Strong surjections from two-complexes with odd order top-cohomology onto the projective plane

Author(s)

Fenille, Marcio Colombo ; Goncalves, Daciberg Lima ; Neto, Oziride Manzoli

• Document type: Journal article
• Source: Journal of Fixed Point Theory and Applications; v. 25, n. 2, p. 13-pg., 2023-06-01.

Abstract

Given a finite and connected two-dimensional CW complex K with fundamental group. and second integer cohomology group H-2 (K; Z) finite of odd order, we prove that: (1) for each local integer coefficient system alpha : Pi -> Aut(Z) over K, the corresponding twisted cohomology group H-2(K;(alpha)Z) is finite of odd order, we say order c* (alpha), and there exists a natural function-which resemble that one defined by the twisted degree-from the set [K; RP2](alpha)* a of the based homotopy classes of based maps inducing alpha on pi(p)1 into H-2(K;(alpha)Z), which is a bi-jection; (2) the set [K; RP2]alpha of the (free) homotopy classes of based maps inducing alpha on pi(1) is finite of order c(alpha) = (c* (alpha)+ 1)/2; (3) all but one of the homotopy classes [f] is an element of [K; RP2](alpha) are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism f * : H-2(RP2; (e)Z) -> H-2 (K; (alpha)Z), where e is the nontrivial local integer coefficient system over the projective plane. Also some calculations of H-2 (K;(alpha)Z) are provided for several two-complexes K and actions alpha, allowing to compare H-2(K; Z) and H-2(K;(alpha)Z) for nontrivial alpha. (AU)

• FAPESP's process: 16/24707-4 - Algebraic, geometric and differential topology
• Grantee: Daciberg Lima Gonçalves
• Support Opportunities: Research Projects - Thematic Grants