The degree of maps of free G-manifolds and the average

Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying map for M, then the degree is one. The condition that the map be equivariant can be relaxed: it is enough to require that it be “nearly equivariant”, up to a positive constant. We will also discuss the G-average construction and show that the requirement that the map be equivariant can be replaced by a somewhat weaker condition involving the average of the map. These results are applied to maps into real, complex and quaternionic Stiefel manifolds. In particular, we show that a nearly equivariant map of a complex or quaternionic Stiefel manifold into itself has degree one. Dedicated to the memory of Jean Leray     

The fibrewise Leray–Schauder index

The index constructed by Leray and Schauder in 1934 admits generalizations in two directions to infinite-dimensional fixed-point and vector field indices. We present the constructions of fibrewise equivariant indices of both types and illustrate the definitions by applications to the stable homo-topy Fuller index and Seiberg–Witten invariant. Dedicated to the memory of Jean Leray     

Equivariant fixed-point indices of iterated maps

We describe an equivariant version (for actions of a finite group G) of Dold’s index theory, [10], for iterated maps. Equivariant Dold indices are defined, in general, for a G-map U → X defined on an open G-subset of a G-ANR X (and satisfying a suitable compactness condition). A local index for isolated fixed-points is introduced, and the theorem of Shub and Sullivan on the vanishing of all but finitely many Dold indices for a continuously differentiable map is extended to the equivariant case. Homotopy Dold indices, arising from the equivariant Reidemeister trace, are also considered.      

On sectioning multiples of vector bundles and more general homomorphism bundles

We derive in a simple way some higher estimates of the number of every-where linearly independent cross-sections for even multiples of real vector bundles, and, more generally, for a certain class of homomorphism bundles, supposing that there exists at least one nowhere vanishing cross-section.     

The Lyusternik-Shnirel’man category, vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff spaceX an interplay between its span, its possible splittings, and the Lyusternik-Shnirel’man category ofX. The other main result, also on vector bundles and the Lyusternik-Shnirel’man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik-Shnirel’man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest.     

The Lyusternik–Shnirel'man category,¶vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff space X an interplay between its span, its possible splittings, and the Lyusternik–Shnirel'man category of X. The other main result, also on vector bundles and the Lyusternik–Shnirel'man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik–Shnirel'man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest. Received: 4 September 1997     

A note on nonstable monomorphisms¶of vector bundles

In this paper we consider the question of the existence of a nonstable vector bundle monomorphism u:α→β over a closed, connected and smooth manifold M, when dimension of α= 3, dimension of β= dimension of M=n≡ 0(4). The singularity method provides the full obstruction to this problem and under some homological hypothesis we can compute it in terms of well known invariants. Received: 31 May 1999     

Fold maps on 4-manifolds

We give a complete characterization of those closed orientable 4-manifolds which admit smooth maps into R ³ with only fold singularities. We also clarify the relationship between the existence problem of fold maps and that of linearly independent vector fields on manifolds.     

Cyclic actions on some Klein bottle bundles over S1

Leth be a cyclic action of periodn onM, whereM is eitherS ¹×K, K is the Klein bottle or on , the twisted Klein bottle bundle overS ¹, such that there is a fiberingq:MS ¹ with fiber a Klein bottleK or a torusT with respect to which the action is fiber preserving. We classify all such actions and show that they might be distinguished by their fixed points or by their orbit spaces.     

A one-dimensional embedding complex

We give the first explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E¹ term is defined in terms of familiar Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E¹ term is zero, and make calculations of E² in a finite range.     

On fixed points of stratified maps

Nielsen fixed point theory deals with the fixed point sets of self maps on compact polyhedra. In this note, we shall extend it to stratified maps, to consider fixed points on (noncompact) strata. The extension was motivated by our recent work on the braid forcing problem in which the deleted symmetric products are indispensable. The stratified viewpoint is theoretically as natural as the equivariant Nielsen fixed point theory, while it can be more tractable computationally and more flexible in applications. This work was partially supported by an NSFC grant and a BMEC grant.     

Real, complex and quaternionic equivariant vector fields on spheres

The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J-groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Önder [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.     

Euler homology

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring of a topological space X. This homology theory Eh _* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh _*, generalizing the equivariant Euler characteristic.     

On exotic characteristic classes for spherical fibrations

We define for every so-called admissible relation r in the Steenrod algebra $\text{A}$ and for every oriented spherical fibration ξ over a CW-space an exotic characteristic class (mod 2) ε(r)(ξ), which is primitive and vanishes for sphere bundles. The set of exotic classes associated with the universal spherical fibration and the admissible Adem relations are compared with the algebra generators of H¹(BSG;$\text{Z}$₂) due to Milgram. Moreover, their behaviour under the action of $\text{A}$ is computed. Finally, we give a secondary Wu formula for exotic classes of special Poincaré duality spaces.     

New perspectives on self-linking

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.     

Periodic equivariant realK-theories have rational Tate theory

Summary We study a generalized equivariantK-theory introduced by M. Karoubi. We prove, that it is anRO (G, U)-graded cohomology-theory and that the associated Tate spectrum is rational whenG is finite. This implies that for finite groups, the Atiyah-Segal Real equivariantK-theories have rational Tate theory. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag     

The Hopf invariant of a Haefliger knot

We show that for any given differentiable embedding of the three-sphere in six-space there exists a Seifert surface (in six-space) with arbitrarily prescribed signature. This implies, according to our previous paper, that given such a (6,3)-knot endowed with normal one-field, we can construct a Seifert surface so that the outward normal field along its boundary coincides with the given normal one-field. This aspect enables us to understand the resemblance between Ekholm–Szűcs’ formula for the Smale invariant and a formula in our previous paper for differentiable (6,3)-knots. As a consequence, we show that an immersion of the three-sphere in five-space can be regularly homotoped to the projection of an embedding in six-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: “A geometric formula for Haefliger knots” [Topology 43: 1425–1447 2004].      

Integer-valued fixed point index for compositions of acyclic maps

An integer-valued fixed point index for compositions of acyclic multivalued maps is constructed. This index has the additivity, homotopy invariance, normalization, commutativity, and multiplicativity properties. The acyclicity is with respect to the Čech cohomology with integer coefficients. The technique of chain approximation is used. Dedicated to the memory of Jean Leray     

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.     

Coincidences of maps into homogeneous spaces

Let f,g:X→M be maps between two closed connected orientable n-manifolds where M=G/K is the homogeneous space of left cosets of a compact connected Lie group G by a finite subgroup K. In this note, we obtain a simple formula for the Lefschetz coincidence number L(f,g) in terms of topological degree, generalizing some previously known formulas for fixed points. Our approach, by means of Nielsen root theory, also allows us to give a simpler and more geometric proof of the fact that all coincidence classes of f and g have coincidence index of the same sign. Received: 3 March 1998 / Revised version: 29 June 1998