The 2-sphere is Wecken for n-valued maps

Brown, Robert F.; Crabb, M. C.; Ericksen, Adam; Stimpson, Matthew T.

doi:10.1007/s11784-019-0691-1

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Circle Maps and Linear Maps On Tori1

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2021

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“…The equality in question is related to the Wecken problem for n-valued maps: for which spaces and selfmaps f : X → D n (X) will we have N (f ) = MF(f )? This can be a difficult question even for simple spaces, see [3] which proves the Wecken property for any n-valued map of the sphere S 2 . Whenever N (f ) = MF(f ), we will automatically have MF(F/q) = MF(f ).…”

Section: Circle Maps and Linear Maps On Torimentioning

confidence: 99%

Partitions of $n$-valued maps

Staecker

2021

Preprint

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An n-valued map is a set-valued continuous function f such that f (x) has cardinality n for every x. Some n-valued maps will "split" into a union of n single-valued maps. Characterizations of splittings has been a major theme in the topological theory of n-valued maps.In this paper we consider the more general notion of "partitions" of an n-valued map, in which a given map is decomposed into a union of other maps which may not be single-valued. We generalize several splitting characterizations which will describe partitions in terms of mixed configuration spaces and mixed braid groups, and connected components of the graph of f . We demonstrate the ideas with some examples on tori.We also discuss the fixed point theory of n-valued maps and their partitions, and make some connections to the theory of finite-valued maps due to Crabb.

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Section: Circle Maps and Linear Maps On Torimentioning

confidence: 99%