nLab
shape fibration

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A shape fibration is an adaptation of the notion of Hurewicz fibration from the setup of homotopy theory to shape theory (of locally compact metrisable spaces).

A map between metric compacta p:EBp : E\to B is called a shape fibration, provided it is induced by a level map p:EBp : E\to B of ANR-sequences satisfying the approximate homotopy lifting property with respect to any metric space XX and the lifting index, and lifting mesh do not depend on XX.

If, more generally, EE and BB are locally compact metric spaces. A proper map p:EBp : E\to B is a shape fibration if for any compact CBC\subset B the restriction of pp to a map p 1(C)Cp^{-1}(C)\to C is a shape fibration between metric compacta.

If EE and BB are locally compact metric ANR’s then the notion of shape fibration coincides with a notion of an approximate fibration.

  • S. Mardešić, T. B. Rushing, Shape fibrations. I. General Topology Appl. 9 (1978), no. 3, 193–215.

  • S. Mardešić, J. Segal, (1982) Shape Theory, North Holland.

  • T. B. Rushing, Cell-like maps, approximate fibrations and shape fibrations

Revised on December 24, 2009 02:19:45 by Toby Bartels (151.213.42.84)