Ulrich Koschorke

Self-intersections and higher Hopf invariants

IN THIS paper we show how the well known models for loop spaces of Boardman and Vogt [3], James [5], May [9], and Segal[ lo], can be viewed in a natural way as “Thorn spaces for immersions”. Thus homotopy classes of maps into these models correspond to bordism classes of immersed manifolds with certain extra structures. By considering the multiple points of such immersions we obtain operations in homotopy theory. Special cases are the generalised and higher Hopf invariants of James[6], the Hopf ladder of Boardman and Steer[2], and the cohomotopy operations of Snaith [ 121, and Segal[ I I]. In 01 we establish the connection between models of loop spaces and structured immersions. In 02 we describe the process of “taking k-tuple points” in what might be termed an “external Hopf invariant”, and set out its properties in Theorem 2.2. We then get the Segal and Snaith operations by composing with a suitable “forgetful” function. A similar procedure is followed in 03 where the James operations are described. We are grateful to the referee for many helpful comments.

Link maps and the geometry of their invariants

The geometry of two types of link homotopy invariants of a link map f:Sp∐Sq→Sm is discussed. The first one is the α-invariant which greatly generalizes the classical notion of linking number. The second one, the β-invariant, is closely related to the linking behaviour of f|sp with only the double point set of f|Sq, and therefore measures (to some extend) the obstruction to embedding Sq. These invariants are related by a Hopf invariant homomorphism. In many cases link maps are classified up to link homotopy here, and a setting is provided e.g. for future injectivity results for α. Also the image of α is studied, yielding an interesting double filtration of stable homotopy groups of spheres.

A generalization of Milnor's μ-invariants to higher-dimensional link maps

Abstract In this paper we generalize Milnors μ-invariants (which were originally defined for “almost trivial” classical links in R3) to (a corresponding large class of) link maps in arbitrary higher dimensions. The resulting invariants play a central role in link homotopy classification theory. They turn out to be often even compatible with singular link concordances. Moreover, we compare them to linking coefficients of embedded links and to related invariants of Turaev and Nezhinskij. Along the way we also study certain auxiliary but important “Hopf homomorphisms”.

Nielsen coincidence theory in arbitrary codimensions

Abstract Given two maps f 1, f 2 : M m → N n between manifolds of the indicated arbitrary dimensions, when can they be deformed away from one another? More generally: what is the minimum number MCC(f 1, f 2) of pathcomponents of the coincidence space of maps f 1 ′, f 2 ′ where f i ′ is homotopic to f i , i = 1, 2? Approaching this question via normal bordism theory we define a lower bound N(f 1, f 2) which generalizes the Nielsen number studied in classical fixed point and coincidence theory (where m = n). In at least three settings N(f 1, f 2) turns out to coincide with MCC(f 1, f 2): (i) when m < 2n – 2; (ii) when N is the unit circle; and (iii) when M and N are spheres and a certain injectivity condition involving James-Hopf invariants is satisfied. We also exhibit situations where N(f 1, f 2) vanishes, but MCC(f 1, f 2) is strictly positive.

On link maps and their homotopy classification

is called a link map if the two spheres S p and S o have disjoint images, f l (S p) nf2(S ~) = 0. Two such link maps are called link homotopic if they are homotopic through link maps. This type of equivalence relation was introduced by Milnor [M] in the classical case m = 3, p = q . . . . . 1 and has recently found much renewed interest (see e.g. [MR; FR; K i l 3 ; Ko2-5]). In this paper we study and determine in a number of cases the set LM~. q of link homotopy classes of link maps f as above.

Multiple Points of Immersions, and the Kahn-Priddy Theorem

Classical methods of Pontrjagin-Thom [-17] and of Hirsch [7] allow us to s of spheres with the bordism group of identify the n-th stable homotopy group zc n oriented n-manifolds smoothly immersed into IR n+l. Thus an analysis of the selfintersections of such immersed manifolds leads to potentially interesting invariants in homotopy theory. In this paper we study, as a first example, the homomorphism

Geometric and homotopy theoretic methods in Nielsen coincidence theory

In classical fixed point and coincidence theory, the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC (and MC , resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are . Furthermore we deduce finiteness conditions for MC . As an application, we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space into path components. Its higher-dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps.

Nonstabilized Nielsen coincidence invariants and Hopf-Ganea homomorphisms

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs .f1;f2/ of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC.f1;f2/ (and MC.f1;f2/, resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to .f1;f2/. Furthermore, we deduce finiteness conditions for MC.f1;f2/. As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E.f1;f2/ into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf‐Ganea homomorphisms which turn out to yield finiteness obstructions for MC . 55M20, 55Q25, 55S35, 57R90; 55N22, 55P35, 55Q40

Nonstable and stable monomorphisms of vector bundles

Abstract In this paper we discuss the existence and classification problem both for nonstable and for stable monomorphisms between given vector bundles. The singularity method supplies invariants (complete in a metastable dimension range), together with methods how to compute them. Particular attention is also paid to stabilization, e.g., to the question how many different nonstable homotopy classes of monomorphisms lie in a stable one.

Minimizing coincidence numbers of maps into projective spaces

In this paper we continue to study (‘strong’) Nielsen coincidence numbers (which were introduced recently for pairs of maps between manifolds of arbitrary dimensions) and the corresponding minimum numbers of coincidence points and pathcomponents. We explore compatibilities with fibrations and, more specifically, with covering maps, paying special attention to selfcoincidence questions. As a sample application we calculate each of these numbers for all maps from spheres to (real, complex, or quaternionic) projective spaces. Our results turn out to be intimately related to recent work of D Goncalves and D Randall concerning maps which can be deformed away from themselves but not by small deformations; in particular, there are close connections to the Strong Kervaire Invariant One Problem. 55M20; 55Q40, 57R22