Maps into the torus and minimal coincidence sets for homotopies

Volume 172 / 2002

D. L. Goncalves, M. R. Kelly Fundamenta Mathematicae 172 (2002), 99-106 MSC: 55M20, 57M05. DOI: 10.4064/fm172-2-1


Let $X,Y$ be manifolds of the same dimension. Given continuous mappings $f_i,g_i :X\to Y$, $i=0,1$, we consider the $1$-parameter coincidence problem of finding homotopies $f_t,g_t$, $0\leq t\leq 1$, such that the number of coincidence points for the pair $f_t,g_t$ is independent of $t$. When $Y$ is the torus and $f_0,g_0$ are coincidence free we produce coincidence free pairs $f_1,g_1$ such that no homotopy joining them is coincidence free at each level. When $X$ is also the torus we characterize the solution of the problem in terms of the Lefschetz coincidence number.


  • D. L. GoncalvesDepartamento de Matemática – IME-USP
    Caixa Postal 66281 – Ag. Cidade de São Paulo
    CEP: 05315-970
    São Paulo, SP Brazil
  • M. R. KellyDepartment of Mathematics
    and Computer Science
    Loyola University
    6363 St Charles Avenue
    New Orleans, LA 70118, U.S.A.

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