Showing papers on "Equivariant map published in 1972"

The equivariant Plateau problem and interior regularity

Abstract: Let M C R\" be a compact submanifold of Euclidean space which is invariant by a compact group G C SO (zz). When dim (M) = zz 2, it is shown that there always exists a solution to the Plateau problem for M which is invariant by G and, furthermore, that uniqueness of this solution among G-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for M within the class of G-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space R/G where, for \"big\" groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds M, the cone C(M) = \\tx; x € M and 0 < t < l! is the unique solution to the Plateau problem for M, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type S\" x S\" C R n+ , zz > 3, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in R and examples in R\" for zz > 10 with any prescribed Betti number nonzero.

Equivariant Pontrjagin Classes and Applications to Orbit Spaces

Abstract: L-classes of rational homology manifolds.- L-classes of symmetric froducts.- The G-signature theorem and some elementary number theory.

Equivariant Pontrjagin Classes and Applications to Orbit Spaces: Applications of the G-signature Theorem to Transformation Groups, Symmetric Products and Number Theory

Abstract: L-classes of rational homology manifolds.- L-classes of symmetric froducts.- The G-signature theorem and some elementary number theory.

Equivariant bordism and cyclic groups

Abstract: For a finite cyclic group G the equivariant complex bordism module Q (G) is shown to be a free module over Qu

Harmonic mapping of spheres

Abstract: This thesis is addressed to the following fundamental problem: given a homotopy class of maps between compact Riemannian manifolds N and M, is there a harmonic representative of that class? Eells and Sampson have given a general existence theorem for the case that M has no positive sectional curvatures [ESJ. Otherwise, very little is known. Certainly no counter-example has ever been established. The most important contributions of this dissertation are two: firstly, we have a direct construction technique for producing some essential harmonic maps between Euclidean spheres. Topologically, this consists simply of joining two harmonic polynomial mappings (e.g., the Hopf fibrations). Analytically, however, this method has a novel physical motivation: we study the equation of motion of an exotic pendulum driven by a gravity which chances sign. If this system has an exceptional trajectory of the right sort, it defines a harmonic map of spheres. One consequence or our theorem is that πn(Sn) is represented by harmonic maps for n= 1,...,7. Finally, the rudiments of an equivariant theory of harmonic maps having been set out earlier, we find that our examples can also be put in this framework. The second significant result which arose from this study is a strong candidate for a counterexample: suppose Sn is stretched to a length b in one direction to make an ellipsoid En(b). Then if n > 3 and b is large enough, there is no harmonic stretching (of degree one) of Sn onto En(b). However, if b=1 the identity is such a harmonic map, so it certainly appears that the existence of a harmonic representative in a homotopy class can depend upon the metric. We also examine here a large collection of examples of harmonic maps of spheres which are defined by harmonic polynomials and orthogonal multiplications. The last chapter takes up the study of the Morse theory of a harmonic map: amongst several pleasing results, we have an example of a simple map whose index and degeneracy can be made arbitrarily large by equally simple changes in the metrics.

A commutative diagram and an application to differentiable transformation groups

Abstract: A commutative diagram is presented which relates the groups of concordance classes of diffeomorphisms f(S2n), F(CPU) and f(S2n+l). This diagram is applied to show that every equivariant diffeomorphism of S7 is concordant to the identity. It follows that the exotic 8-sphere, Y8, admits no smooth semifree SI-action with exactly two fixed points. Introduction. In this paper we shall present a commutative diagram (Theorem 1) involving the stable homotopy group 1711 and the groups r(Sn), r+(CPn) defined below. This diagram is then applied to show that every orientation preserving diffeomorphism of S7 which is equivariant with respect to the standard free action of S' on S7 iS concordant to the identity. Finally we show, using a result of R. Lee [3], that the exotic 8sphere, E8, does not admit a smooth action by S' which is semifree with exactly two fixed points. The paper concludes with a brief discussion of possible further applications of Theorem 1 to construction of smooth actions of S' on exotic spheres such that the actions are semifree with exactly two fixed points. The results of this paper are contained in the author's doctoral dissertation written at the University of Massachusetts. The author wishes to express his indebtedness to Professor J. C. Su for his generous help while the author was a graduate student. The author also acknowledges his gratitude for suggestions by a referee which resulted in a considerable shortening of the proof of Theorem 1. Preliminaries. Iffo and fi are diffeomorphisms of M onto N, M and N CO-manifolds, we sayfo is concordant tof1 if there is a diffeomorphism F:M x I->-N x I,I= [0, 1], suchthatforallxeMwehaveF(x, 0) = (fo(x), 0) and F(x, 1) = (f1(x), 1). The relation of concordance is an equivalence relation on the set of all diffeomorphisms of M onto N. If f :M -* N is a diffeomorphism its concordance class will be denoted [f]. If M is an orientable manifold there is a group, r(M), of all concordance Received by the editors March 15, 1971. AMS 1970 subject classifications. Primary 55E45, 57D60, 57E25.

Equivariant integrality theorems for differentiable manifolds

Abstract: In differential topology it is often useful to be able to find restrictions on the possible vector bundles over a given manifold. For the non-equivariant case these restrictions usually state that some rational multinomial in the various charac teristic classes is an integral multiple of the fundamental cocyle.

equivariant bordism of maps

Abstract: It is shown that A-equivariant maps may be classified up to bordism where the acting groups are abelian ot odd order.