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!


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.

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

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