Minimalidade de coincidências do tipo Borsuk-Ulam em superfícies utilizando tranças

Abstract

Given topological spaces $M$ and $N$, with $M$ equipped with a free involution $\tau$, we define the set of Borsuk-Ulam coincidences as

$$ \operatorname{BUCoin}(f; \tau) = \{ \{ x, \tau(x) \} \ | \ f(x) = f(\tau(x)) \}. $$

In this work, we want to exhibit $g \colon M \to N$ homotopic to $f$ such that it minimizes the cardinality of set of Borsuk-Ulam coincidences. In the cases where spaces are surfaces, we will show an algebraic technique involving braid groups of surface that is useful to study such a problem. This technique also displays the index of each Borsuk-Ulam coincidence, seen as coincidences classes. We will give an explicit answer to the problem when both spaces are Klein bottles.

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