There are explicit perturbations of Morse-Bott functions $f:X\to\mathbb{R}$ used in the literature (ex: Austin-Braam, Banyaga-Hurtubise, Bourgeois) to help solve various problems (ex: building Morse homology, and constructing nondegenerate contact forms). Namely, Morse functions $f_i$ are chosen on the critical submanifolds $C_i\subset crit(f)$ and then $0<\epsilon<<1$ is chosen so that $f+\epsilon\sum_i\rho_if_i$ is Morse (whose critical points have become $crit(f_i)$ on each $C_i$). Here $\rho_i$ is a bump function which has support on a tubular neighborhood of $C_i$ and takes value 1 in a smaller tubular neighborhood.

I am questioning how "rigid" this approach is, i.e. if I can replace the above description with something similar, to achieve a Morse function whose critical points live within the original $\operatorname{crit}(f)$. From the description, the $f_i$ are extended to be constant in the normal direction of $C_i\subset X$ (before being cut-off to zero). Is this "constancy condition" crucial?

Can I instead extend $f_i$ to a small tubular neighborhood in any fashion (there is at least one way by Whitney's extension theorem) and then cut it off to zero? This extension would have no requirement to be constant in the normal directions to $C_i$, and I can still choose $\epsilon$ arbitrarily small and have the critical manifolds $C_i$ be perturbed into $crit(f_i)$. In other words, the original approach pulls back $f_i$ to the normal bundle $N_{C_i}$ of $C_i$ so that locally the perturbation is $(c,x)\mapsto f_i(c)$, and I'm looking for other functions $(c,x)\mapsto \tilde{f}_i(c,x)$ satisfying $\tilde{f}_i(c,0)=f_i(c)$.

The more I think about this though, the more I understand why the original description is used (so that $\nabla f\perp\nabla f_i$ away from $C_i$, helping to ensure no new critical points appear).

  • $\begingroup$ No, it's not crucial. I think people do that because it reflects the minimal amount of input you need to get the machine working. Perhaps there are applications I'm not thinking about, but just to get Morse theory working, you can use all kinds of perturbations. $\endgroup$ Mar 3 '17 at 0:32
  • $\begingroup$ But it could be the case that new critical sets are created away from the original $C_i$? Or can I ensure that the gradients of $f$ and the extension of $f_i$ don't interfere with each other. $\endgroup$ Mar 3 '17 at 3:14
  • $\begingroup$ I think I need to know what application you are considering to be able to quantify what "interfere" might mean. Is there a reason you want to have just Morse functions, rather than Bott-style Morse functions (attaching disc bundles over submanifolds)? $\endgroup$ Mar 3 '17 at 5:22
  • $\begingroup$ I always found Morse theory frustrating, as any section $f$ will do... except for the ones you can write down. I'm always amazed how you can write an entire textbook without writing down a single equation or manifold! Sounds awfully theoretical to me. But that's just my opinion. $\endgroup$ Mar 3 '17 at 5:47
  • $\begingroup$ @RyanBudney One of the applications in mind boils down to wanting a new Morse function having critical points lying inside the original critical manifolds, and agreeing with the original function outside a neighborhood of those critical manifolds. $\endgroup$ Mar 3 '17 at 6:06

Here is a local model for a Bott-style Morse function on $\mathbb R^n$:

$$f(x_1, \cdots, x_n) = a + x_1^2 + \cdots + x_k^2 - x_{k+1}^2 - \cdots - x_{k+j}^2$$

The critical-point set is $x_1=x_2=\cdots = x_{k+j} = 0$, a linear subspace of dimension $n-k-j$.

Let $\beta$ be a bump-function in the variables $x_1, \cdots, x_{k+j}$ centred at the origin, then if I understand the suggestion you are making, the idea is to replace $f$ with

$$g(x_1, \cdots, x_n) = a + x_1^2 + \cdots + x_k^2 - x_{k+1}^2 - \cdots - x_{k+j}^2 + \beta(x_1,\cdots,x_{k+j})( \pm x_{k+j+1}^2 + \cdots + \pm x_n^2 )$$

So the function is unchanged outside of a neighbourhood of the critical submanifold. The critical submanifold of this function is just a single point (the origin). The flow of gradient preserves the original critical-point set $x_1 = x_2 = \cdots = x_{k+j} = 0$.

I take it you want to understand the flowlines in the support of $\beta$. The way I've set things up, there's no magic. No flowlines stay in the support of $\beta$, (using both forward and backward time) other than the original critical-point set $x_1 = \cdots = x_{k+j} = 0$.

Does that help at all?

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    $\begingroup$ OK I kind of like this -- it's not exactly what I'm after but it's given me a better understanding of how to manipulate critical sets. $\endgroup$ Mar 3 '17 at 23:05

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