Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is equal to $k$. Let us pick $\varepsilon>0$ such that $c$ is the only critical value of $F$ in $(c-\varepsilon, c+\varepsilon)$.

I would like to find an explicit (preferably from a book) reference for (one of) the following statements. Denote by $M_{<c'}$ the sub-level set $F^{-1}(-\infty,c')$.

1) The image of the embedding $i_*: H_k(M_{<c-\varepsilon},\mathbb R)\to H_k(M_{<c+\varepsilon},\mathbb R)$ has codimension $\le 1$.

2) Alternatively, the function $F$ can be perturbed in a small neighbourhood of $M_c$ to become a Morse function that has exactly one critical point of index $k$ in a small neighbourhood of $M_c$ and so that all other critical points in this neighbourhood have index $>k$.