## A pullback theorem for locally-equiconnected spaces.(English)Zbl 0595.55007

A space X is locally equiconnected (L.E.C.) if the inclusion of the diagonal $$\Delta$$ X in $$X\times X$$ is a cofibration. This paper contains the following result. Given any fibration $$p: E\to B$$ and map $$f: X\to B$$ where E, B and X are all LEC, then the pullback space $$X\varsubsetneq E=\{(x,e):\quad f(x)=p(e)\}$$ is also LEC. This can be applied to construct for fibrations with path connected fibres, translation functions between fibres that are base point preserving [c.f. the author, Pac. J. Math. 117, 267-289 (1985; Zbl 0571.55002)].
Reviewer: T.Porter

### MSC:

 55P05 Homotopy extension properties, cofibrations in algebraic topology 54F99 Special properties of topological spaces 55R05 Fiber spaces in algebraic topology

Zbl 0571.55002
Full Text:

### References:

 [1] Dyer, E., Eilenberg, S.,An Adjunction Theorem for locally equiconnected spaces. Pacific. J. Math. 41, (1972) 669,685 · Zbl 0237.55007 [2] Furey, R., Heath, P.R.,Note on Toppair, Top *,and Regular Fibrations, Canadian Math. Bulletin, 24(3), 1981, 317-329 · Zbl 0486.55012 [3] Heath P.R.,Product Formlae for Nielsen Numbers of Fibre Maps Pacific J. Math. Vol 117, No. 2, (1985), 267-289 · Zbl 0571.55002 [4] Heath, P.R., Norton, G.H.,Equiconnectivity and Cofibrations I, Manuscr. Math., 35, 1981, 53-68 · Zbl 0483.55005 [5] Lewis, L. Gaunce, Jr.,When is the natural map X???X a cofibration? Trans. Amer. Math. Soc. 273 (1982) No 1, 145-155 · Zbl 0508.55010 [6] Lillig, J.,A Union Theorem for Cofibrations. Arch. Math. 24, (1973) 410-415 · Zbl 0274.55008 [7] Strøm, A.,Note on cofibrations Math. Scand. 19 (1966), 11-14 · Zbl 0145.43604 [8] Strøm, A.,Note on cofibrations II Math. Scand. 22 (1968), 130-112 · Zbl 0181.26504 [9] Strøm,The homotopy category is a homotopy category Arch. Math. Vol XXIII (1972), 435-441. 130-112 · Zbl 0261.18015 [10] tom Dieck, T., Kamps, K. H., and Puppe, D.,Homotopietheory Lecture Notes in Mathematics, 157, Springer-Verlag, Berlin, Heidelberg, New York (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.