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On G -disconnected injective models

Marek Golasiński — 2003

Annales de l’institut Fourier

Let G be a finite group. It was observed by L.S. Scull that the original definition of the equivariant minimality in the G -connected case is incorrect because of an error concerning algebraic properties. In the G -disconnected case the orbit category 𝒪 ( G ) was originally replaced by the category 𝒪 ( G , X ) with one object for each component of each fixed point simplicial subsets X H of a G -simplicial set X , for all subgroups H G . We redefine the equivariant minimality and redevelop some results on the rational homotopy...

Injective models of G -disconnected simplicial sets

Marek Golasiński — 1997

Annales de l'institut Fourier

We generalize the results by G.V. Triantafillou and B. Fine on G -disconnected simplicial sets. An existence of an injective minimal model for a complete 𝕀 -algebra is presented, for any E I -category 𝕀 . We then make use of the E I -category 𝒪 ( G , X ) associated with a G -simplicial set X to apply these results to the category of G -simplicial sets. Finally, we describe the rational homotopy type of a nilpotent G -simplicial set by means of its injective minimal model.

On generalized “ham sandwich” theorems

Marek Golasiński — 2006

Archivum Mathematicum

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let A 1 , ... , A m n be subsets with finite Lebesgue measure. Then, for any sequence f 0 , ... , f m of -linearly independent polynomials in the polynomial ring [ X 1 , ... , X n ] there are real numbers λ 0 , ... , λ m , not all zero, such that the real affine variety { x n ; λ 0 f 0 ( x ) + + λ m f m ( x ) = 0 } simultaneously bisects each of subsets A k , k = 1 , ... , m . Then some its applications are studied.

Co-H-structures on equivariant Moore spaces

Martin ArkowitzMarek Golasiński — 1994

Fundamenta Mathematicae

Let G be a finite group, 𝕆 G the category of canonical orbits of G and A : 𝕆 G 𝔸 b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with E x t n - 1 ( A , A A ) . Then the case G = p k leads to an example of infinitely...

Residue class rings of real-analytic and entire functions

Marek GolasińskiMelvin Henriksen — 2006

Colloquium Mathematicae

Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...

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