A fixed point theorem in o-minimal structures

The Lefschetz coincidence theorem in o-minimal expansions of fields

“…As we pointed out in [10], the o-minimal version of the Eilenberg-Zilber theorem (Proposition 3.2 in [12]) gives, as in [7] Chapter VI, Section 12 and Chapter VII, Section 2 respectively, the following two theorems: Theorem 5.1 (Künneth Formula for Homology, [10]) Let (X, A) and (Y, B) be pairs of definable sets with A ⊆ X and B ⊆ Y such that (X × Y ; A × Y, X × B) is an excisive triad. Then, for all n ∈ Z, there is an isomorphism…”

“…This result implies an o-minimal Lefschetz fixed point theorem for definable continuous maps on orientable, definably compact definable manifolds as in [1]. For a more general o-minimal Lefschetz-Hopf fixed point theorem generalizing Brumfiel's Hopf fixed point theorem for semi-algebraic maps in [4] see [10].…”

On the Euler characteristic of definable groups

“…We end the paper with the following observations: -Corollary 2.2 was previously proved in o-minimal expansions of fields in [1] using the o-minimal Lefschetz fixed point theorem ( [2], [11] and [14]). …”

Selection problems in O-minimal structures

“…We will prove the above theorem by using the following results: Theorem 4.10 (M. J. Edmundo [11,Corollary 3.6]). Let X ⊆ R n be definable closed and bounded and f : X → X be definable and continuous.…”