Cohomology of the free loop space of a complex projective space. (English) Zbl 1132.55004
Topology Appl. 155, No. 3, 127-129 (2007); addendum ibid. 156, No. 4, 847 (2009).
For a connected based space \((X,x_0)\), let \(\Lambda (X)\) denote the space consisting of all continuous maps \(\omega :S^1\to X\) with the compact open topology and let \(ev:\Lambda (X)\to X\) be the the evaluation map defined by \(ev(\omega )=\omega (x_0)\). It is known that this gives the evaluation fibration sequence \(\xi_X:\Omega X\to \Lambda (X) @>ev>> X\) with cross section.
It follows from a result of Ziller that the Serre spectral sequence for the case \(X=\mathbb C \text{P}^n\) does not collapse at \(E_2\). In this paper, the author shows this fact by using only an elementary diagram chasing. The proof is very short and easy to understand.
It follows from a result of Ziller that the Serre spectral sequence for the case \(X=\mathbb C \text{P}^n\) does not collapse at \(E_2\). In this paper, the author shows this fact by using only an elementary diagram chasing. The proof is very short and easy to understand.
Reviewer: Kohhei Yamaguchi (Tokyo)
MSC:
55P35 | Loop spaces |
55T10 | Serre spectral sequences |
55R20 | Spectral sequences and homology of fiber spaces in algebraic topology |
References:
[1] | Smith, L., On the characteristic zero cohomology of the free loop space, Amer. J. Math., 103, 887-910 (1981) · Zbl 0475.55004 |
[2] | Vigué-Poirrier, M., Dans le fibré de l’espace des lacets libres, la fibre n’est pas, en général, totalement non cohomologue à zéro, Math. Z., 181, 537-542 (1982) · Zbl 0505.55006 |
[3] | Ziller, W., The free loop space of globally symmetric spaces, Inv. Math., 41, 1-22 (1977) · Zbl 0338.58007 |
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