TY - THES
T1 - E_n-cohomology as functor cohomology and additional structures
A1 - Ziegenhagen,Stephanie
Y1 - 2014/08/26
N2 - This thesis studies E_n-homology and E_n-cohomology. These are invariants associated to algebraic analogues of n-fold loop spaces: Iterated loop spaces can be described via topological operads, from which one can construct corresponding operads in differential graded modules. Algebras over such an algebraic operad are called E_n-algebras. More concretely, an E_n-algebra is a differential graded module equipped with a product which is associative up to a coherent system of higher homotopies for associativity, but commutative only up to homotopies of a certain level, depending on n. In particular, every commutative algebra over a commutative unital ring is an E_n-algebra.
Using the operadic description, one can construct suitable homological invariants for E_n-algebras, called E_n-homology and -cohomology.
For n=1 and n=\infty this gives rise to familiar invariants: E_1-homology and -cohomology coincide with Hochschild homology and cohomology, while for n=\infty one retrieves Gamma-homology and -cohomology. Note that in characteristic zero Gamma-homology and -cohomology equal AndrĂ¨-Quillen-homology and -cohomology.
Although Hochschild homology and AndrĂ¨-Quillen-homology are classical invariants and have been extensively studied, very little is known in the intermediate cases.
In this thesis we extend results known for special cases of E_n-homology and -cohomology to a broader context.
We use these extensions to examine E_n-cohomology for additional structures.
Benoit Fresse proved that E_n-homology with trivial coefficients can be computed via a generalized iterated bar construction.
By unpublished work of Fresse, if one assumes that the E_n-algebra in question is strictly commutative, this is also possible for cohomology and for coefficients in a symmetric bimodule.
We give the details of a proof of this result based on a sketch of a proof by Benoit Fresse.
Hochschild homology and cohomology can be interpreted as functor homology and cohomology. Muriel Livernet and Birgit Richter proved that this is always possible for E_n-homology of commutative algebras with trivial coefficients. We extend the category defined by Livernet and Richter in their work to a category which also incorporates the action of a commutative algebra A on a symmetric A-bimodule M. We then show that E_n-homology as well as E_n-cohomology of A with coefficients in M can be calculated as functor homology and cohomology.
Hence E_n-cohomology of such objects is representable in a derived sense. In this case the Yoneda pairing yields a natural action of the E_n-cohomology of the representing object on E_n-cohomology.
We prove that E_n-cohomology of the representing object is trivial, therefore no operations arise this way.
Livernet and Richter showed in that E_n-homology of commutative algebras with trivial coefficients coincides with higher order Hochschild cohomology. We extend this result to cohomology and to coefficients in a symmetric bimodule.
It is well known that for a suitable choice of a chain complex calculating E_n-cohomology of an algebra with coefficients in the algebra itself, this chain complex is an E_{n+1}-algebra. For n=1 this is the classical Deligne conjecture. For n>1, the constructions of the E_{n+1}-action given so far have not been very explicit. We show that in characteristic two the chain complex defined via the n-fold bar construction admits at least a part of an E _{n+1}-structure, namely a homotopy for the cup product, and give an explicit formula for this homotopy.
KW - E_n-Homologie
KW - Funktorhomologie
KW - Iterierte Barkonstruktion
KW - Operaden
KW - Hochschildhomologie
CY - Hamburg
PB - Staats- und UniversitĂ¤tsbibliothek Hamburg
AD - Von-Melle-Park 3, 20146 Hamburg
UR - http://ediss.sub.uni-hamburg.de/volltexte/2014/6950
ER -