Mac Lane method in the investigation of magnetic translation groups

Central extensions of the three-dimensional translation group T=Z^3 by the unitary group U(1) (a group of factors) are considered within the frame of the Mac~Lane method. All nonzero vectors t in T are considered to be generators of T. This choice leads to very illustrative relations between the Mac~Lane method and Zak's approach to magnetic translation groups. It is shown that factor systems introduced by Zak and Brown can be realized only for the unitary group U(1) and for some of its finite subgroups.Comment: 8 pages, 1 fig. in text, romp_sty.tex attached at the beginning Presented at 28 Symp. on Math. Phys., Torun 2-6 Dec 199

Finite domination and Novikov rings. Iterative approach

Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules, if and only if the two chain complexes C((x)) and C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.Comment: 15 pages; diagrams typeset with Paul Taylor's "diagrams" macro package. Version 2: clarified proof of main theorem, fixed minor typos; Version 3: expanded introduction, now 16 pages; Version 4: corrected mistake on functoriality of mapping tor

Database queries and constraints via lifting problems

Previous work has demonstrated that categories are useful and expressive models for databases. In the present paper we build on that model, showing that certain queries and constraints correspond to lifting problems, as found in modern approaches to algebraic topology. In our formulation, each so-called SPARQL graph pattern query corresponds to a category-theoretic lifting problem, whereby the set of solutions to the query is precisely the set of lifts. We interpret constraints within the same formalism and then investigate some basic properties of queries and constraints. In particular, to any database $\pi$ we can associate a certain derived database \Qry(\pi) of queries on $\pi$. As an application, we explain how giving users access to certain parts of \Qry(\pi), rather than direct access to $\pi$, improves ones ability to manage the impact of schema evolution

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper20.abs.html Version 3: Reference to Salvatore added (see footnote 3, page 834

Magnetic translation groups as group extension

Extensions of a direct product T of two cyclic groups Z_n1 and Z_n2 by an Abelian (gauge) group G with the trivial action of T on G are considered. All possible (nonequivalent) factor systems are determined using the Mac Lane method. Some of resulting groups describe magnetic translation groups. As examples extensions with G=U(1) and G=Z_n are considered and discussed.Comment: 10 page

I-adic towers in topology

A large variety of cohomology theories is derived from complex cobordism MU^*(-) by localizing with respect to certain elements or by killing regular sequences in MU_*. We study the relationship between certain pairs of such theories which differ by a regular sequence, by constructing topological analogues of algebraic I-adic towers. These give rise to Higher Bockstein spectral sequences, which turn out to be Adams spectral sequences in an appropriate sense. Particular attention is paid to the case of completed Johnson--Wilson theory E(n)-hat and Morava K-theory K(n) for a given prime p.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-65.abs.htm

Topological Representation of Geometric Theories

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models and semantical groupoids. Technically a variant of the syntax-semantics duality constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids---reflecting the use of `indexed' models in the representation theorem---which in several respects simplifies the construction and allows for an intrinsic characterization of the semantic side.Comment: 32 pages. This is the first pre-print version, the final revised version can be found at http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting of which is not allowed by Wiley). Changes in v2: updated comment

Complete Bredon cohomology and its applications to hierarchically defined groups

By considering the Bredon analogue of complete cohomology of a group, we show that every group in the class \LHFF of type Bredon-\FP_\infty admits a finite dimensional model for \EFG. We also show that abelian-by-infinite cyclic groups admit a $3$-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for \mF, the class of virtually cyclic groups, the class of \LHFF-groups contains all locally virtually soluble groups and all linear groups over $\mathbb C$ of integral characteristic.Comment: 14 page

Explicit tensor network representation for the ground states of string-net models

The structure of string-net lattice models, relevant as examples of topological phases, leads to a remarkably simple way of expressing their ground states as a tensor network constructed from the basic data of the underlying tensor categories. The construction highlights the importance of the fat lattice to understand these models.Comment: 5 pages, pdf figure