## Homotopy type of mapping spaces and existence of geometric exponents.(English)Zbl 1192.55010

A simple space $$Z$$ is said to have a homotopy exponent at the prime $$p$$ if there is an integer $$k\geq 0$$ such that $$p^{k}\pi_{l}(Z)_{(p)}=0$$ for all $$l$$, where $$G_{(p)}$$ denotes the $$p$$-primary subgroup of a group $$G$$. For a homotopy associative $$H$$-space $$Z$$, it is said to have an $$H$$-space exponent if there exists an integer $$b\geq 0$$ such that the map $$b:Z\to Z$$ given by $$z\mapsto z^b$$ is nullhomotopic. Let $$Y_{(p)}$$ and $$map_{*}(X,Z)$$ denote the $$p$$-localization of a simple space $$Y$$ and the space of based maps $$f:X\to Z$$, respectively.
In this paper, the author proves that $$map_*(S^m[p^{-1}],Y)$$ has the homotopy type of a CW complex if and only if the $$m$$-fold loop space $$\Omega^mY_{(p)}$$ admits an $$H$$-space exponent for all big enough $$m$$.

### MSC:

 55P15 Classification of homotopy type 55P10 Homotopy equivalences in algebraic topology 55P45 $$H$$-spaces and duals 55P35 Loop spaces
Full Text:

### References:

 [1] DOI: 10.1007/BF01110331 · Zbl 0185.51101 [2] DOI: 10.2307/1971238 · Zbl 0443.55009 [3] Cohen F. R., Wash. pp 1– (1985) [4] DOI: 10.2307/1970341 · Zbl 0203.25402 [5] DOI: 10.2307/1993265 · Zbl 0087.38203 [6] Geoghegan R., Topology Proc. 4 pp 99– (1979) [7] DOI: 10.2307/2007107 · Zbl 0064.41505 [8] DOI: 10.2307/1969666 · Zbl 0077.36502 [9] DOI: 10.2307/2045037 · Zbl 0511.57013 [10] DOI: 10.1007/BF02566349 · Zbl 0538.55010 [11] DOI: 10.2307/2046328 · Zbl 0594.55006 [12] DOI: 10.2307/2007071 · Zbl 0552.55014 [13] DOI: 10.2307/1993204 · Zbl 0084.39002 [14] Neisendorfer J. A., Ont. pp 343– (1981) [15] Serre J.-P., Applications. Ann. of Math. 54 (2) pp 425– (1951) [16] Smrekar J., Topology Appl. 130 pp 291– (2003) [17] DOI: 10.1016/0040-9383(63)90006-5 · Zbl 0123.39705 [18] DOI: 10.1307/mmj/1028999711 · Zbl 0145.43002 [19] Strøm A., Math. Scand. 22 pp 130– (1968) [20] Strøm A., Arch. Math. (Basel) 23 pp 435– (1972) [21] DOI: 10.2307/1970841 · Zbl 0355.57007 [22] DOI: 10.1007/BF02785672 · Zbl 0638.55020 [23] Zabrodsky A., NJ pp 228– (1983)
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.