Intersections and Transformations of Complexes and Manifolds

Lefschetz, Solomon

doi:10.2307/1989171

Cited by 44 publications

(47 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We define the intersection sign of the simplices as follows (this can be seen as a very simple special case of Lefschetz intersection theory [14]). …”

Section: Intersection Numbersmentioning

confidence: 99%

See 1 more Smart Citation

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Mabillard

Wagner

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) selfintersection points of maps from finite simplicial complexes (compact polyhedra) into R d and study conditions under which such multiple points can be eliminated.The most classical case is that of embeddings (i.e., maps without double points) of a k-dimensional complex K into R 2k . For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen obstruction). For k ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f : K → R 2k , which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f . One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign.We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y ∈ R d an r-fold Tverberg point of a map f :The analogue of (non-)embeddability that we study is the problem Tverberg r k→d : Given a k-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f :Tverberg point? Here, we show that for fixed r, k and d of the form d = rm and k = (r − 1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings).Our main tool is an r-fold analogue of the Whitney trick: * Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org. SoCG'14, June 8-11, 2014, Kyoto, Japan. Copyright is held by the owner/author(s). Publication rights licensed to ACM. ACM 978-1-4503-2594-3/14/06 ...$15.00.Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y− of opposite intersection sign, we can eliminate y+ and y− by a local isotopy of f .In a subsequent paper, we plan to develop this further and present a generalization of the classical Haefliger-Weber Theorem (which yields a necessary and sufficient condition for embeddability of k-complexes into R d for a wider range of dimensions) to intersection points of higher multiplicity.

show abstract

“…We define the intersection sign of the simplices as follows (this can be seen as a very simple special case of Lefschetz intersection theory [14]). …”

Section: Intersection Numbersmentioning

confidence: 99%

“…Note that here the intersections σ1 ∩ σi are given suitable orientations compatible with orientations of σ1 and σi and the orientation of the 'ambient space', i.e., in our case the standard orientation of R d , see [14] (our earlier definition of pairwise intersection signs is a special case of this).…”

Section: Piping and Connectivitymentioning

confidence: 99%

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Mabillard

Wagner

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

show abstract

“…In mathematics, there are many theorems for studying fixed point theory such as Brouwer fixed point theorem, Lefschetz fixed point theorem, Banach fixed point theorem, Schauder's fixed point theorem, Nielsen fixed point theorem and so forth [2,25,26]. Indeed, using these theorems, we can recognize the existence of a fixed point of a compact mapping in terms of traces of the induced mappings on the algebraic topological tools such as homology groups of X. Owing to the usage of homology groups, it is well known that the fixed point property (FPP for short) is both a topological and a homotopy invariant [26].…”

Section: Introductionmentioning

confidence: 99%

Fixed Point Theorems for Digital Images

Han¹

2015

Honam Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…Fixed point index theory was initiated in 1926/1927 by S. Lefschetz [27] with his celebrated fixed point formula on manifolds (with triangulation), and it was generalized one year later by H. Hopf [23] to compact polyhedra. (For a good survey of fixed point index theory we recommend [14].)…”

mentioning

confidence: 99%

Fixed point index and chain approximations

Siegberg¹,

Skordev²

1982

Pacific J. Math.

View full text Add to dashboard Cite

Localizing the Lef schetz number of certain chain approximations of upper semi continuous multivalued mappings a new approach to fixed point index is given. It turns out that this fixed point index satisfies the commutativity property as well as the mod-p property (known from the singlevalued case). In particular, in the single-valued case the proof of the mod-p property is a natural consequence of a corresponding property of (global) Lefschetz number.

show abstract

Intersections and Transformations of Complexes and Manifolds

Cited by 44 publications

References 0 publications

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Fixed Point Theorems for Digital Images

Fixed point index and chain approximations

Contact Info

Product

Resources

About