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.

$\text{Dev}(M, (G,X))= \{\text{dev}: \tilde M \xrightarrow{\text{smooth}} X : \text{dev is developing map of }(G,X) \text{ structure on M}\}$

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.

Previous Post: Lecture 15: Tautological Line Bundle