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my question is the following: given a smooth manifold $M$, take a homotopy of maps $f_{t}:M \rightarrow \mathbb{R}, \quad t \in [0,1]$ such that every $f_{t}$ is a Morse function. Do $f_{0}$ and $f_{1}$ have the same critical points' structure? In that case, is it possible to generalize the result to Morse-Bott functions?

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What do you mean by "critical points structure"? For example, under such a homotopy the relative height of critical points can switch. An index $i$ critical point could move to be higher than an index $j$ one. So it very much depends on what you consider the "structure" to be. – Ryan Budney Dec 6 '12 at 19:24
For Morse-Bott functions, I think the answer is no. Imagine a circle sitting in the $xy$-plane, and rotate it out of the plane. Then the $z$-coordinate is a Morse-Bott function at all times, but the critical point structure changes. – Ian Agol Dec 6 '12 at 19:36
You're right, to be more precise by critical points' structure I meant the collection of the critical points together with their indices. I wasn't considering that Morse functions are Morse-Bott as well, so in order to state this question I have to restrict the class of possible homotopies. Thank you! – Alessandro Gentile Dec 11 '12 at 10:24
up vote 7 down vote accepted

To see why the answer is yes for the first question use the implicit function theorem. For any $t_0$ you can find $\newcommand{\ve}{\varepsilon}$ $\ve >0$ and smooth maps

$$\gamma_1,\dotsc, \gamma_N:(t_0-\ve,t_0+\ve)\to M$$ such that for any $t\in (t_0-\ve,t_0+\ve)$ the points $\gamma_1(t),\dotsc, \gamma_n(t)\in M$ are pairwise distinct and the critical set of $f_t$ consists precisely of these points. Clearly the index of $\gamma_i(t)$ is independent of $t$, for any $i$.

If we denote consider the set

$$ C:=\bigl\lbrace (t,x)\in [0,1]\times M;\;\;df_t(x)=0\bigr\rbrace, $$

then the above fact shows that the natural projection

$$ C\ni (t,x)\mapsto t\in [0,1] $$

is a covering map.

As for the second question, I will give an example. Denote by $\DeclareMathOperator{\Gr}{Gr}$ by ${\Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of the complex Hermitian space $V$. For any selfadjoint operator $A: V\to V$ we have a smooth function

$${\Gr}_k(V)\ni L \mapsto f_A(L)= {\rm Re}\; {\rm tr}(AP_L)\in \mathbb{R}, $$

where $P_L$ denotes the orthogonal projection onto $L$. This function is Morse-Bott for any $A$, but it is Morse if and only if the eigenvalues of $A$ are simple, i.e., pairwise distinct. You can now easily construct a smooth family of selfadjoit operators $A_t$, $t\in [0,1]$ such that $A_0=\boldsymbol{1}_V$ but $A_t$ has simple eigenvalues for any $t>0$.

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