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 $$M_{ the sub-level set $$F^{-1}(-\infty,c')$$.
1) The image of the embedding $$i_*: H_k(M_{ 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 $$>k$$.
• 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. – Thomas Rot Nov 16 '18 at 16:22
• Obviously you can work in a coordinate chart, assume that $c=0$ and $$F(x)=-x_1^2-\dots -x_k^2+x_{k+1}^2+\dots+x_{n}^2=-\vert x'\vert^2+\vert x''\vert^2,$$ $$M_{-\epsilon}=\{(x',x''), \vert x''\vert^2+\epsilon\le\vert x'\vert^2 \},\quad M_{\epsilon}=\{(x',x''), \vert x''\vert^2\le\vert x'\vert^2 +\epsilon\}.$$ – Bazin Nov 23 '18 at 22:04