Some things I find interesting:

• Riemann mapping theorem for homeomorphisms
• The 0th homology of a path-connected space
• Differentiable structures on R^3
• Why is complex projective space triangulable?
• http://www.map.mpim-bonn.mpg.de/Exotic_spheres
• wreath product and matrix presentation
• Smoothness of distance function in Riemannian Manifolds
• http://math.stackexchange.com/questions/440475/definition-of-the-higher-dimensional-mapping-tori
• http://math.stackexchange.com/questions/396608/does-x-times-s1-cong-y-times-s1-imply-that-x-times-mathbb-r-cong-y-times-ma
• http://math.stackexchange.com/questions/441149/exotic-spheres-and-homotopy-groups-sanity-check
• http://math.stackexchange.com/questions/433706/using-geometric-arguments-to-solve-an-analysis-problem
• Are non-PL manifolds CW-complexes?
• http://math.stackexchange.com/questions/11175/is-the-serre-spectral-sequence-a-special-case-of-the-leray-spectral-sequence
• http://math.stackexchange.com/questions/450368/preimage-of-open-pathwise-connected-set-is-pathwise-connected
• http://math.stackexchange.com/questions/453454/a-property-of-analytic-functions
• Künneth theorem for fibred products
• http://math.stackexchange.com/questions/459536/the-two-sided-simplicial-bar-construction-is-reedy-cofibrant
• (A question about)${}^3$ 3-dimensional convex bodies
• Functoriality of base change
• http://math.stackexchange.com/questions/345004/is-it-possible-to-define-a-ring-as-a-category
• http://math.stackexchange.com/questions/143165/the-jacobson-radical-and-quasi-regular-elements-in-polynomial-rings-trouble-u
• http://math.stackexchange.com/questions/514578/what-is-the-homotopy-type-of-the-affine-space-in-the-zariski-topology
• http://math.stackexchange.com/questions/394896/why-learning-modern-algebraic-geometry-is-so-complicated