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Lifting of homeomorphisms to branched coverings of a disk

Bronisław Wajnryb, Agnieszka Wiśniowska-Wajnryb (2012)

Fundamenta Mathematicae

We consider a simple, possibly disconnected, d-sheeted branched covering π of a closed 2-dimensional disk D by a surface X. The isotopy classes of homeomorphisms of D which are pointwise fixed on the boundary of D and permute the branch values, form the braid group Bₙ, where n is the number of branch values. Some of these homeomorphisms can be lifted to homeomorphisms of X which fix pointwise the fiber over the base point. They form a subgroup L π of finite index in Bₙ. For each equivalence class...

Light paths with an odd number of vertices in polyhedral maps

Stanislav Jendroľ, Heinz-Jürgen Voss (2000)

Czechoslovak Mathematical Journal

Let P k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2 -manifold 𝕄 with Euler characteristic χ ( 𝕄 ) 0 contains a path P k such that each vertex of this path has, in G , degree k 5 + 49 - 24 χ ( 𝕄 ) 2 . Moreover, this bound is attained for k = 1 or k 2 , k even. In this paper we prove that for each odd k 4 3 5 + 49 - 24 χ ( 𝕄 ) 2 + 1 , this bound is the best possible on infinitely many compact 2 -manifolds, but on infinitely many other compact 2 -manifolds the upper bound can be lowered to ( k - 1 3 ) 5 + 49 - 24 χ ( 𝕄 ) 2 .

Link bordism skein modules

Uwe Kaiser (2004)

Fundamenta Mathematicae

We compute link bordism skein modules of colored oriented links in oriented 3-manifolds. A Hurewicz theorem relating link bordism and link homotopy skein modules is proved.

Link homology and Frobenius extensions

Mikhail Khovanov (2006)

Fundamenta Mathematicae

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

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