The demonstration of the theorem 3.2 in the book Morse theory by Milnor
THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index $\lambda$. Setting $f(p)=c$; suppose that $f^{-1}[c-\epsilon,c+\epsilon]$ is compact, and contains no critical point of $f$ other then $p$, for some $\epsilon\gt0$. Then, for all sufficiently small $\epsilon$, the set $M^{C+\epsilon}$ has the homotopy type of $M^{C-\epsilon}$ with a $\lambda$-cell attached.
is given in the special case whene the manifold is the Torus ,
My question is : can i prove it in the case where the manifold is a manifold with dimension 1 ?
this is a part of proof , in dimension 1 we chose only $(u^1)$ as a coordinate systeme, the index of the critical point is 0 or 1 ,so $f=c±(u^1)^2$
my first probleme is when i must define $e^{\lambda}$ ,I can't say that is the set of points in $U$ with : $u^1)^2\leq \varepsilon$ ,and $u^1 = 0$ !
so how to define $e^{\lambda}$
and in the book ,they applicated this to the torus , how to applicated it on a manifold with dimension 1 ?
Please
Thank you
@Yvoz
I just give an idea to prove the result when $M$ is a polygonal line embedded in $\mathbb{R}^3$ and $\,f\,\colon\, M \to \mathbb{R}$ is the height function defined by $\,f(x,y,z) = z$.
The critical points of $\,f$ of index $0$ (resp. 1) are just its local minima (resp. maxima). Since $\,f$ is linear, the set of its critical points is a subset of the vertices of $M$; since $\,f$ is Morse, there can't be consecutive vertices at the same height (i.e. no edge of $M$ is parallel to the $xy$-plane).
In the hypothesis of the theorem, consider the set $S = \,f^{-1}[c-\varepsilon,c+\varepsilon] =M \cap \{c-\varepsilon \leq z \leq c+ \varepsilon\}$. Choosing sufficiently small $\varepsilon$ (and slightly perturbing $M$ if necessary) we can assume that $\,p$ is the only vertex contained in $S$. Now, it's easy to see that $\,S = P \sqcup Q$, where $P$ is made up of disjoint straight segments joining $\{z = c-\varepsilon\}$ with $\{z = c+\varepsilon\}$ and $Q$ is composed of two edges with common vertex in $\,p$ and the remaining two vertices in $\{z = c+\varepsilon\}$ (resp. $\{z = c-\varepsilon\}$) if $\lambda = 0$ (resp. $\lambda = 1$). In particular $Q$ is a one-dimensional cell.
Now recall that $\,M^{c - \varepsilon} = M \cap \{z \leq c - \varepsilon\}$ and $\,M^{c + \varepsilon} = M^{c - \varepsilon} \cup S$. By retracting the straight lines on the lower base, we get that $M^{c-\varepsilon} \cup P$ has the same homotopy type of $M^{c-\varepsilon}$ and so $M^{c+\varepsilon} \cong M^{c-\varepsilon} \cup Q$ where
So $Q$ is attached to $M^{c - \varepsilon}$ precisely as a $\lambda$-cell.