The Coincidence Nielsen Number on Non-Orientable Manifolds

Dobreńko, R.; Jezierski, Jerzy

doi:10.1216/rmjm/1181072611

Cited by 47 publications

(26 citation statements)

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“…As an application of Theorem 5.4, using the averaging formula we can easily compute the Nielsen coincidence number of any pair of self maps on the Klein bottle. In [5], Dobreńko and Jezierski showed the same result on the Klein bottle using the fiber structure of the Klein bottle. In the following section we will consider other examples.…”

Section: If Two Smooth Mapsmentioning

confidence: 62%

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Averaging formula for Nielsen coincidence numbers

Kim

Lee²

2007

Nagoya Mathematical Journal

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Section: If Two Smooth Mapsmentioning

confidence: 62%

“…We first recall the definition of the semi-index [5]. Let V 1 , V 2 be finitedimensional real vector spaces, and let α, β be the orientations determined by the ordered bases {v 1 , .…”

Section: Semi-indexmentioning

confidence: 99%

Averaging formula for Nielsen coincidence numbers

Kim

Lee²

2007

Nagoya Mathematical Journal

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“…To compute the Nielsen number of a map f : K → K given by f # (α) = α r , f # (β) = α s β t we have the following result of [DJ93]. Theorem 1.2.…”

Section: 2mentioning

confidence: 99%

Fixed points on Klein bottle fiber bundles over the circle

2009

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Abstract. The main purpose of this work is to study fixed points of fiber-preserving maps over the circle S 1 for spaces which are fiber bundles over S 1 and the fiber is the Klein bottle K. We classify all such maps which can be deformed fiberwise to a fixed point free map. The similar problem for torus fiber bundles over S 1 has been solved recently.Introduction. Given a fiber bundle E → B and a fiber-preserving map f : E → E over B, the question whether f can be deformed over B (by a fiberwise homotopy) to a fixed point free map has been considered by many, Fadell and Husseini showed that the above problem can be stated in terms of obstructions (including higher ones). This was done under the hypothesis that the base space, the total space and the fiber F are manifolds, and the dimension of F is greater than or equal to 3. The case where the fiber has dimension 2 was considered in [GPV04], where a few generalities were discussed and the fixed point problem over B as defined above was completely solved for any torus fiber bundles over the circle S 1 . In the present work we study the fixed point problem over B for Klein bottle fiber bundles over S 1 .Recall that a Klein bottle bundle over S 1 has as total space the mapping torus M (φ) where φ : K → K is a homeomorphism. A relevant step in solving the problem is to determine, for each fiber bundle M (φ) → S 1 , the set of homotopy classes of maps f over S 1 such that f restricted to the fiber can be deformed to a fixed point free map. This is done in Theorem 2.4. The main result of the paper is Theorem 6.26, which gives a classification of the homotopy classes of fiber-preserving maps given by Theorem 2.4 which can be deformed over S 1 to a fixed point free map. Our method is to study solutions of a system of equations in a free group either by providing an explicit solution or by considering the system in some quotients of this group.

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“…In [2,6] the Nielsen coincidence theory was extended to maps between nonorientable topological manifolds. The main idea to do this is the notion of semi-index (a nonnegative integer) for a coincidence set.…”

Section: Introductionmentioning

confidence: 99%

“…This definition makes sense, since it does not depend on a decomposition (c.f. [2,6]). Moreover the semi-index is homotopy invariant, it is well defined for all continuous maps, and if U ⊂ M is an open subset such that Coin( f ,g) ∩ U is compact, we can extend this definition to that of the semi-index of a pair on the subset U, which is denoted by | ind|( f ,g;U).…”

Section: Introductionmentioning

confidence: 99%