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1

Finite-dimensional spaces in resolving classes

100%

Strom J.

Fundamenta Mathematicae

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2012

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tom 217

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nr 2

171-187

EN

Using the theory of resolving classes, we show that if X is a CW complex of finite type such that $map⁎(X,S^{2n+1}) ~ ∗$ for all sufficiently large n, then map⁎(X,K) ∼ ∗ for every simply-connected finite-dimensional CW complex K; and under mild hypotheses on π₁(X), the same conclusion holds for all finite-dimensional complexes K. Since it is comparatively easy to prove the former condition for X = Bℤ/p (we give a proof in an appendix), this result can be applied to give a new, more elementary proof of the Sullivan conjecture.

2

Miller spaces and spherical resolvability of finite complexes

100%

Strom J.

Fundamenta Mathematicae

|

2003

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tom 178

|

nr 2

97-108

EN

Let 𝒜 be a fixed collection of spaces, and suppose K is a nilpotent space that can be built from spaces in 𝒜 by a succession of cofiber sequences. We show that, under mild conditions on the collection 𝒜, it is possible to construct K from spaces in 𝒜 using, instead, homotopy (inverse) limits and extensions by fibrations. One consequence is that if K is a nilpotent finite complex, then ΩK can be built from finite wedges of spheres using homotopy limits and extensions by fibrations. This is applied to show that if map⁎(X,Sⁿ) is weakly contractible for all sufficiently large n, then map⁎(X,K) is weakly contractible for any nilpotent finite complex K.

3

The Gray filtration on phantom maps

63%

Hà L., Strom J.

Fundamenta Mathematicae

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2001

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tom 167

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nr 3

251-268

EN

This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail. McGibbon and Roitberg have shown that if f* is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if f* is surjective just in dimension k, then f induces a surjection on a certain subquotient of the phantom set. If the condition holds for all k, we recover McGibbon and Roitberg's theorem. There is a dual result, and a theorem on phantom maps into spheres which holds one dimension at a time as well. Finally, we examine the set of phantom maps whose Gray index is infinite. The main theorem is a partial verification of our conjecture that if X and Y are nilpotent and of finite type, then every phantom map f: X → Y must have finite index.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Strom J.

1 Hà L.

1 2012

1 2003

1 2001

Finite-dimensional spaces in resolving classes

Miller spaces and spherical resolvability of finite complexes

The Gray filtration on phantom maps