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Strong surjectivity of mappings of some 3-complexes into 3-manifolds

100%
Aniz C.
Fundamenta Mathematicae
|
2006
|
tom 192
|
nr 3
195-214
EN
Let K be a CW-complex of dimension 3 such that H³(K;ℤ) = 0, and M a closed manifold of dimension 3 with a base point a ∈ M. We study the problem of existence of a map f: K → M which is strongly surjective, i.e. such that MR[f,a] ≠ 0. In particular if M = S¹ × S² we show that there is no f: K → S¹ × S² which is strongly surjective. On the other hand, for M the non-orientable S¹-bundle over S² there exists a complex K and f: K → M such that MR[f,a] ≠ 0.
2
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Strong surjectivity of mappings of some 3-complexes into $$ M_{Q_8 } $$

100%
Aniz C.
Open Mathematics
|
2008
|
tom 6
|
nr 4
497-503
EN
Let K be a CW-complex of dimension 3 such that H 3(K;ℤ) = 0 and $$ M_{Q_8 } $$ the orbit space of the 3-sphere $$ \mathbb{S}^3 $$ with respect to the action of the quaternion group Q 8 determined by the inclusion Q 8 ⊆ $$ \mathbb{S}^3 $$. Given a point a ∈ $$ M_{Q_8 } $$, we show that there is no map f:K → $$ M_{Q_8 } $$ which is strongly surjective, i.e., such that MR[f,a]=min{#(g −1(a))|g ∈ [f]} ≠ 0.
3
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A topological invariant for pairs of maps

63%
Polezzi M., Aniz C.
Open Mathematics
|
2006
|
tom 4
|
nr 2
294-303
EN
In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝn, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.
4
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The minimizing of the Nielsen root classes

63%
Gonçalves D., Aniz C.
Open Mathematics
|
2004
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tom 2
|
nr 1
112-122
EN
Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M 2)−χ(M 10/(1−χ(M 2)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M 2)−χ(M 2)/(1−χ(M 2)). Also we construct, for each integer n≥3, an example of a map f: K n→N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.
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