10,606 research outputs found
A Combinatorial Analog of a Theorem of F.J.Dyson
Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the
case n=2 was proposed by Tucker in 1945. Numerous generalizations and
applications of the Lemma have appeared since then. In 2006 Meunier proved the
Lemma in its full generality in his Ph.D. thesis. There are generalizations and
extensions of the Borsuk-Ulam theorem that do not yet have combinatorial
analogs. In this note, we give a combinatorial analog of a result of Freeman J.
Dyson and show that our result is equivalent to Dyson's theorem. As with
Tucker's Lemma, we hope that this will lead to generalizations and applications
and ultimately a combinatorial analog of Yang's theorem of which both
Borsuk-Ulam and Dyson are special cases.Comment: Original version: 7 pages, 2 figures. Revised version: 12 pages, 4
figures, revised proofs. Final revised version: 9 pages, 2 figures, revised
proof
Twisted conjugacy classes in nilpotent groups
A group is said to have the property if every automorphism has an
infinite number of twisted conjugacy classes. We study the question whether
has the property when is a finitely generated torsion-free
nilpotent group. As a consequence, we show that for every positive integer
, there is a compact nilmanifold of dimension on which every
homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product,
we give a purely group theoretic proof that the free group on two generators
has the property. The property for virtually abelian
and for -nilpotent groups are also discussed.Comment: 22 pages; section 6 has been moved to section 2 and minor
modification has been made on exposition; to be published in Crelle
On Fox spaces and Jacobi identities
In 1945, R. Fox introduced the so-called Fox torus homotopy groups in which
the usual homotopy groups are embedded and their Whitehead products are
expressed as commutators. A modern treatment of Fox torus homotopy groups and
their generalization has been given and studied. In this note, we further
explore these groups and their properties. We discuss co-multiplications on Fox
spaces and a Jacobi identity for the generalized Whitehead products and the
-Whitehead products.Comment: 16 page
Equivariant evaluation subgroups and Rhodes groups
In this paper, we define equivariant evaluation subgroups of the higher
Rhodes groups and study their relations with Gottlieb-Fox groups.Comment: 13 page
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