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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 $>k$.

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  • $\begingroup$ Are you also interested in a proof of $2$, or you really want a citable reference? I don't think it is too hard and could write you one tomorrow. $\endgroup$ – Thomas Rot Nov 16 '18 at 16:22
  • $\begingroup$ Thomas, I think that this is such a classical statement, that it should be contained in any good book on the topic. Just that I can not find it, I really need a reference $\endgroup$ – aglearner Nov 16 '18 at 16:59
  • $\begingroup$ Obviously you can work in a coordinate chart, assume that $c=0$ and $$ F(x)=-x_1^2-\dots -x_k^2+x_{k+1}^2+\dots+x_{n}^2=-\vert x'\vert^2+\vert x''\vert^2, $$ $$ M_{-\epsilon}=\{(x',x''), \vert x''\vert^2+\epsilon\le\vert x'\vert^2 \},\quad M_{\epsilon}=\{(x',x''), \vert x''\vert^2\le\vert x'\vert^2 +\epsilon\}. $$ $\endgroup$ – Bazin Nov 23 '18 at 22:04

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