198 research outputs found
Twisty itsy bitsy topological field theory
We extend the topological field theory (``itsy bitsy topological field
theory"') of our previous work from mod-2 to twisted coefficients. This
topological field theory is derived from sutured Floer homology but described
purely in terms of surfaces with signed points on their boundary (occupied
surfaces) and curves on those surfaces respecting signs (sutures). It has
information-theoretic (``itsy'') and quantum-field-theoretic (``bitsy'')
aspects. In the process we extend some results of sutured Floer homology,
consider associated ribbon graph structures, and construct explicit admissible
Heegaard decompositions.Comment: 52 pages, 26 figure
Sutured Floer homology, sutured TQFT and non-commutative QFT
We define a "sutured topological quantum field theory", motivated by the
study of sutured Floer homology of product 3-manifolds, and contact elements.
We study a rich algebraic structure of suture elements in sutured TQFT, showing
that it corresponds to contact elements in sutured Floer homology. We use this
approach to make computations of contact elements in sutured Floer homology
over of sutured manifolds where is
finite. This generalises previous results of the author over
coefficients. Our approach elaborates upon the quantum field theoretic aspects
of sutured Floer homology, building a non-commutative Fock space, together with
a bilinear form deriving from a certain combinatorial partial order; we show
that the sutured TQFT of discs is isomorphic to this Fock space.Comment: v.2: 49 pages, 13 figures. Improved and expanded exposition, some
minor corrections. Sections on torsion, annuli, and tori moved to a separate
pape
The hyperbolic meaning of the Milnor–Wood inequality
AbstractWe introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor–Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor’s function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains
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