3,793 research outputs found
Measure homology and singular homology are isometrically isomorphic
Measure homology is a variation of singular homology designed by Thurston in
his discussion of simplicial volume. Zastrow and Hansen showed independently
that singular homology (with real coefficients) and measure homology coincide
algebraically on the category of CW-complexes. It is the aim of this paper to
prove that this isomorphism is isometric with respect to the l^1-seminorm on
singular homology and the seminorm on measure homology induced by the total
variation. This, in particular, implies that one can calculate the simplicial
volume via measure homology -- as already claimed by Thurston. For example,
measure homology can be used to prove the proportionality principle of
simplicial volume.Comment: 20 pages, typos corrected, see also
http://www.math.uni-muenster.de/u/clara.loeh/preprints.html, accepted by
Mathematische Zeitschrift -- the original publication is available at
www.springerlink.com (http://dx.doi.org/10.1007/s00209-005-0905-7
The Fundamental Crossed Module of the Complement of a Knotted Surface
We prove that if is a CW-complex and is its 1-skeleton then the
crossed module depends only on the homotopy type of as a
space, up to free products, in the category of crossed modules, with
. From this it follows that, if is a finite crossed module
and is finite, then the number of crossed module morphisms can be re-scaled to a homotopy invariant , depending only on the
homotopy 2-type of . We describe an algorithm for calculating
as a crossed module over , in the case when
is the complement of a knotted surface in and is
the handlebody made from the 0- and 1-handles of a handle decomposition of .
Here is presented by a knot with bands. This in particular gives us a
geometric method for calculating the algebraic 2-type of the complement of a
knotted surface from a hyperbolic splitting of it. We prove in addition that
the invariant yields a non-trivial invariant of knotted surfaces in
with good properties with regards to explicit calculations.Comment: A perfected version will appear in Transactions of the American
Mathematical Societ
Resonance varieties and Dwyer-Fried invariants
The Dwyer-Fried invariants of a finite cell complex X are the subsets
\Omega^i_r(X) of the Grassmannian of r-planes in H^1(X,\Q) which parametrize
the regular \Z^r-covers of X having finite Betti numbers up to degree i. In
previous work, we showed that each \Omega-invariant is contained in the
complement of a union of Schubert varieties associated to a certain subspace
arrangement in H^1(X,\Q). Here, we identify a class of spaces for which this
inclusion holds as equality. For such "straight" spaces X, all the data
required to compute the \Omega-invariants can be extracted from the resonance
varieties associated to the cohomology ring H^*(X,\Q). In general, though,
translated components in the characteristic varieties affect the answer.Comment: 39 pages; to appear in "Arrangements of Hyperplanes - Sapporo 2009,"
Advanced Studies in Pure Mathematic
Geometric and homological finiteness in free abelian covers
We describe some of the connections between the Bieri-Neumann-Strebel-Renz
invariants, the Dwyer-Fried invariants, and the cohomology support loci of a
space X. Under suitable hypotheses, the geometric and homological finiteness
properties of regular, free abelian covers of X can be expressed in terms of
the resonance varieties, extracted from the cohomology ring of X. In general,
though, translated components in the characteristic varieties affect the
answer. We illustrate this theory in the setting of toric complexes, as well as
smooth, complex projective and quasi-projective varieties, with special
emphasis on configuration spaces of Riemann surfaces and complements of
hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics
and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201
The period-index problem for twisted topological K-theory
We introduce and solve a period-index problem for the Brauer group of a
topological space. The period-index problem is to relate the order of a class
in the Brauer group to the degrees of Azumaya algebras representing it. For any
space of dimension d, we give upper bounds on the index depending only on d and
the order of the class. By the Oka principle, this also solves the period-index
problem for the analytic Brauer group of any Stein space that has the homotopy
type of a finite CW-complex. Our methods use twisted topological K-theory,
which was first introduced by Donovan and Karoubi. We also study the cohomology
of the projective unitary groups to give cohomological obstructions to a class
being represented by an Azumaya algebra of degree n. Applying this to the
finite skeleta of the Eilenberg-MacLane space K(Z/l,2), where l is a prime, we
construct a sequence of spaces with an order l class in Br, but whose indices
tend to infinity.Comment: To appear in Geometry & Topology; minor cosmetic change
On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups
We give an interpretation of Yetter's Invariant of manifolds in terms of
the homotopy type of the function space , where is a crossed
module and is its classifying space. From this formulation, there
follows that Yetter's invariant depends only on the homotopy type of , and
the weak homotopy type of the crossed module . We use this interpretation to
define a twisting of Yetter's Invariant by cohomology classes of crossed
modules, defined as cohomology classes of their classifying spaces, in the form
of a state sum invariant. In particular, we obtain an extension of the
Dijkgraaf-Witten Invariant of manifolds to categorical groups. The
straightforward extension to crossed complexes is also considered.Comment: 45 pages. Several improvement
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