Mod 2 instanton homology and 4-manifolds with boundary

Using instanton homology with coefficients in $Z/2$ we construct a homomorphism $q_2$ from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton $h$--invariant and the Heegaard Floer correction term $d$. If an oriented homology $3$--sphere $Y$ bounds a smooth, compact, negative definite $4$--manifold without $2$--torsion in its homology then $q_2(Y)\ge0$, with strict inequality if the intersection form is non-standard.Comment: 88 pages, 2 figure

Revisjonsplikten for norskregistrerte utenlandske foretak : en empirisk studie av små foretak

Vi ønsker med denne masterutredningen å komme med et empirisk begrunnet innspill i debatten om revisjonsplikt for små selskap. På bakgrunn av en empirisk sammenligning av regnskapskvalitet mellom reviderte aksjeselskap og ikke reviderte Norskregistrerte Utenlandske Foretak (NUF) finner vi ikke grunnlag for å hevde at det finnes noen signifikant forskjell i regnskapskvalitet mellom disse to selskapsformene. Dette resultatet tyder på at revisjon ikke nødvendigvis bidrar til å øke kvaliteten på små selskapers regnskap. Videre finner vi at brukergruppene som Finansdepartementet påpeker at revisjonen er ment å tjene, i stor grad er fraværende for små selskap. Vi supplerer med en lønnsomhetsanalyse for å undersøke hvorvidt det finnes tegn på at selskapsformen NUF benyttes til profittskifting til utlandet, det vil si skjuling av skattepliktige inntekter. Motsatt av hva den generelle oppfatningen i samfunnet later til å være, nemlig at NUF ofte er et skalkeskjul for useriøse aktiviteter, gjenspeilet i NOU 2009:4 sin omtale av NUF som ”… spesielt velegnet til å skjule skatteunndragelser og annen økonomisk kriminalitet.”, finner vi ikke indikasjoner på at NUF utfører profittskifting, men at de har signifikant høyere skattbart resultat enn sammenlignbare aksjeselskap. På bakgrunn av dette, samt et uttalt behov for harmonisering med de internasjonale regelverk og et ønske om forenkling av regelverket for små selskap, finner vi ikke tilstrekkelig grunnlag til å anbefale en opphevelse av fritaket fra revisjonsplikten for NUF. Vår anbefaling er å avvente en eventuell revisjonsplikt til en grundigere utredning kan sannsynliggjøre dens behov

Heegaard Floer homology and alternating knots

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper6.abs.htm

Symplectic fillings and positive scalar curvature

Let X be a 4-manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b_2^+(X)>0 or the boundary of X is disconnected. As an application we show that the Poincare homology 3-sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semi-fillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3-manifold which is not symplectically semi-fillable. Using work of Froyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3-spheres having positive scalar curvature metrics.Comment: 14 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper6.abs.htm

Seiberg-Witten-Floer homology of a surface times a circle for non-torsion spin-c structures

We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-manifolds which are connected sums along surfaces and also we reprove the higher type adjunction inequalities previously obtained by Oszv\'ath and Szab\'o.Comment: 26 pages, no figures, Latex2e, to appear in Math. Nac

Monopoles over 4-manifolds containing long necks, I

We study moduli spaces of Seiberg-Witten monopoles over spin^c Riemannian 4-manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg-Witten invariants.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper1.abs.htm

Seiberg-Witten-Floer stable homotopy type of three-manifolds with b_1=0

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a relative invariant of four-manifolds with boundary which generalizes the Bauer-Furuta stable homotopy invariant of closed four-manifolds.Comment: v4, added errata: In Section 9, the Coulomb-Neumann condition should be replaced by a double Coulomb condition, as in Khandhawit's paper (arxiv:1401.7590). Other minor errors are fixed. The main results are unchanged. Version 3 was published in Geom. Topol. 7(2003) 889-932; current version contains appended errata. v5: added item (4) to the errat