Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2012 (vol 65 pp. 575)
The Geometry and Fundamental Group of Permutation Products and Fat Diagonals Permutation products and their various ``fat diagonal'' subspaces are
studied from the topological and geometric point of view. We describe
in detail the stabilizer and orbit stratifications related to the
permutation action, producing a sharp upper bound for its depth and
then paying particular attention to the geometry of the diagonal
stratum. We write down an expression for the fundamental group of any
permutation product of a connected space $X$ having the homotopy type
of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then
prove that the fundamental group of the configuration space of
$n$-points on $X$, of which multiplicities do not exceed $n/2$,
coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving
conditions for when fat diagonal subspaces of manifolds can be
manifolds again. Various examples and homological calculations are
included.
Keywords:symmetric products, fundamental group, orbit stratification Categories:14F35, 57F80 |
2. CJM 2012 (vol 65 pp. 544)
Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The non-uniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 |
3. CJM 2011 (vol 63 pp. 1345)
Pointed Torsors This paper gives a characterization of homotopy fibres of inverse
image maps on groupoids of torsors that are induced by geometric
morphisms, in terms of both pointed torsors and pointed cocycles,
suitably defined. Cocycle techniques are used to give a complete
description of such fibres, when the underlying geometric morphism is
the canonical stalk on the classifying topos of a profinite group
$G$. If the torsors in question are defined with respect to a constant
group $H$, then the path components of the fibre can be identified with
the set of continuous maps from the profinite group $G$ to the group
$H$. More generally, when $H$ is not constant, this set of path components
is the set of continuous maps from a pro-object in sheaves of
groupoids to $H$, which pro-object can be viewed as a ``Grothendieck
fundamental groupoid".
Keywords:pointed torsors, pointed cocycles, homotopy fibres Categories:18G50, 14F35, 55B30 |
4. CJM 2011 (vol 63 pp. 1388)
Nonabelian $H^1$ and the Ãtale Van Kampen Theorem
Generalized Ã©tale homotopy pro-groups $\pi_1^{\operatorname{Ã©t}}(Ä{C}, x)$
associated with pointed, connected, small Grothendieck
sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois
theory and the theory of pointed torsors for discrete
groups is explained.
Applications include new rigorous proofs of some folklore results around $\pi_1^{\operatorname{Ã©t}}(Ã©t(X), x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new Ã©tale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows. Keywords:Ã©tale homotopy theory, simplicial sheaves Categories:18G30, 14F35 |
5. CJM 2008 (vol 60 pp. 140)
On the Geometry of $p$-Typical Covers in Characteristic $p$ For $p$ a prime, a $p$-typical cover of a connected scheme on which $p=0$ is a finite
\'etale cover whose monodromy group (\emph{i.e.,} the Galois group of its
normal closure) is a $p$-group.
The geometry of such covers exhibits some unexpectedly pleasant
behaviors; building on work of Katz, we demonstrate some of these.
These include a criterion for when a morphism induces an isomorphism of
the $p$\nobreakdash-typi\-cal quotients of the \'etale fundamental groups,
and a decomposition theorem for $p$-typical covers of polynomial rings
over an algebraically closed field.
Category:14F35 |
6. CJM 2003 (vol 55 pp. 133)
On the Zariski-van Kampen Theorem Let $f \colon E\to B$ be a dominant morphism, where $E$ and $B$ are
smooth irreducible complex quasi-projective varieties, and let $F_b$
be the general fiber of $f$. We present conditions under which the
homomorphism $\pi_1 (F_b)\to \pi_1 (E)$ induced by the inclusion is
injective.
Category:14F35 |
7. CJM 2003 (vol 55 pp. 157)
Zariski Hyperplane Section Theorem for Grassmannian Varieties Let $\phi \colon X\to M$ be a morphism from a smooth irreducible
complex quasi-projective variety $X$ to a Grassmannian variety $M$
such that the image is of dimension $\ge 2$. Let $D$ be a reduced
hypersurface in $M$, and $\gamma$ a general linear automorphism of
$M$. We show that, under a certain differential-geometric condition
on $\phi(X)$ and $D$, the fundamental group $\pi_1 \bigl( (\gamma
\circ \phi)^{-1} (M\setminus D) \bigr)$ is isomorphic to a central
extension of $\pi_1 (M\setminus D) \times \pi_1 (X)$ by the cokernel
of $\pi_2 (\phi) \colon \pi_2 (X) \to \pi_2 (M)$.
Categories:14F35, 14M15 |