On G-maps between cohomology spheres and a representation of the Reeb Graph as a subcomplex of a manifold

Author(s)

Nelson Antonio Silva

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2016-04-29

Examining board members

Denise de Mattos; Alice Kimie Miwa Libardi; Waclaw Boleslaw Marzantowicz; Caio Jose Colletti Negreiros; Pedro Luiz Queiroz Pergher

• Advisor: Denise de Mattos; Waclaw Boleslaw Marzantowicz

Abstract

Bartsch (BARTSCH, 1993) introduced a numerical cohomological index theory, known as the length, for G-spaces, where G is a compact Lie group. We present the length of G-spaces which are cohomology spheres and G = (Z2)k, (Zp)k or (S1)k, k ≥ 1. As consequences, we obtain a Borsuk-Ulam theorem in this context and we give a sucient condition for the existence of G-maps between a cohomological sphere and a representation sphere when G = (Zp)k. Also, a Bourgin-Yang version of the Borsuk-Ulam theorem is presented. As a second part of this thesis, a new definition of the Reeb graph R( f ) of a smooth function f : M → R with isolated critical points as a subcomplex of M is given. For that, a 1-dimensional complex Γ ( f ) embedded into M and homotopy equivalent to R( f ) is constructed. As consequence it is shown that for every function f on a manifold with finite fundamental group, the Reeb graph of f is a tree. If π 1 (M) is an abelian group, or more generally, an amenable group2, then R( f ) contais at most one loop. Finally, it is proved that the number of loops of the Reeb graph of every function on a surface Mg is estimated from above by g, the genus of Mg. The results of this second part is published in (KALUBA; MARZANTOWICZ; SILVA, 2015). (AU)

• FAPESP's process: 11/23610-3 - Topological invariants of minimax problems with symmetry
• Grantee: Nelson Antonio Silva
• Support Opportunities: Scholarships in Brazil - Doctorate