4,734 research outputs found

### Gromov's macroscopic dimension conjecture

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

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

We consider the family $\mathrm{MP}_d$ of affine conjugacy classes of
polynomial maps of one complex variable with degree $d \geq 2$, and study the
map $\Phi_d:\mathrm{MP}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d /
\mathfrak{S}_d$ which maps each $f \in \mathrm{MP}_d$ to the set of fixed-point
multipliers of $f$. We show that the local fiber structure of the map $\Phi_d$
around $\bar{\lambda} \in \widetilde{\Lambda}_d$ is completely determined by
certain two sets $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$ which are
subsets of the power set of $\{1,2,\ldots,d \}$. Moreover for any
$\bar{\lambda} \in \widetilde{\Lambda}_d$, we give an algorithm for counting
the number of elements of each fiber $\Phi_d^{-1}\left(\bar{\lambda}\right)$
only by using $\mathcal{I}(\lambda)$ and $\mathcal{K}(\lambda)$. 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

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

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

### Transfer maps and nonexistence of joint determinant

Transfer Maps, sometimes called norm maps, for Milnor's $K$-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
$K$-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

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