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!