Fundamental dimension and suspension of approximate fibrations

Duvall, P. F.; Husch, L. S.

doi:10.1090/S0002-9939-1980-0560597-5

Fundamental dimension and suspension of approximate fibrations
HTML articles powered by AMS MathViewer

by P. F. Duvall and L. S. Husch PDF

Proc. Amer. Math. Soc. 79 (1980), 122-126 Request permission

Abstract:

Given an approximate fibration $f:X \to Y$ between ANR’s a new approximate fibration Sf is constructed with the property that the fibers of Sf are the suspensions of the fibers of f. This construction is used together with an approximation theorem of T. Chapman to give conditions under which ${\text {Fd}}\;X = {\text {Fd}}\;Y + {\text {Fd}}\;{f^{ - 1}}(x)$.

References

Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
Karol Borsuk, Theory of shape, Monografie Matematyczne, Tom 59, PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0418088
T. A. Chapman, Approximating maps into fiber bundles by homeomorphisms, Rocky Mountain J. Math. 10 (1980), no. 2, 333–350. MR 575307, DOI 10.1216/RMJ-1980-10-2-333
D. S. Coram and P. F. Duvall Jr., Approximate fibrations, Rocky Mountain J. Math. 7 (1977), no. 2, 275–288. MR 442921, DOI 10.1216/RMJ-1977-7-2-275
Donald Coram and Paul Duvall, Approximate fibrations and a movability condition for maps, Pacific J. Math. 72 (1977), no. 1, 41–56. MR 467745
J. Dydak, S. Nowak, and M. Strok, On the union of two FANR-sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 7, 485–489 (English, with Russian summary). MR 413036

A locally compact ANR is a Hilbert cube manifold factor

David A. Edwards and Ross Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. (2) 101 (1975), 521–535. MR 375330, DOI 10.2307/1970939
Steve Ferry, Approximate fibrations with nonfinite fibers, Proc. Amer. Math. Soc. 64 (1977), no. 2, 335–345. MR 461505, DOI 10.1090/S0002-9939-1977-0461505-8
Sławomir Nowak, On the fundamental dimension of approximatively $1$-connected compacta, Fund. Math. 89 (1975), no. 1, 61–79. MR 377832, DOI 10.4064/fm-89-1-61-79
Frank Raymond, Local triviality for Hurewicz fiberings of manifolds, Topology 3 (1965), 43–57. MR 159337, DOI 10.1016/0040-9383(65)90069-8
Jean-Pierre Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425–505 (French). MR 45386, DOI 10.2307/1969485