We spent today's class with students presenting solutions and/or half-baked ideas about exercises. We had a complete solution to all of the various $S^1 \times S^2$ problems. The problem of showing that the space of metrics adapted to a fixed Morse function is connected (by which I really meant path-connected) was reduced to the following question:

Let $f$ be a standard Morse model function $f = \sum \pm x_i^2$ on $\mathbb{R}^n$ and let $\phi : \mathbb{R}^n \to \mathbb{R}^n$ be any orientation-preserving diffeomorphism sending $0$ to $0$ and respecting $f$, i.e. $f \circ \phi = f$. Show that $\phi$ is isotopic to the identity through a $1$-parameter family of maps $f_t$ with $f_t(0)=0$ and $f_t \circ \phi = f_t$.

A suggestion for showing that $f(x)=x^2$ is stable was to use the fact that any function (in particular, a $1$-parameter family $f_t$) can be approximated by polynomials. Another approach suggested was to show that $f_t$, for small $t$, has "the same kind of singularity" that $f$ has, where "same kind" means $f'(x_t)=0$ and $f''(x_t)>0$.

That's it.