4,734 research outputs found

    Gromov's macroscopic dimension conjecture

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    In this note we construct a closed 4-manifold having torsion-free fundamental group and whose universal covering is of macroscopic dimension 3. This yields a counterexample to Gromov's conjecture about the falling of macroscopic dimension.Comment: This is the version published by Algebraic & Geometric Topology on 14 October 200

    Resolutions of p-stratifolds with isolated singularities

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    Recently M. Kreck introduced a class of stratified spaces called p-stratifolds [M. Kreck, Stratifolds, Preprint]. He defined and investigated resolutions of p-stratifolds analogously to resolutions of algebraic varieties. In this note we study a very special case of resolutions, so called optimal resolutions, for p-stratifolds with isolated singularities. We give necessary and sufficient conditions for existence and analyze their classification.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-36.abs.htm

    The Moduli Space of Polynomial Maps and Their Fixed-Point Multipliers

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    We consider the family MPd\mathrm{MP}_d of affine conjugacy classes of polynomial maps of one complex variable with degree d2d \geq 2, and study the map Φd:MPdΛ~dCd/Sd\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d / \mathfrak{S}_d which maps each fMPdf \in \mathrm{MP}_d to the set of fixed-point multipliers of ff. We show that the local fiber structure of the map Φd\Phi_d around λˉΛ~d\bar{\lambda} \in \widetilde{\Lambda}_d is completely determined by certain two sets I(λ)\mathcal{I}(\lambda) and K(λ)\mathcal{K}(\lambda) which are subsets of the power set of {1,2,,d}\{1,2,\ldots,d \}. Moreover for any λˉΛ~d\bar{\lambda} \in \widetilde{\Lambda}_d, we give an algorithm for counting the number of elements of each fiber Φd1(λˉ)\Phi_d^{-1}\left(\bar{\lambda}\right) only by using I(λ)\mathcal{I}(\lambda) and K(λ)\mathcal{K}(\lambda). It can be carried out in finitely many steps, and often by hand.Comment: 40pages; Revised expression in Introduction a little, and added proofs for some propositions; results unchange

    Moduli spaces of vector bundles over a real curve: Z/2-Betti numbers

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    Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obtained by Liu and Schaffhauser.Comment: 33 pages. Implemented referee suggestions and simplified exposition in the introduction. Comments welcom

    Arithmetic of Unicritical Polynomial Maps

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    This note will study complex polynomial maps of degree n2n\ge 2 with only one critical point.Comment: 9 pages incl. references, 2 figure

    Transfer maps and nonexistence of joint determinant

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    Transfer Maps, sometimes called norm maps, for Milnor's KK-theory were first defined by Bass and Tate (1972) for simple extensions of fields via tame symbol and Weil's reciprocity law, but their functoriality had not been settled until Kato (1980). On the other hand, functorial transfer maps for the Goodwillie group are easily defined. We show that these natural transfer maps actually agree with the classical but difficult transfer maps by Bass and Tate. With this result, we build an isomorphism from the Goodwillie groups to Milnor's KK-groups of fields, which in turn provides a description of joint determinants for the commuting invertible matrices. In particular, we explicitly determine certain joint determinants for the commuting invertible matrices over a finite field, the field of rational numbers, real numbers and complex numbers into the respective group of units of given field.Comment: Some minor revisions have been made after the 1st versio