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 \(\rho.\)


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

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.

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