Bounded homotopy equivalences of Hilbert cube manifolds

Hughes, C. Bruce

doi:10.1090/S0002-9947-1985-0768729-X

Bounded homotopy equivalences of Hilbert cube manifolds
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by C. Bruce Hughes

Trans. Amer. Math. Soc. 287 (1985), 621-643

DOI: https://doi.org/10.1090/S0002-9947-1985-0768729-X

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Abstract:

Let $M$ and $F$ be Hilbert cube manifolds with $F$ compact. The purpose of this paper is to study homotopy equivalences $f:M \to {{\mathbf {R}}^m} \times F$ which have bounded control in the ${{\mathbf {R}}^m}$-direction. Roughly, these homotopy equivalences form a semi-simplicial complex $\mathcal {W}\mathcal {H}({{\mathbf {R}}^m} \times F)$, the controlled Whitehead space. Using results about approximate fibrations, $\mathcal {W}\mathcal {H}({{\mathbf {R}}^m} \times F)$ is related to the semi-simplicial complex of bounded concordances on ${{\mathbf {R}}^m} \times F$. Then the homotopy groups of $\mathcal {W}\mathcal {H}({{\mathbf {R}}^m} \times F)$ are computed in terms of the lower algebraic $K$-theoretic functors ${K_{ - i}}$.

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