Superlevel sets of a parametrized quadratic forms

Question

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.

Now consider the quadratic form $\Omega(a)=\sum_{l\in\mathbb{Z}/N\mathbb{Z}}(\frac{1}{\tan{a}}q_l^2-\frac{1}{\sin{a}}q_lq_{l+1})$ on $(\mathbb{R}^n)^N$ where $q_l\in\mathbb{R}^n$ for $l\in\mathbb{Z}/N\mathbb{Z}$ when $a\neq 0$, and $\Omega(0)=0$ defined only on the diagonal $\{ (q,\cdots,q)|q\in\mathbb{R}^n\}$. Then for nonzero $a$ it has eigenvalues $\lambda_0=\cot{a}-\csc{a}$ with multiplicity $n$ and $\lambda_l=\lambda_{N-l}=\cot{a}-\csc{a}\cos{\frac{2\pi l}{N}}$ for $l=1,\cdots,\frac{N-1}{2}$ with multiplicity $2n$.

Let $\rho:\mathbb{R}\times(\mathbb{R}^n)^N\rightarrow \mathbb{R}$ be the canonical projection given by $(a,\{q_l\})\mapsto a$ and let $W=\{(a,\{q_l\})|\Omega(a)\geq 0\}\subset \mathbb{R}\times(\mathbb{R}^n)^N$.

In Chiu's paper "Non-squeezing property of contact balls", he says that $W'=\rho^{-1}(\{0\leq -a\leq \frac{2T}{NR^2}\})\cap W$ is homotopic to the Euclidean space $\mathbb{R}^{D+1}$ where $D+1$ is the number of positive eigenvalues of $\Omega(-\frac{2T}{NR^2})$, or equivalently $D+1=2nI+n$ where $I$ is the number of solutions $\lambda_l>0$ among $l=1,\cdots,\frac{N-1}{2}$. Moreover their compactly supported cohomologies are isomorphic.

In addition, he says that $\rho^{-1}(\{0< -a\leq \frac{2T}{NR^2}\})\cap W$ is homotopic to $S^{D-n}\times\mathbb{R}^n\times\mathbb{R}_{>0}$.

I wanted to check all Chiu's assertions but I could't. Please tell me if you know about that.

Answer 1

I am happy to see the question which bothered me too for a long time. But now I know what he did, hope it could help.

Actually, Chiu want to compute $R\Gamma_c(W\cap\rho^{-1}(\{0\leq -a\leq \frac{2T}{NR^2}\}),\mathbb{K})$. Let's take $F=\mathbb{K}_{W\cap\rho^{-1}(\{0\leq -a\leq \frac{2T}{NR^2}\})}$, and $\pi(a,\{q_l\})=(q_l)$.

The cohomology Fubini theorem shows $$R\Gamma(\mathbb{R}_a\times \mathbb{R}^{nN},F)\cong R\Gamma(\mathbb{R}^{nN},R\pi_!F).$$

The Begle-Vietoris theorem tell us $R\pi_!F\cong \mathbb{K}_{\pi\left(W\cap\rho^{-1}(\{0\leq -a\leq \frac{2T}{NR^2}\})\right)}$. In this case, you can find $$\pi\left(W\cap\rho^{-1}(\{0\leq -a\leq \frac{2T}{NR^2}\})\right)=\{(q_l): \Omega(a)\geq 0, a\in [-\frac{2T}{NR^2},0] \}.$$

Now, like what Chiu said, you can choose $a=-\frac{2T}{NR^2}$, and compute cohomology of $$\{(q_l): \Omega(-\frac{2T}{NR^2})\geq 0\}.$$ This space is a quadratic cone, which is properly homotopic to the maximal positive defined subspace of the quadratic form $\Omega(-\frac{2T}{NR^2})$. In particular, it is $\mathbb{R}^{D+1}$.

(In fact, if $Q(x,y)=|x|^2-|y|^2$, then $f_t(x,y)=(x,(1-t)y)$ is the homotopy you want.)

Actually, it is still possible to write down a homotopy directly to compute the cohomology without the Begle-Vietoris theorem, but I don't think it is a good point of view. In my work (arXiv 2103.05143) on generalizing Chiu's result to toric domains, it looks simpler to use Begle-Vietoris.

The second part works similarly. A quadratic cone without a center axis will homotopy to some sphere, right?