Lecture 16: Openness of Deformation Space

Geometric topology: a space of $(G,X)$ structure on a manifold $M$.

What does it mean for two structures to be nearby?

Consider the space $C_w^\infty(M^m,N^n) = \{f: M \xrightarrow{\text{smooth}} N \}$ with the smooth weak topology, where $M^m$ and $N^n$ are smooth manifolds with dimension $m,n$ respectively.

Definition. for smooth weak topology. Choose $k >0$, $\varepsilon >0, K \subset M$ where $K$ is compact, finitely many charts covering $K$ $\{\phi_n\}$. Then, $U(k, \varepsilon, K, \{\phi_n\})\subset C_w^\infty(M,N)$ is the set of all $g: M \rightarrow N$ such that on each of the charts, $f,g$ are close in $C^k$, i.e. $|D^\ell f - D^\ell g | < \varepsilon$ for all $\ell \le k$. That is, the partial derivatives of order less than or equal to $k$ differ by less than $\varepsilon$.

Now, let $M$ be a closed $n$-manifold. A $(G,X)$ structure is $\text{dev} : \tilde M \rightarrow X$, $\rho: \pi_1 M \rightarrow G$ such that $\forall \alpha \in \pi_1 M$, $\forall x \in \tilde M$, $\text{dev}(\tau_\alpha \cdot x) = \rho(\alpha) \cdot \text{dev} x$.

So, dev determines $\rho.$

Definition.

Then, notice $\text{Dev}(M,(G,X)) \subset C_w^\infty (\tilde M, X)$, so we can give it the subspace topology.

Special case. Suppose $\text{dev} \in \text{Dev}(M, (G,X))$ is injective. Let $U = \text{dev} (\tilde M)$. Identify $\tilde M$ with $U \subset X$. Then, dev becomes the inclusion $i : U \hookrightarrow X$. Then, $\text{dev}': U \rightarrow X$ is nearby in the weak topology if on a compact set $K \subset U$, $\text{dev}'$ is close the the identity in the $C^k$ topology.

We have a smalll neighborhood in the weak topology if we have a big $K$ and a small $\varepsilon$.

This is a stronger version of the Holonomy Theorem: If $M$ is a closed manifold, $\text{Hol} :\text{Dev}(M, (G,X)) \rightarrow \text{Hom}(\pi_1(M), G)$ is open.

Lemma 3.3 Let $M$ be a properly convex manifold. Let $N = \xi M$, the tautological line bundle. Let $C: N \rightarrow \mathbb R$ be a flow function, i.e. there is a flow $\Phi_t$ such that $C(\Phi_t(x)) = C(x) +t$. Then $C$ is Hessian convex ($d^2C >0$ (positive definite matrix)) if and only if $S = C^{-1}(0)$ is a Hessian convex surface.

Example Consider $f =x^4$. This is convex in the old sense, because every secant lies on one side. But, it’s not Hessian convex because $f''(0) = 0$.

Definition A smooth hypersurface $F^{n-1} \subset \mathbb R^n$ is Hessian convex if $\forall x \in F$, $P$ the tangent plane to $F$ at $x$, there exists a neighborhood $U \subset P$ of $x$, $g: U \xrightarrow{\text{smooth}} \mathbb R$ above $U$, where $F$ is the graph of $g$ and $D^2g >0$ (positive definite).

Now, we give our main result of today:

Openness for Properly Convex Structures (Koszul \~ 1962) Let $m$ be a closed $n$-manifold.
$\text{Dev}_C(M, \mathbb P) = \{ \text{dev} \in \text{Dev}(M, (\mathsf{PGL}(n+1), \mathbb R), \mathbb R P^n): \text{dev}(\tilde M) \text{ is properly convex and dev is injective}\}$ Then $\text{Dev}_C(M, \mathbb P) \rightarrow \text{Hom}(\pi_1M, \mathsf{PGL}(n+1, \mathbb R))$ is open.

Sketch proof: Suppose $\text{dev}_\rho \in \text{Dev}_c(M)$, so $\text{dev}_\rho: \tilde M \xrightarrow{\text{injective}} \mathbb R P^n$, holonomy $\rho : \pi_1 M \rightarrow \mathsf{PGL}(n+1, \mathbb R)$. Suppose $\sigma \in \text{Hom}(\pi_1 M, \mathsf{PGL}(n+1), \mathbb R))$ close to $\rho$. Let $\Omega_\rho = \text{dev}_\rho (\tilde M)$, $\sigma$ close to $\rho$. This implies that nearby $\text{dev}_\sigma : \tilde M _\sigma \equiv \Omega_\rho \rightarrow \mathbb R P^n$.

We can to the same thing for $\xi_1 M_\rho$ and $\xi_1 M_\sigma$.

Similarly, there exist nearby dev maps, $\widetilde{\xi_1 M_\rho}, \widetilde{\xi_1 M_\sigma} \xrightarrow{\text{smooth}} \mathbb R^{n+1}$. Then, notice $\widetilde{\xi_1 M_\rho} \equiv \mathcal C \Omega_\rho = \xi \Omega_\rho$, where $\mathcal C \Omega_\rho$ denote the cone over $\Omega_\rho$.

So, there exists $h: \xi_1 \tilde M_\rho \xrightarrow{\text{diffeo}} \xi \tilde M_\sigma$ with lift $\tilde h : \mathcal C \Omega_\rho \rightarrow \mathbb R^{n+1} - \{0\}$ where $\tilde h$ is close the the identity on a large compact set in $C^k$. Thus $d^2 \tilde h$ is close to 0, and $d \tilde h$ is close the the identity.

Define $S \subset \xi_2 \tilde M \rho$, a level set of the convexity function $c: \xi_1 M_\rho \rightarrow S^1$. Then, $S$ is a hypersurface, and $S$ is Hessian convex by the lemma.

So, $h(S) \subset \xi_1 \tilde M_\sigma$. Since $d^2 h \approx 0$ and $dh \approx$ identity, we use the chain rule to see that $h(S)$ is Hessian convex in $\xi_1 \tilde M _\sigma$, since $M$ is compact. Then, by the lemma, we know that $h(S)$ is the zero set of a convexity flow function on $\xi_1 M _\sigma$, and from the theorem from last time, we know that $M_\sigma$ is properly convex.

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