Suppose we construct the free loop space $\Lambda M$ of $H^1$-Sobolev class as follows. If $E: \Lambda M\to\mathbb{R}$ is a Morse-Bott energy functional, with $p$ a non-degenerate critical point of index $\gamma$ and $E(p)=q$, and if $E^{-1}[q-\varepsilon,q+\varepsilon]$, for $\varepsilon>0$, is compact containing only one critical point $p$, then $E^{-1}(-\infty, q+\varepsilon]$ is diffeomorphic and homotopy equivalent to $\left(E^{-1}(-\infty, q-\varepsilon]\right)\cup I^{\gamma}$ for $I^{\gamma}$ a $\gamma$-cell (and also $E^{-1}(-\infty, q-\varepsilon]$ with a $\gamma$-handle $H^{\gamma}$ attached), from which we can interpret the loop space as a CW complex. Note, we let $f$ have $\alpha$-many non-degenerate critical points.

From this, how would we compute Čech homology of $\Lambda M$ explicitly? Also does the Mayer-Vietoris sequence of this space upon passing a critical point of $E$ give us useful information in this calculation?

Of course, Čech homology is isomorphic to singular homology. Let $E^{-1}(-\infty,c]=\Lambda^{c}$.

Then does the isomorphism on singular homology $H_k(\Lambda^{q+\varepsilon};\mathbb{Z})\cong H_k(\Lambda^{q-\varepsilon};\mathbb{Z})\oplus\mathbb{Z}$ hold?

Let $\Lambda^{q+\varepsilon}\cong \Lambda^{q-\varepsilon}\cup H^{\gamma}$, for the $\gamma$-handle $H^{\gamma}=D^{\gamma}\times D^{\dim\Lambda^{q-\varepsilon}-\gamma}$ where $\Lambda^{q-\varepsilon}\cap H^{\gamma}\cong S^{\gamma-1}$. Consider the inclusion maps $i:\Lambda^{q-\varepsilon}\cap H^{\gamma}\hookrightarrow \Lambda^{q-\varepsilon}$, $j:\Lambda^{q-\varepsilon}\cap H^{\gamma}\hookrightarrow H^{\gamma}$, $k:\Lambda^{q-\varepsilon}\hookrightarrow \Lambda^{q+\varepsilon}$, and $\ell: H^{\gamma} \hookrightarrow\Lambda^{q+\varepsilon}$. Then the Mayer-Vietoris short-exact sequence is: $$\dots\longrightarrow H_{n+1}(\Lambda^{q+\varepsilon})\overset{\partial_{*}}{\longrightarrow} H_n(\Lambda^{q-\varepsilon}\cap H^{\gamma})\overset{(i_{*},j_{*})}{\longrightarrow} H_n({\Lambda}^{q-\varepsilon})\oplus H_n(H^{\gamma})\overset{k_{*}-\ell_{*}}{\longrightarrow} H_n(\Lambda^{q+\varepsilon})\overset{\partial_{*}}{\longrightarrow} H_{n-1}(\Lambda^{q-\epsilon}\cap H^{\gamma})\longrightarrow\dots \longrightarrow H_0(\Lambda^{q-\varepsilon})\oplus H_0(D^{\gamma})\otimes H_0(D^{n-\gamma})\overset{k_*-\ell_*}\longrightarrow H_0(\Lambda^{q+\varepsilon})\longrightarrow\dots$$

Any help would be much appreciated. Thanks in advance!

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    $\begingroup$ No matter how much math I learn, there will always be questions like this, where I know maybe 5% of the terminology. $\endgroup$ – The Count Aug 30 '18 at 3:33
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    $\begingroup$ @TheCount that was precisely my reaction. $\endgroup$ – Pete Caradonna Aug 30 '18 at 3:40
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    $\begingroup$ Free loop space makes sense for a CW complex (use continuous paths). The Hilbert manifold model as loops of given Sobolev class does not, as Sobolev class of map does not make sense. When you say "A Morse function", you probably want to use the energy functional, which has finite-dimensional negative eigenspaces. Cech homology is isomorphic to singular homology. In the end the stuff at the start is not really relevant and the question is "How do you compute the homology of a free loop space?" This is a hard question and it's not obvious to me there's even some algorithm. $\endgroup$ – user98602 Aug 30 '18 at 3:47
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    $\begingroup$ You can see this question, but I guess you already have. $\endgroup$ – user98602 Aug 30 '18 at 3:48

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