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Dissertation zugänglich unter
URN: urn:nbn:de:gbv:18-69504
URL: http://ediss.sub.uni-hamburg.de/volltexte/2014/6950/

E_n-cohomology as functor cohomology and additional structures

E_n-Kohomologie als Funktorkohomologie und Zusatzstrukturen

Ziegenhagen, Stephanie

 Dokument 1.pdf (718 KB) 

Freie Schlagwörter (Deutsch): E_n-Homologie , Funktorhomologie , Iterierte Barkonstruktion , Operaden , Hochschildhomologie
Freie Schlagwörter (Englisch): E_n-homology , Functor homology , Iterated bar construction , Operads , Hochschild homology
Basisklassifikation: 31.61
Institut: Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Dissertation
Hauptberichter: Richter, Birgit (Prof. Dr.)
Sprache: Englisch
Tag der mündlichen Prüfung: 02.07.2014
Erstellungsjahr: 2014
Publikationsdatum: 26.08.2014
Kurzfassung auf Englisch: 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 Andrè-Quillen-homology and -cohomology.

Although Hochschild homology and Andrè-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By unpublished work of Fresse, if one assumes that the E_n-algebra in question is strictly commutative, this is also possible for cohomology and for coefficients in a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.


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