Resolutions and the group cohomology of the sapphire manifolds and of the virtually cyclic groups

• Grant number: 13/07510-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: August 01, 2013
• End date: April 29, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Daciberg Lima Gonçalves
• Grantee: Sérgio Tadao Martins
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated scholarship(s): 13/21394-7 - Diagonal maps and cohomology rings of sol manifolds and virtually cyclic groups, BE.EP.PD

Abstract

The project consists of the study of two problems: the first one is to study the cohomology rings of 3-manifolds known as ``torus semi-bundles''. This problem is a continuation of the results obtained in the candidate's PhDthesis, where the cohomology ring of torus bundles was determined. The study of the cohomology of such spaces is of interestsince they admit one of Thurston's eight geometries.The second problem is to determine the cohomology rings of some virtually cyclic groups. The interest in studying such groups and their cohomology comes from the fact that they act on homotopy spheres. Swan has proved that a finite groups having periodic cohomology acts freely on a finite complex X ~S^n. Since then, there has been an effort to generalize that resultto infinite groups. Among the infinite groups, those that have periodic cohomology are the first natural candidates in this generalization,and the groups we'll study have this periodicity.Virtually cyclic groups are also relevant in the study of the braid groups of surfaces.

Scientific publications

(5)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MARTINS, SERGIO TADAO. Diagonal approximation and the cohomology ring of torus fiber bundles. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 25, n. 3, p. 493-530, MAY 2015. (13/21394-7, 13/07510-4)

GONCALVES, DACIBERG LIMA; MARTINS, SERGIO TADAO. Diagonal approximation and the cohomology ring of the fundamental groups of surfaces. EUROPEAN JOURNAL OF MATHEMATICS, v. 1, n. 1, p. 122-137, MAR 2015. (13/21394-7, 13/07510-4)

GONCALVES, DACIBERG LIMA; MARTINS, SERGIO TADAO. The groups Aut and Out of the fundamental group of a closed Sol 3-manifold. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. N/A, p. 29-pg., 2022-07-07. (16/24707-4, 12/24454-8, 13/07510-4)

GONCALVES, DACIBERG LIMA; MARTINS, SERGIO TADAO; SOARES, MARCIO DE JESUS. The cohomology ring of certain families of periodic virtually cyclic groups. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 27, n. 7, p. 793-818, NOV 2017. (12/24454-8, 13/07510-4)

GONCALVES, DACIBERG LIMA; MARTINS, SERGIO TADAO. The cohomology ring of the sapphires that admit the Sol geometry. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 28, n. 3, p. 365-380, MAY 2018. (13/21394-7, 13/07510-4)