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1
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Homotopy decompositions of orbit spaces and the Webb conjecture

100%
Słomińska J.
Fundamenta Mathematicae
|
2001
|
tom 169
|
nr 2
105-137
EN
Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $(B𝓐_p(G))/G$ of the classifying space of the category associated to the G-poset $𝓐_p(G)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{Eₙ}/(NE₀ ∩ ... ∩ NEₙ)$ defined over the poset $(sd𝓐_p(G))/G$, where sd is the barycentric subdivision. We also investigate some other equivariant homotopy and homology decompositions of X and prove that if G is a compact Lie group with a non-trivial p-subgroup, then the map $EG ×_G B𝓐_p(G) → BG$ induced by the G-map $B𝓐_p(G) → ∗$ is a mod p homology isomorphism.
2
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Graph Cohomology, Colored Posets and Homological Algebra in Functor Categories

100%
Słomińska J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2012
|
tom 60
|
nr 3
219-239
EN
The homology theory of colored posets, defined by B. Everitt and P. Turner, is generalized. Two graph categories are defined and Khovanov type graph cohomology are interpreted as Ext* groups in functor categories associated to these categories. The connection, described by J. H. Przytycki, between the Hochschild homology of an algebra and the graph cohomology, defined for the same algebra and a cyclic graph, is explained from the point of view of homological algebra in functor categories.
3
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Morita Equivalences of Functor Categories and Decompositions of Functors Defined on a Category Associated to Algebras with One-Side Units

100%
Słomińska J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 1
33-40
EN
Conditions which imply Morita equivalences of functor categories are described. As an application a Dold-Kan type theorem for functors defined on a category associated to associative algebras with one-side units is proved.
4
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Hecke structure on Bredon cohomology

100%
Słomińska J.
Fundamenta Mathematicae
|
1991-1992
|
tom 140
|
nr 1
1-30
EN
We construct a Hecke structure on equivariant Bredon cohomology with local coefficients and then describe some of its properties. We compare this structure with the Mackey structure defined by T. tom Dieck and with the Illman transfer.
5
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Decompositions of the category of noncommutative sets and Hochschild and cyclic homology

100%
Słomińska J.
Open Mathematics
|
2003
|
tom 1
|
nr 3
327-331
EN
In this note we show that the main results of the paper [PR] can be obtained as consequences of more general results concerning categories whose morphisms can be uniquely presented as compositions of morphisms of their two subcategories with the same objects. First we will prove these general results and then we will apply it to the case of finite noncommutative sets.
6
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G-functors, G-posets and homotopy decompositions of G-spaces

64%
Jackowski S., Słomińska J.
Fundamenta Mathematicae
|
2001
|
tom 169
|
nr 3
249-287
EN
We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group G acting on a poset W and an isotropy presheaf d:W → 𝓢(G) we construct a natural G-map $hocolim_{𝒲_{d}}G/d(-) → |W|$ which is a (non-equivariant) homotopy equivalence, hence $hocolim_{𝒲_{d}}EG × _GF_{d} → EG ×_G |W|$ is a homotopy equivalence. Different choices of G-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy presheaves $𝒲_{d}$; in some important cases they vanish in dimensions greater than the length of W and can be explicitly calculated in low dimensions. We prove a cofinality theorem for functors F: 𝓒 → 𝒪(G) into the category of G-orbits which guarantees that the associated map $α_F: hocolim_{𝓒} EG ×_G F(-) → BG$ is a mod-p-homology decomposition.
7
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André-Quillen cohomology for commutative coalgebras

44%
Słomińska J.
Colloquium Mathematicum
|
1974-1975
|
tom 32
|
nr 2
167-184
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