On the number of countably compact group topologies on a free Abelian group

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

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Approximate inverse limits and (m,n)-dimensions

Glasnik Matematicki ◽

10.3336/gm.56.1.11 ◽

2021 ◽

Vol 56 (1) ◽

pp. 175-194

Author(s):

James F. Peters ◽

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Keyword(s):

Fixed Point ◽

Abelian Group ◽

Free Abelian Group ◽

Betti Numbers ◽

Boundary Region ◽

Inverse Limits ◽

Jordan Curve Theorem ◽

Group Representations ◽

Approximate Inverse ◽

Fixed Cell

This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.

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A note on the duality of locally compact groups

Glasgow Mathematical Journal ◽

10.1017/s0017089500000343 ◽

1968 ◽

Vol 9 (2) ◽

pp. 87-91 ◽

Cited By ~ 2

Author(s):

J. W. Baker

Keyword(s):

Abelian Group ◽

Compact Group ◽

Locally Compact Group ◽

Locally Compact Groups ◽

Compact Groups ◽

Group Topology ◽

Locally Compact ◽

Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

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The Abelian Case of Solitar's Conjecture on Infinite Nielsen Transformations

Canadian Mathematical Bulletin ◽

10.4153/cmb-1985-026-x ◽

1985 ◽

Vol 28 (2) ◽

pp. 223-230 ◽

Cited By ~ 1

Author(s):

Olga Macedonska-Nosalska

Keyword(s):

Abelian Group ◽

Free Abelian Group ◽

Infinite Rank ◽

Abelian Case ◽

Countably Infinite ◽

Identical Transformation

AbstractThe paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.

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Planar pure braids on six strands

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500974 ◽

2020 ◽

Vol 29 (01) ◽

pp. 1950097

Author(s):

Jacob Mostovoy ◽

Christopher Roque-Márquez

Keyword(s):

Abelian Group ◽

Fundamental Group ◽

Free Product ◽

Free Group ◽

Configuration Space ◽

Braid Group ◽

Free Abelian Group ◽

Braid Groups ◽

Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

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