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Homology of function spaces. (English) Zbl 0621.55005

We compute the homology of the space of pointed maps from X to Y under certain conditions on X and Y. One such set of conditions is that X be the suspension of a connected finite CW complex and that Y be the m-fold suspension of a connected CW complex, with \(m\geq\) (dimension of \(X+\) the connectivity of \(X+2).\) Under these conditions, the space Map\((X,Y)\) has the mod p homology of a product of spaces \(\Omega ^ iY\), the i-fold loops of Y. If we choose a basis for the mod p homology of X, there is one copy of \(\Omega ^ iY\) for each basis element of \(H_ i(X;F_ p)\). Under the above conditions, the homology of \(\Omega ^ iY\) is known, so this really does give a calculation of \(H_ *(\text{Map}(X,Y);F)\) for any field F.

MSC:

55N99 Homology and cohomology theories in algebraic topology
55P40 Suspensions
54C35 Function spaces in general topology
55P35 Loop spaces
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References:

[1] Bott, R., Samelson, H.: On the Pontrjagin product in the space of paths. Comment. Math. Helv.27, 320-337 (1953) · Zbl 0052.19301 · doi:10.1007/BF02564566
[2] Campbell, H.E.A., Cohen, F.R., Peterson, F.P., Selick, P.S.: The space of maps of Moore spaces into spheres (in print) · Zbl 0699.55005
[3] Cohen, F.R., Lada, T.J., May, J.P.: The homology of iterated loop spaces. Lecture Notes in Math.533. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0334.55009
[4] Cohen, F.R., Moore, J.C., Neisendorfer, J.A.: Torsion in homotopy groups. Ann. Math.104, 128-168 (1979) · Zbl 0405.55018
[5] Cohen, F.R., Neisendorfer, J.A.: A construction ofp-localH-spaces. Algebraic Topology Aarhus 1982. Lecture Notes in Math.1051, pp. 351-359. Berlin-Heidelberg-New York: Springer 1984
[6] Cohen, F.R., Taylor, L.R.: The homology of function spaces. Proceedings of the Northwestern Homotopy Theory Conference. Contemp. Math.19, 39-50 (1983) · Zbl 0518.55004
[7] James, I.M.: Reduced product spaces. Ann. Math.62, 170-197 (1955) · Zbl 0064.41505 · doi:10.2307/2007107
[8] Jacobson, N.: Lie Algebras. New York: Dover 1962 · Zbl 0121.27504
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