Classifying compact 4 -manifolds via generalized regular genus and G -degree
Annales de l’Institut Henri Poincaré D, Tome 10 (2023) no. 1, pp. 121-158.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

(d+1)-colored graphs, i.e., edge-colored graphs that are (d+1)-regular, have already been proved to be a useful representation tool for compact PL d-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact d-manifolds, such as handlebodies, products of closed manifolds by the interval and 𝔻 2 -bundles over 𝕊 2 . The main results of the paper concern dimension 4, where we obtain the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.

Publié le :
DOI : 10.4171/aihpd/128
Classification : 57-XX
Mots-clés : compact 4-manifolds, edge-colored graphs, PL-invariants, regular genus, G-degree
@article{AIHPD_2023__10_1_121_0,
     author = {Casali, Maria Rita and Cristofori, Paola},
     title = {Classifying compact $4$-manifolds via generalized regular genus and $G$-degree},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {121--158},
     volume = {10},
     number = {1},
     year = {2023},
     doi = {10.4171/aihpd/128},
     mrnumber = {4548772},
     zbl = {1521.57024},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/aihpd/128/}
}
TY  - JOUR
AU  - Casali, Maria Rita
AU  - Cristofori, Paola
TI  - Classifying compact $4$-manifolds via generalized regular genus and $G$-degree
JO  - Annales de l’Institut Henri Poincaré D
PY  - 2023
SP  - 121
EP  - 158
VL  - 10
IS  - 1
UR  - http://archive.numdam.org/articles/10.4171/aihpd/128/
DO  - 10.4171/aihpd/128
LA  - en
ID  - AIHPD_2023__10_1_121_0
ER  - 
%0 Journal Article
%A Casali, Maria Rita
%A Cristofori, Paola
%T Classifying compact $4$-manifolds via generalized regular genus and $G$-degree
%J Annales de l’Institut Henri Poincaré D
%D 2023
%P 121-158
%V 10
%N 1
%U http://archive.numdam.org/articles/10.4171/aihpd/128/
%R 10.4171/aihpd/128
%G en
%F AIHPD_2023__10_1_121_0
Casali, Maria Rita; Cristofori, Paola. Classifying compact $4$-manifolds via generalized regular genus and $G$-degree. Annales de l’Institut Henri Poincaré D, Tome 10 (2023) no. 1, pp. 121-158. doi : 10.4171/aihpd/128. http://archive.numdam.org/articles/10.4171/aihpd/128/

Cité par Sources :