Some things I find interesting: http://mathoverflow.net/questions/66048/riemann-mapping-theorem-for-homeomorphisms http://mathoverflow.net/questions/41299/the-0th-homology-of-a-path-connected-space http://mathoverflow.net/questions/24930/differentiable-structures-on-r3 http://mathoverflow.net/questions/20664/why-is-complex-projective-space-triangulable http://www.map.mpim-bonn.mpg.de/Exotic_spheres