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Realization of primitive branched coverings over closed surfaces following the hurwitz approach

100%

Bogatyi S., Gonçalves D., Kudryavtseva E., Zieschang H.

Open Mathematics

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2003

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tom 1

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nr 2

184-197

EN

Let V be a closed surface, H⊑π1(V) a subgroup of finite index l and D=[A 1,...,A m] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A i ∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11]. The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].

2

The minimizing of the Nielsen root classes

100%

Gonçalves D., Aniz C.

Open Mathematics

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2004

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tom 2

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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.

3

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

80%

Toda M.

Open Mathematics

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2004

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tom 2

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nr 2

218-249

EN

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B 4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B 4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B 4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number.

Ograniczanie wyników

3 Open Mathematics

2 Gonçalves D.

1 Aniz C.

1 Bogatyi S.

1 Kudryavtseva E.

1 Toda M.

1 Zieschang H.

2 2004

1 2003

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

The minimizing of the Nielsen root classes

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold