Universal operations in Hochschild homology
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Standard
Universal operations in Hochschild homology. / Wahl, Nathalie.
I: Journal fuer die Reine und Angewandte Mathematik, Bind 2016, Nr. 720, 2016, s. 81-127.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - Universal operations in Hochschild homology
AU - Wahl, Nathalie
PY - 2016
Y1 - 2016
N2 - We provide a general method for finding all natural operations on the Hochschild complex of E-algebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A_infty-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich-Soibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler-Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.
AB - We provide a general method for finding all natural operations on the Hochschild complex of E-algebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A_infty-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich-Soibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler-Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.
KW - math.AT
KW - math.QA
U2 - 10.1515/crelle-2014-0037
DO - 10.1515/crelle-2014-0037
M3 - Journal article
VL - 2016
SP - 81
EP - 127
JO - Journal fuer die Reine und Angewandte Mathematik
JF - Journal fuer die Reine und Angewandte Mathematik
SN - 0075-4102
IS - 720
ER -
ID: 45323759