector bundles that account for constancy phenomena and certain jet bundles, arising out of the vector bundles by prolongation, that generate the differential invariants characterizing higher form perception. The resultant theory of perceptual-cognitive processing has been termed “geometric psychology,” in analogy to MacLane's “geometrical mechanics” and Brockett–Hermann–Mayne's “geometry of systems,” the mathematical structure being very similar in all three instances. Functorial maps from the category G v FB(S) of equivariant fibre bundles to the simplicial category and the category of simplicial objects complete the theory by extending the perceptual system to cognitive phenomena and information-processing psychology.

Semifree actions on homotopy spheres

In this paper, we study the semifree Z m {Z_m} actions on homotopy sphere pairs. We show that in some cases the equivariant normal bundle to the fixed point set is equivariantly stably trivial. We compute the rank of the torsion free part of the group of semifree actions on homotopy sphere pairs in some cases. We also show that there exist infinitely many semifree Z 4 s {Z_{4s}} actions on even dimensional homotopy sphere pairs.