Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

Non-symplectic smooth circle actions on symplectic manifolds

We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.     

Chern and Pontryagin Numbers in Perfect Symmetries of Spheres

The paper presents a procedure for constructing smooth actions of finite perfect groups on spheres with fixed point sets having certain prescribed properties (Theorem A); in particular, having any prescribed configuration of Chern and Pontryagin numbers (Corollary C). The main ingredients used are equivariant thickening and equivariant surgery.     

Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks

The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawaowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finiteG CW complex, and an element in a K theoretic group arising as an obstruction class of geometric object f. It often holds that (1-[C])^m becomes trivial for large integers m where [C] is the element represented byC in the Burnside ring (G). One expects that the algebraic object (1 - [C])^m is realizable as the obstruction class of G connected sum of f's related to (1 - [C])^m. Since it is true for the case here, we can kill the obstruction by taking G connected sum off's     

A converse to the P. A. Smith theorem for nonunitary homology spheres

If p is an odd prime and F is the fixed point set of a smooth Z_p action on Sⁿ or Dⁿ, then F is a smooth manifold with a unitary structure. Conversely; most Z_p homology disks or spheres with unitary structures are fixed point sets of smooth Z_p actions on Dⁿ or Sⁿ for suitable n. The results of this paper show that an arbitrary oriented mod p homology disk or sphere is the fixed point set of a smooth Z_p action on some Z[l/2]-homology disk or sphere. This result is in general the best possible.Partially supported by NSF Grants MCS 81-04852 and MCS 83-00669     

Torus actions,fixed-point formulas,elliptic genera and positive curvature

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.     

Equivariant Chow groups for torus actions

We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.     

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

     

Torsion in equivariant cohomology and Cohen-Macaulay G-actions

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.     

On the topological classification of pseudofree group actions on 4-manifolds: II

This paper is concerned with the algebraic aspects of the classification of pseudofree, locally linear group actions on a simply connected 4-manifold, particularly with the splitting and stability properties of the associated Hermitian intersection module and its isometry group. Our main result is the proof of stability of the equivariant intersection form for a large class of pseudofree actions. We also prove a topological rigidity theorem stating that two locally linear, pseudofree actions on a closed, oriented, simply connected 4-manifold, with the equivariant intersection forms indefinite and of rank at least 3 at each irreducible character, are topologically conjugate by an orientation preserving homeomorphism if and only if their oriented local representations at the corresponding fixed points are linearly equivalent.Partially supported by the N.S.F.     

Transitive actions on Hopf homogeneous spaces

In this paper we investigate transitive actions of compact connected Lie groups on certain spaces X which are not spheres, whose dimension is not too small and whose rational cohomology algebra is an exterior algebra on homogeneous generators of odd degree. In case X is a simply connected classical group, a 3-connected real or a 5-connected complex or a quaternionic Stiefel manifold, we obtain (in principle) the classification of the transitive actions on X up to equivariant homeomorphism.     

On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.     

六阶对流Cahn-Hilliard方程解的整体适定性

In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then by combining some a priori estimates,continuity argument,the local smooth solution is extended step by step to all t>0 provided that the L1 norm of initial data is suitably small and the smooth nonlinear functions f(u)and g(u)satisfy certain local growth conditions at some fixed point■.     

On differentiable compactifications of the hyperbolic plane and algebraic actions of $${\rm SL}_2(\mathbb{R})$$ on surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2;R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. The online version of the original article can be found under doi: .     

G-invariante Morse-funktionen

If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index are in a one-one correspondence to the -cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.     

On Differentiable Compactifications of the Hyperbolic Plane and Algebraic Actions of SL2(\mathbb R) on Surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2,R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. An erratum to this article can be found at     

The Universal Functorial Equivariant Lefschetz Invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K₀ of the category of “ ϕ -endomorphisms of finitely generated free RΠ(G, X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds. Received: February 2006     

On the fixed point sets of smooth involutions on the products of spheres

In this paper, we have, under some conditions on cohomology, that the fixed point set of a smooth involution on a product of spheres is of constant dimension.

     

Integral Deligne cohomology for real varieties

We develop an integral version of Deligne cohomology for smooth proper real varieties. For this purpose the role played by singular cohomology in the complex case has to be replaced by the ordinary bigraded Gal(\mathbbC/\mathbbR){Gal(\mathbb{C}/{\mathbb{R}})}-equivariant cohomology of Lewis et al. (Bull Am Math Soc (N.S.) 4(2):208–212, 1981), the equivariant counterpart of singular cohomology. The theory is aimed at giving more precise information about the 2-primary components of regulators. We establish basic properties and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2.