Evaluation at base point induces a splitting of homology of free loop space $LM$ of a compact manifold $M$, i.e. $H_*(LM)\cong H_*(M) \oplus H_*(LM, M)$. Can such splitting be realised on cellular chain level as direct sum of two subcomplexes, possibly given by some suitable Morse function on $LM$?

  • $\begingroup$ How do you give the cellular structure on $LM$? $\endgroup$ – user43326 Apr 21 '19 at 13:59
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    $\begingroup$ For a splitting in homology, you only need that $M$, regarded as subspace of constant loops, is a retract of $LM$, evaluation being the retraction map. $\endgroup$ – Sebastian Goette Apr 21 '19 at 18:21
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    $\begingroup$ @user43326 Milnor shows that $LM$ has the homotopy type of a CW type, see Trans AMS 90 (1959), 272-280. $\endgroup$ – Sebastian Goette Apr 21 '19 at 18:23
  • $\begingroup$ @SebastianGoette I know that it has a homotopy type of a CW complex, but my question is what does the CW structure look like concretely? Or even that of $Map(N,M)$ of arbitrary $N$? Can we do this by simply adding cells on M, to start with? $\endgroup$ – user43326 Apr 22 '19 at 9:35

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