Bimodules in bordered Heegaard Floer homology

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of A(F). We also prove a duality theorem relating the two versions of the 3-manifold invariant. Finally, in the case of a genus one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.Comment: 153 pages, 29 figures; v4: Address referee comment

DG-algebras and derived A-infinity algebras

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga.Comment: v3: 27 pages. Minor corrections, to appear in Crelle's Journa

Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective

The search for higher homotopy Hopf algebras (known today as A_\infty-bialgebras) began in 1996 during a conference at Vassar College honoring Jim Stasheff in the year of his 60th birthday. In a talk entitled "In Search of Higher Homotopy Hopf Algebras", I indicated that a DG Hopf algebra could be thought of as some (unknown) higher homotopy structure with trivial higher order structure and deformed using a graded version of Gerstenhaber and Schack's bialgebra deformation theory. In retrospect, the bi(co)module structure encoded in Gerstenhaber and Schack's differential defining deformation cohomology detects some (but not all) of the A_infty-bialgebra structure relations. Nevertheless, this motivated the discovery of A_infty-bialgebras by S. Saneblidze and myself in 2005.Comment: 17 pages; 7 figures; this revision cleans up the bibliography and replaces the term "matrahedron" with "biassociahedron

Islamist groups in the UK and recruitment

Since 2001 and 7/7 the search to find out why and how Muslims born in Europe join political and violence orientated Islamist groups has occupied policy makers and social scientist. The search has produced explanations that suggest social grievance, Islam and physiological problems are the motivations for why some Muslims join and act on behalf of Islamist groups in the UK. However, the approaches tend not to focus the role emotions generated from events that involve Muslim suffering play in some individuals becoming interested in acquiring and acting upon them. These events are often experienced variously by Muslims living in Europe through the media and are used by Islamist groups as resources to recruit. Consequently, this paper is based on interviews carried out with Islamists in the UK and tentatively discusses two process that take into account the emotional effect of events that concern Muslims in order to make sense of how some Muslims become compelled to acquire extreme ideas, act upon extreme ideas (independently or behalf of a group) or join Islamist groups.Publisher PD