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Torsion in graph homology

Laure Helme-Guizon, Józef H. Przytycki, Yongwu Rong (2006)

Fundamenta Mathematicae

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsion. When the underlying algebra is ℤ[x]/(x²), we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov-Rozansky...

Torsion in one-term distributive homology

Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra (2014)

Fundamenta Mathematicae

The one-term distributive homology was introduced in [Prz] as an atomic replacement of rack and quandle homology, which was first introduced and developed by Fenn-Rourke-Sanderson [FRS] and Carter-Kamada-Saito [CKS]. This homology was initially suspected to be torsion-free [Prz], but we show in this paper that the one-term homology of a finite spindle may have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely....

Towards one conjecture on collapsing of the Serre spectral sequence

Markl, Martin (1990)

Proceedings of the Winter School "Geometry and Physics"

[For the entire collection see Zbl 0699.00032.] A fibration F E B is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if H * ( E ; k ) H * ( F ; k ) is surjective. This is equivalent to saying that π 1 ( B ) acts trivially on H * ( F ; k ) and the Serre spectral sequence collapses at E 2 . S. Halperin conjectured that for c h a r ( k ) = 0 and F a 1-connected rationally elliptic space (i.e., both H * ( F ; 𝒬 ) and π * ( F ) 𝒬 are finite dimensional) such that H * ( F ; k ) vanishes in odd degrees, every fibration F E B is TNCZ. The author proves this being the case...

Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2

Tran Ngoc Nam (2008)

Annales de l’institut Fourier

On détermine la dimension d’une représentation du groupe linéaire définie par un sous-espace vectoriel de l’algèbre à puissances divisées, puis on explicite l’image du transfert algébrique en degré générique et celle du transfert algébrique quadruple, et finalement on identifie les indécomposables de degré pair de l’algèbre polynomiale à quatre variables, vue comme module sur l’algèbre de Steenrod.

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