Paths in spaces of homeomorphisms on the plane (Q714760)

The path components and connected components are determined for the space \(\mathcal H(\mathbb C)\) of homeomorphisms on the complex plane \(\mathbb C\) when endowed with the pointwise, compact-open and fine topology, respectively (denoted as \( \mathcal H_p(\mathbb C)\), \(\mathcal H_k(\mathbb C)\), and \( \mathcal H_f(\mathbb C)\)); the space \(\mathcal H_u(\mathbb C)\) -- i.e. \( \mathcal H(\mathbb C)\) with the uniform topology -- is also considered, but characterizations of these components are left as open problems. One of the main results shows that \(\mathcal H_k(\mathbb C)\) has exactly two path components, the set of orientation preserving homomorphisms \(\mathcal H^+(\mathbb C)\) and the set of orientation reversing homomorphisms \(\mathcal H^-(\mathbb C)\), moreover, \( \mathcal H^+(\mathbb C)\) is clopen in \( \mathcal H_k(\mathbb C)\) so it is also the connected component of \( \mathcal H_k(\mathbb C)\) containing the identity homeomorphism, and \( \mathcal H^-(\mathbb C)\) is the other connected component of \( \mathcal H_k(\mathbb C)\). For a contrast, \(\mathcal H^+(\mathbb C)\) is dense (with a dense complement \(\mathcal H^-(\mathbb C)\)) in \( \mathcal H_p(\mathbb C)\); in fact, \( \mathcal H_p(\mathbb C)\) is itself pathwise connected. On the other hand, \(\mathcal H_u(\mathbb C)\) and \(\mathcal H_f(\mathbb C)\) have infinitely many path components and connected components. It is also shown that the arc components in each of the spaces \( \mathcal H_p(\mathbb C)\), \(\mathcal H_k(\mathbb C)\), \( \mathcal H_f(\mathbb C)\)), \(\mathcal H_u(\mathbb C)\) are the same as the path components, and at least in each of the spaces \( \mathcal H_p(\mathbb C)\), \(\mathcal H_k(\mathbb C)\), and \( \mathcal H_f(\mathbb C)\)), the connected components are the same as the path components.