Publications

An A∞-coalgebra structure on a closed compact surface.

Faculty Author(s): Umble, Ronald
Student Author(s): -
Department: MATH
Publication: Georgian Mathematical Journal
Year: 2018
Abstract: Let P be an n-gon with n ≥ 3 {n\geq 3}. There is a formal combinatorial A ∞ {A_{\infty}} -coalgebra structure on cellular chains C * ⁢ (P) {C_{*}(P)} with non-vanishing higher order structure when n ≥ 5 {n\geq 5}. If X g {X_{g}} is a closed compact surface of genus g ≥ 2 {g\geq 2} and P g {P_{g}} is a polygonal decomposition, the quotient map q : P g → X g {q\colon P_{g}\to X_{g}} projects the formal A ∞ {A_{\infty}} -coalgebra structure on C * ⁢ (P g) {C_{*}(P_{g})} to a quotient structure on C * ⁢ (X g) {C_{*}(X_{g})} , which persists to homology H ∗ ⁢ (X g ; ℤ 2) {H_{\ast}(X_{g};\mathbb{Z}_{2})} , whose operations are determined by the quotient map q, and whose higher order structure is non-trivial if and only if X g {X_{g}} is orientable with g ≥ 2 {g\geq 2} or unorientable with g ≥ 3 {g\geq 3}. But whether or not the A ∞ {A_{\infty}} -coalgebra structure on homology observed here is topologically invariant is an open question. [ABSTRACT FROM AUTHOR] Copyright of Georgian Mathematical Journal is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Link: An A∞-coalgebra structure on a closed compact surface.