# Splitting of chains of loop space

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 $$LM$$?

• How do you give the cellular structure on $LM$? – user43326 Apr 21 at 13:59
• 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. – Sebastian Goette Apr 21 at 18:21
• @user43326 Milnor shows that $LM$ has the homotopy type of a CW type, see Trans AMS 90 (1959), 272-280. – Sebastian Goette Apr 21 at 18:23
• @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? – user43326 Apr 22 at 9:35