Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem

Gonçalves, Daciberg Lima; Randall, Duane

doi:10.1016/j.crma.2006.01.016

Cited by 10 publications

(8 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Definition 1.2. (compare [DG], p. 293-296 and [GR2], §1, where a different terminology is used for the same concepts; see also [K7], 5.3).…”

Section: Introductionmentioning

confidence: 99%

Selfcoincidences and roots in Nielsen theory

Koschorke

2007

J. fixed point theory appl.

View full text Add to dashboard Cite

Abstract. Given two maps f 1 and f 2 from the sphere S m to an n-manifold N , when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On one hand the resulting bordism class of coincidence data and the corresponding Nielsen numbers are strong looseness obstructions. On the other hand the values which these invariants may possibly assume turn out to satisfy severe restrictions, e.g. the Nielsen numbers can only take the values 0, 1 or the cardinality of the fundamental group of N . In order to show this we compare different Nielsen classes in the root case (where f 1 or f 2 is constant) and we use the fact that all but possibly one Nielsen class are inessential in the selfcoincidence case (where f 1 = f 2 ). Also we deduce strong vanishing results. Mathematics Subject Classification (2000). Primary 54H25, 55M20; Secondary 55N22, 55P35.

show abstract

“…Definition 1.2. (compare [DG], p. 293-296 and [GR2], §1, where a different terminology is used for the same concepts; see also [K7], 5.3).…”

Section: Introductionmentioning

confidence: 99%

Selfcoincidences and roots in Nielsen theory

Koschorke

2007

J. fixed point theory appl.

View full text Add to dashboard Cite

show abstract

“…classical self-coincidence theory of maps from spheres to real projective spaces) answers to Questions 1.1 and 1.2 even involve the Kervaire invariant one problem or divisibility questions for Hopf invariants (cf. [4,13,14]).…”

Section: Introduction and Outline Of Resultsmentioning

confidence: 99%

“…For m, n 2 the group Ω m−n+1 (M ; ϕ) fits into a long exact Gysin sequence [12,Theorem 5.4]). It is obtained from the normal bordism sequence of the pair (M, M − F M ) of spaces, using identifications via the Thom-Gysin isomorphisms 4) and the Thom-Pontryagin isomorphism between the framed bordism groups Ω fr * of a point and the stable homotopy groups π S * of spheres. …”

Section: The Invariants ω B and Deg Bmentioning

confidence: 99%

Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

Gonçalves¹,

Koschorke²,

Libardi³

2013

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

When can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.In the setting of sphere bundles, a certain degree map deg B turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set F of fibrewise homotopy classes of fibrewise maps. We introduce an addition on F , which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.

show abstract

“…[1,8,12,13,[22][23][24][25][26]28] and others) and includes the fixed point question as the special case where M = N and f 2 is the identity map id.…”

Section: ])mentioning

confidence: 99%

Fixed Points and Coincidences in Torus Bundles

Koschorke

2011

J. Topol. Anal.

View full text Add to dashboard Cite

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper, we investigate fiberwise analoga and present a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace E B and a natural generalization of Nielsen numbers.As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensions over spheres and, in particular, over the unit circle. Our results are based on a careful analysis of the geometry of generic coincidence manifolds. They also allow a simple algebraic description in terms of the Reidemeister invariant (a certain selfmap of an abelian group) and its orbit behavior (e.g. the number of odd order orbits which capture certain nonorientability phenomena). We carry out several explicit sample computations, e.g. for fixed points in (S 1 ) 2 -bundles. In particular, we obtain existence criteria for fixed point free fiberwise maps.

show abstract

Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem

Cited by 10 publications

References 11 publications

Selfcoincidences and roots in Nielsen theory

Selfcoincidences and roots in Nielsen theory

Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

Fixed Points and Coincidences in Torus Bundles

Contact Info

Product

Resources

About