0
$\begingroup$

On Page 76 in Y. Matsumoto's "An Introduction to Morse Theory" he introduces an $m$-dimensional $\lambda$-handle including the following figure:

enter image description here

he then says

The lightly shaded area corresponds to the inequalities

$$x_1^2+\cdots+ x_\lambda^2-x_{\lambda+1}^2-\cdots - x_m^2 \le \varepsilon $$ $$x_{\lambda+1}^2+\cdots+ x_m^2\le \delta $$

Remark: the lightly shaded area is supposed to be the "bridge" between the black areas.

My question is: shouldn't the first inequality rather be

$$x_1^2+\cdots+ x_\lambda^2 \le \varepsilon $$

such that the slightly shaded area is described by the inequalities

$$x_1^2+\cdots+ x_\lambda^2 \le \varepsilon $$ $$x_{\lambda+1}^2+\cdots+ x_m^2\le \delta $$

Thanks for any consideration!

$\endgroup$
1
$\begingroup$

I think not.

I don't have Matsumoto's book in front of me, but I do have Milnor's Morse Theory, whose figures 6 and 7 (pp. 18-19) seem to describe the same thing. The general topic is how $M^a$ is obtained from $M^0$, where $M^t=\{x\in M: f(x)\le t\}$ is the sub-level set (my made-up term; Milnor does't seem to provide one) for some $a>0$. The union of the two black regions represents $M^0$ and their union with the lightly shaded region represents $M^a$.

Pictorially, the shaded region is close to the horizontal "axis", deformable to a tube connecting the two black regions. The inequality not in dispute is $x_{\lambda+1}^2+\cdots x_m^2\le\delta$, which forces the shaded region to be a subset of the tube around the horizontal axis. M's inequality $x_1^2+\cdots x_\lambda^2-x_{\lambda+1}^2\cdots -x_m^2 \le \epsilon$ says $f(x)\le \epsilon$, that is, cuts out $M^\epsilon$.

The OP's interpretation that the constraint should be $x_{\lambda+1}^2+\cdots +x_m^2 \le \epsilon$ would restrict the shaded region to a tube about the "vertical" axis, as well, so the shaded region would be the the intersections of the vertical tube and the horizontal tube, a small blob about the origin, not connecting the two black regions.

I'm sure the OP understands the general thrust of the "lowering sea levels creates a land bridge or causeway (the shaded region) between Asia and America (the black regions)" story given in popular expositions of Morse theory. The $\epsilon$ inequality says that in this picture the causeway connects the two continents; the $\delta$ inequality says that the connection is narrow.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ thank you very much @kimchi lover! $\endgroup$ – Zest Dec 24 '19 at 13:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.