MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Does anyone know a concrete example of a Morse function on some manifold that is perfect with respect to some field but not with respect to $\mathbb Z_2$?

share|cite|improve this question
Sure, take a lens space with odd order fundamental group with a standard Morse function (coming from a genus 1 Heegaard splitting). – Ian Agol Sep 14 '11 at 15:19
Ok, I see. Thank you. – Stephan Wiesendorf Sep 15 '11 at 9:17
up vote 7 down vote accepted

I'll expand on my comment, since you requested a "concrete example" of a Morse function. A (3-dimensional) lens space $L(p,q)$ is obtained from the unit sphere $S^3=\{(z_1,z_2)\in \mathbb{C}^2 | |z_1|^2+|z_2|^2 =1\}$ by quotienting by the action $(z_1, z_2) \mapsto (e^{2\pi i/p} z_1,e^{2\pi i q/p})$, which will have fundamental group $\mathbb{Z}/p$ when $gcd(p,q)=1$. There is a Morse-Bott function on $S^3$ given by $|z_1|^2$, which has two critical circles of index $0$ and $2$ when $z_1=0, z_2=0$. This function is obviously invariant under the cyclic group action, so it descends to a Morse-Bott function on $L(p,q)$ with two critical circles. One may perturb a Morse-Bott function in a neighborhood of a critical submanifold to be a Morse function, by adding a bump function on the normal coordinate times a Morse function on the critical submanifold, where the index adds. So one may perturb the Morse-Bott function on $L(p,q)$ near the critical cirles (when $z_1=0$, it will be a Morse function on $z_2$ times a bump function of $z_1$) to get index $0$ and $1$ critical points at one circle, and index $2$ and $3$ critical points at the other circle (a bump function of $z_2$ times a Morse function on $z_1$ near $z_2=0$). This is not quite explicit, but hopefully is concrete enough for you. Then $H_i(L(p,q);\mathbb{Z}/p)=\mathbb{Z}/p$ for $i=0,1,2,3$, so the Morse function is perfect with respect to $\mathbb{Z}/p$ coefficients.

share|cite|improve this answer
I had suspected that Lens spaces might provide an example, but I was not familiar with their description via the Heegaard splitting. When I thought about your first comment, it was not clear to me what is meant by a standard Morse function coming from such a splitting, but I assumed that it should be obtained on each solid torus from a perfect Morse function on the core circle extended in such a way that it is constant on the boundary. I was pretty sure that this can be done and if I understand your expansion right this is exactly what you describe (reversed). So, thank you again. – Stephan Wiesendorf Sep 17 '11 at 7:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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