Geodesic
Szczarba’s twisting cochain is comultiplicative
Homology, homotopy, and applications, Tome 26 (2024) no. 1, pp. 287-317.

Voir la notice de l'article provenant de la source International Press of Boston

We prove that Szczarba’s twisting cochain is comultiplicative. In particular, the induced map from the cobar construction $\Omega C(X)$ of the chains on a $1$-reduced simplicial set $X$ to $C(GX)$, the chains on the Kan loop group of $X$, is a quasiisomorphism of $\operatorname{dg}$ bialgebras. We also show that Szczarba’s twisted shuffle map is a $\operatorname{dgc}$ map connecting a twisted Cartesian product with the associated twisted tensor product. This gives a natural $\operatorname{dgc}$ model for fibre bundles.We apply our results to finite covering spaces and to the Serre spectral sequence.
DOI : 10.4310/HHA.2024.v26.n1.a18
Classification : 55U10, 55R20, 55T10
Keywords: Szczarba’s twisting cochain, Kan loop group, extended cobar construction, homotopy Gerstenhaber coalgebra, twisted Cartesian product, twisted tensor product, $\operatorname{dgc}$ model 
@article{HHA_2024_26_1_a17,
     author = {Matthias Franz},
     title = {Szczarba{\textquoteright}s twisting cochain is comultiplicative},
     journal = {Homology, homotopy, and applications},
     pages = {287--317},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2024},
     doi = {10.4310/HHA.2024.v26.n1.a18},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4310/HHA.2024.v26.n1.a18/}
}
TY  - JOUR
AU  - Matthias Franz
TI  - Szczarba’s twisting cochain is comultiplicative
JO  - Homology, homotopy, and applications
PY  - 2024
SP  - 287
EP  - 317
VL  - 26
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4310/HHA.2024.v26.n1.a18/
DO  - 10.4310/HHA.2024.v26.n1.a18
LA  - en
ID  - HHA_2024_26_1_a17
ER  - 
%0 Journal Article
%A Matthias Franz
%T Szczarba’s twisting cochain is comultiplicative
%J Homology, homotopy, and applications
%D 2024
%P 287-317
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4310/HHA.2024.v26.n1.a18/
%R 10.4310/HHA.2024.v26.n1.a18
%G en
%F HHA_2024_26_1_a17
Matthias Franz. Szczarba’s twisting cochain is comultiplicative. Homology, homotopy, and applications, Tome 26 (2024) no. 1, pp. 287-317. doi : 10.4310/HHA.2024.v26.n1.a18. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2024.v26.n1.a18/

Cité par Sources :