Showing papers on "Equivariant map published in 1977"

Circle actions on simply connected 4-manifolds

Abstract: Locally smooth S'-actions on simply connected 4-manifolds are studied in terms of their weighted orbit spaces. An equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected 4-manifold with S '-action. This is used to show that a simply connected 4-manifold which admits a locally smooth 5'-action must be homotopy equivalent to a connected sum of copies of S4, CP2, CP2, and S2 x S2.

Non-linear fredholm maps and the leray-schauder theory

Abstract: ContentsIntroduction § 1. Banach manifolds and their maps § 2. The degree of a Fredholm map § 3. Equivariant Fredholm maps § 4. Solubility of equations with Fredholm operators § 5. Some applications to existence theorems for differential equations § 6. Complete and local invertibility of proper maps § 7. Intersection indicesReferences

Real and complex singularities, Oslo 1976 : proceedings of the Nordic Summer School/NAVF Symposium in Mathematics, Oslo, August 5-25, 1976

Abstract: Irreducibility of the compactified Jacobian.- Quasianalytic and parametric spaces.- Topologie normaler gewichtet homogener Flachen.- Generic equivariant maps.- Cohomology of a type of affine hypersurfaces.- Residual intersections and the double point formula.- Intersecting cycles on an algebraic variety.- Stratification and flatness.- Exakte Sequenzen fur globale und lokale Poincare-Homomorphismen.- The enumerative theory of singularities.- Some remarks on relative monodromy.- The multiplicity of a holomorphic map at an isolated critical point.- Numerical characters of a curve in projective n-space.- Periodicities in Arnold's lists of singularities.- Mixed Hodge structure on the vanishing cohomology.- The hunting of invariants in the geometry of discriminants.- Counterexamples in stratification theory: two discordant horns.

Stratifications of equivariant varieties

Abstract: Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

The topological structure of $4$ manifolds with effective torus actions (II)

Abstract: Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of S4, S2 X S2, cp2 Cp-2, SI X S3, and three families of elementary 4-manifolds, R(m, n), T(m, n; m', n'), L(n; p, q; m) has been obtained (where m, n, m', n', p, and q are integers). In addition, a stable homeomorphic relation for the manifolds R(m, n) and T(m, n; m', n') can also be found in [9]. But the topological classification of R's, T's, and especially L's were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R's and T's, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds L(n;p, q; m) form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of L(n; p, q; m) is finite cyclic of order n. And it is proved that ,r(L(n;p,q;m)) and irj (L(n; p',q';m')) act identically on the second cohomology of their common universal covering space, (S2 X S2) # ... # (S2 X S2) (n 1 copies), even though L(n; p, q; m) and L(n; p', q'; m') are not homotopically equivalent for some (n;p',q'; m')'s. This family of manifolds is explicitly constructed and completely classified. In addition, a normal form is imposed on a connected sum decomposition mentioned above. These nromal forms are unique. Most of the material of this paper appeared first in the author's doctoral dissertation. The author would like to thank his thesis advisor Professor F. Raymond for his help and encouragement.

Taming free circle actions

Abstract: It is shown that an arbitrary free action of the circle group on a closed manifold of dimension at least six is concordant to a "tame" action (so that the orbit space is a manifold). A consequence is that the concordance classification of arbitrary free actions of the circle on a simply connected manifold is the same as the equivariant homeomorphism classification of free tame actions. Consider a free action of a compact Lie group G of positive dimension on a topological manifold M. Such an action is called tame if the orbit space MIG is a manifold and wild otherwise. Starting with a tame free action on M one may obtain uncountably many inequivalent wild actions on M by collapsing out noncellular arcs in M/G, provided MIG has dimension at least three. For an exposition of this construction see L. Lininger [9]. The existence of such wild actions makes classification theorems difficult without a tameness hypothesis. In this paper a standard concordance equivalence relation on the set of free actions on a manifold is considered. At least for the circle group S', the concordance equivalence relation makes tame classification theorems applicable in the general case. Define a G-concordance to be a G-action on M X [0, 1] preserving M x 0 and M x 1. Actions p and 4 of G on M are G-concordant if there is a Gconcordance 9 on M x [0, 1] such that AI M x 0 = p and AI M X 1 = 4'. MAIN THEOREM. If M is a closed topological n-manifold, n > 6, then any free S'action on M is S -concordant to a tame action. There are both a relative version and a suitable uniqueness statement. One might conjecture a similar result for arbitrary compact Lie groups. C.T.C. Wall [12] has classified equivariant homeomorphism classes of tame free Si-actions on spheres S2,+', n > 3, by classifying the possible orbit spaces using the theory of surgery. Wall conjectured [12, p. 192] that even in the wild case this calculation can be interpreted to give a correct result. The Main Theorem provides an affirmative answer to this conjecture. See Corollary 3.3. Received by the editors April 12, 1976. AMS (MOS) subject classifications (1970). Primary 57E10, 57A99.

Transformation groups : proceedings of the conference in the University of Newcastle upon Tyne, August 1976

Abstract: Part I: 1. Generators and relations for groups of homeomorphisms Herbert Abels 2. Affine embeddings of real Lie groups Nguiffo B. Boyom 3. Equivariant regular neighbourhoods Allan L. Edmonds 4. Characteristic numbers and equivariant spin cobordism V. Giambalvo 5. Equivariant K-theory and cyclic subgroups Stefan Jakowski 6. Z/p manifolds with low dimensional fixed point set Czes Kosniowski 7. Gaps in the relative degree of symmetry Hsu-Tung Ku and Mei-Chin Ku 8. Characters do not lie Arunas Liulevicius 9. Actions of Z/2n on S3 Gerhard X. Ritter 10. Periodic homeomorphisms on non-compact 3 manifolds Gerhard X. Ritter and Bradd E. Clark 11. Equivariant function spaces and equivariant stable homotopy theory Reinhard Schulz 12. A property of a characteristic class of an orbit foliation Haruo Suzuki 13. Orbit structure for Lie group actions on higher cohomology projective spaces Per Tomter 14. On the existence of group actions on certain manifolds Steven H. Weintraub Part II. Summaries and Surveys: 15. Proper transformation groups H. Abels 16. Problems on group actions on Q manifolds R. D. Anderson 17. A non-abelian view of abelian varieties L. Auslander, B. Kolb and R. Tolimieri 18. Non compact Lie groups of transformation and invariant operator measures on homogenous spaces in Hilbert space M. P. Heble 19. Approximation of simplicial G-maps by equivariantly non degenerate maps Soren Illman 20. Equivariant Riemann-Roch type theorems and related topics Jatsuo Kawakubo 21. Knots and diffeomorphisms M. Kreck 22. Some remarks on free differentiable involuetions on homotopy spheres Peter Loffler 23. Compact transitive isometry spaces Gordon Lukesh 24. A problem of Breson concerning homology manifolds W. J. R. Mitchell.