Five papers published by Geometry & Topology Publications
Geometry and Topology
(1) Algebraic & Geometric Topology 10 (2010) 1747-1780
On Davis-Januszkiewicz homotopy types II: Completion and globalisation
by Dietrich Notbohm and Nigel Ray
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p060.xhtml
DOI: 10.2140/agt.2010.10.1747
(2) Algebraic & Geometric Topology 10 (2010) 1781-1806
Gamma-homology of algebras over an operad
by Eric Hoffbeck
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p061.xhtml
DOI: 10.2140/agt.2010.10.1781
Three papers have been published by Geometry & Topology
(3) Geometry & Topology 14 (2010) 1941-1991
The Maskit embedding of the twice punctured torus
by Caroline Series
URL: http://www.msp.warwick.ac.uk/gt/2010/14-04/p040.xhtml
DOI: 10.2140/gt.2010.14.1941
(4) Geometry & Topology 14 (2010) 1993-2045
Perturbative invariants of 3-manifolds with the first Betti number 1
by Tomotada Ohtsuki
URL: http://www.msp.warwick.ac.uk/gt/2010/14-04/p041.xhtml
DOI: 10.2140/gt.2010.14.1993
(5) Geometry & Topology 14 (2010) 2047-2076
Homotopy groups of the moduli space of metrics of positive scalar curvature
by Boris Botvinnik, Bernhard Hanke, Thomas Schick and Mark Walsh
URL: http://www.msp.warwick.ac.uk/gt/2010/14-04/p042.xhtml
DOI: 10.2140/gt.2010.14.2047
Abstracts follow
(1) On Davis-Januszkiewicz homotopy types II: Completion and globalisation
by Dietrich Notbohm and Nigel Ray
For any finite simplicial complex K, Davis and Januszkiewicz defined a
family of homotopy equivalent CW-complexes whose integral cohomology
rings are isomorphic to the Stanley--Reisner algebra of
K. Subsequently, Buchstaber and Panov gave an alternative
construction, which they showed to be homotopy equivalent to the
original examples. It is therefore natural to investigate the extent
to which the homotopy type of a space X is determined by such a
cohomology ring. Having analysed this problem rationally in Part I, we
here consider it prime by prime, and utilise Lannes' T-functor and
Bousfield--Kan type obstruction theory to study the p-completion of
X. We find the situation to be more subtle than for rationalisation,
and confirm the uniqueness of the completion whenever K is a join of
skeleta of simplices. We apply our results to the global problem by
appealing to Sullivan's arithmetic square, and deduce integral
uniqueness whenever the Stanley--Reisner algebra is a complete
intersection.
(2) Gamma-homology of algebras over an operad
by Eric Hoffbeck
The purpose of this paper is to study generalizations of
Gamma-homology in the context of operads. Good homology theories are
associated to operads under appropriate cofibrancy hypotheses, but
this requirement is not satisfied by usual operads outside the
characteristic zero context. In that case, the idea is to pick a
cofibrant replacement Q of the given operad P. We can apply to
P-algebras the homology theory associated to Q in order to define a
suitable homology theory on the category of P-algebras. We make
explicit a small complex to compute this homology when the operad P is
binary and Koszul. In the case of the commutative operad P = Com, we
retrieve the complex introduced by Robinson for the Gamma-homology of
commutative algebras.
(3) The Maskit embedding of the twice punctured torus
by Caroline Series
The Maskit embedding M of a surface Sigma is the space of
geometrically finite groups on the boundary of quasifuchsian space for
which the "top" end is homeomorphic to Sigma, while the "bottom" end
consists of triply punctured spheres, the remains of Sigma when a set
of pants curves have been pinched. As such representations vary in the
character variety, the conformal structure on the top side varies over
the Teichmuller space T(Sigma).
We investigate M when Sigma is a twice punctured torus, using the
method of pleating rays. Fix a projective measure class [mu] supported
on closed curves on Sigma. The pleating ray P_[mu] consists of those
groups in M for which the bending measure of the top component of the
convex hull boundary of the associated 3-manifold is in [mu]. It is
known that P is a real 1-submanifold of M. Our main result is a
formula for the asymptotic direction of P in M as the bending measure
tends to zero, in terms of natural parameters for the complex
$2$--dimensional representation space R and the Dehn--Thurston
coordinates of the support curves to [mu] relative to the pinched
curves on the bottom side. This leads to a method of locating M in R.
(4) Perturbative invariants of 3-manifolds with the first Betti number 1
by Tomotada Ohtsuki
It is known that perturbative invariants of rational homology
3--spheres can be constructed by using arithmetic perturbative
expansion of quantum invariants of them. However, we could not make
arithmetic perturbative expansion of quantum invariants for
3--manifolds with positive Betti numbers by the same method.
In this paper, we explain how to make arithmetic perturbative
expansion of quantum SO(3) invariants of 3-manifolds with the first
Betti number 1. Further, motivated by this expansion, we construct
perturbative invariants of such 3-manifolds. We show some properties
of the perturbative invariants, which imply that their coefficients
are independent invariants.
(5) Homotopy groups of the moduli space of metrics of positive scalar curvature
by Boris Botvinnik, Bernhard Hanke, Thomas Schick and Mark Walsh
We show by explicit examples that in many degrees in a stable range
the homotopy groups of the moduli spaces of Riemannian metrics of
positive scalar curvature on closed smooth manifolds can be
non-trivial. This is achieved by further developing and then applying
a family version of the surgery construction of Gromov-Lawson to
certain nonlinear smooth sphere bundles constructed by Hatcher.
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