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**11 - 18**of**18**### by Paul G. Goerss

, 1996

"... Let A be a simplicial bicommutative Hopf algebra over the field F 2 with the property that 0 A = F 2 . We show that A is a functor of the Andr'e-Quillen homology of A, where A is regarded as an F 2 algebra. Then we give a method for calculating that Andr'e-Quillen homology independent of know ..."

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Let A be a simplicial bicommutative Hopf algebra over the field F 2 with the property that 0 A = F 2 . We show that A is a functor of the Andr'e-Quillen homology of A, where A is regarded as an F 2 algebra. Then we give a method for calculating that Andr'e-Quillen homology independent of knowledge of A. Let G be an abelian group. Since the work of Serre [19] and Cartan [6], we have know that the mod p homology of an Eilenberg-MacLane space K(G;n), n 1, depends only on Tor s (Z=p; G), s = 0; 1. More is true: the structure of H K(G;n) = H (K(G; n); F p ) as an unstable coalgebra over the Steenrod algebra depends only on there Tor groups and the Bockstein fi : Tor 1 (Z=p; G) ! Tor 0 (Z=p; G) = Z=p\Omega G which is the connecting homomorphism of the six term exact sequence obtained by tensoring G with the short exact sequence 0 ! Z=p ! Z=p 2 ! Z=p ! 0: The purpose of this paper to expand on this observation; indeed, our principal result will be that this is an algebraic ...

### "Coalgebra" Structures on 1-Homological Models for Commutative Differential Graded Algebras

"... In [3] "mall" 1-homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [7-9] provides an explicit description of an A1-coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination of the map ..."

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In [3] "mall" 1-homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [7-9] provides an explicit description of an A1-coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination of the map 2 : H ! H H as a first step in the study of this structure. Developing the techniques given in [20] (inversion theory), we get an important improvement in the computation of 2 with regard to the first formula given by HPT. In the case of purely quadratic algebras, we sketch a procedure for giving the complete Hopf algebra structure of its 1-homology.

### An algorithm for computing the first homology groups of CDGAs with linear differential

, 1999

"... We design here a primary platform for computing the basic homological information of Commutative Differential Graded Algebras (briefly, CDGAs), endowed with linear di#erential. All the algorithms have been implemented in the framework settled by Mathematica, so that we can take advantage of the use ..."

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We design here a primary platform for computing the basic homological information of Commutative Differential Graded Algebras (briefly, CDGAs), endowed with linear di#erential. All the algorithms have been implemented in the framework settled by Mathematica, so that we can take advantage of the use of symbolic computation and many other powerful tools this system provides.

### MIXING ACTIONS OF THE RATIONALS

"... Abstract. We study mixing properties of algebraic actions of Q d, showing in particular that prime mixing Q d actions on connected groups are mixing of all orders, as is the case for Z d-actions. This is shown using a uniform result on the solution of S-unit equations in characteristic zero fields d ..."

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Abstract. We study mixing properties of algebraic actions of Q d, showing in particular that prime mixing Q d actions on connected groups are mixing of all orders, as is the case for Z d-actions. This is shown using a uniform result on the solution of S-unit equations in characteristic zero fields due to Evertse, Schlickewei and W. Schmidt. In contrast, algebraic actions of the much larger group Q ∗ are shown to behave quite differently, with finite order of mixing possible on connected groups. Mixing properties of Z d-actions by automorphisms of a compact metrizable abelian group are quite well understood. Roughly speaking, the picture has three facets. Firstly, the one-to-one correspondence between such actions and countably generated modules over the integral group ring Rd = Z[Z d] of the acting group Z d due to Kitchens and K. Schmidt [6] allows any mixing problem to be reduced to the case corresponding to a cyclic module of the form Rd/P for a prime ideal P ⊂ Rd. Secondly, in the connected case P ∩Z = {0}, K. Schmidt and Ward [13] showed that mixing implies mixing of all orders by relating the mixing property to S-unit equations and exploiting a deep result of Schlickewei on solutions of such equations [11] (see also [4] and [14]). Finally, in the totally disconnected case P ∩ Z = pZ for some rational prime p, Masser [9] has shown that the order of mixing is determined by the mixing behaviour of shapes, reducing the problem – in principle – to an algebraic one. Our purpose here is to show how some of this changes for algebraic actions of infinitely generated abelian groups. The algebra is more involved, so for simplicity we restrict attention to the simplest extreme examples: actions of Q ×>0 (isomorphic to the direct sum of countably many copies of Z) and actions of Q d (which is a torsion extension of Z d). These groups are the simplest non-trivial examples chosen from the ‘dual ’ categories of free abelian and infinitely divisible groups in the sense of MacLane [8]. The algebraic difficulties mean we cannot present the complete picture found for Z d-actions, and the emphasis is partly on revealing or suggestive examples. Some topological properties (expansiveness and closed invariant sets) for actions of infinitely generated abelian groups have been studied by Berend [1] and Miles [10].

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, 2008

"... certifies that this is the approved version of the following dissertation: Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories Committee: ..."

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certifies that this is the approved version of the following dissertation: Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories Committee: