Extensions of maps as fibrations and cofibrations

Author:
Frank Quinn

Journal:
Trans. Amer. Math. Soc. **211** (1975), 203-208

MSC:
Primary 55D05

MathSciNet review:
0385847

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Abstract: Suppose is a map of 1-connected spaces. In the ``stable'' range, roughly where the connectivity of *Y* exceeds the homology, or homotopy, dimension of *X*, it is well known that *f* can be extended as a cofibration , or respectively a fibration . A criterion is given for the existence of such extensions in a less restrictive ``metastable'' range. A main result is that if *f* is at least 2-connected and 2 con , then *f* extends as a cofibration if and only if the map factors through *f*.

**[1]**T. Ganea,*A generalization of the homology and homotopy suspension*, Comment. Math. Helv.**39**(1965), 295–322. MR**0179791****[2]**T. Ganea,*Induced fibrations and cofibrations*, Trans. Amer. Math. Soc.**127**(1967), 442–459. MR**0210131**, 10.1090/S0002-9947-1967-0210131-2**[3]**Frank Quinn,*Surgery on Poincaré and normal spaces*, Bull. Amer. Math. Soc.**78**(1972), 262–267. MR**0296955**, 10.1090/S0002-9904-1972-12950-1**[4]**-,*Poincaré spaces*(in preparation).**[5]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[6]**Robert A. Nowlan,*𝐴_{𝑛}-actions on fibre spaces*, Indiana Univ. Math. J.**21**(1971/1972), 285–313. MR**0288767**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1975-0385847-2

Keywords:
Fibration,
cofibration,
metastable extension

Article copyright:
© Copyright 1975
American Mathematical Society