6
$\begingroup$

A substantial part of mathematics studies manifolds which are defined as second countable Hausdorff locally Euclidean topological spaces. That always seemed kind of random to me since what is so special about Euclidean spaces?

I guess the answer is that there is a lot of interesting phenomena involving manifolds: the classification of topological surfaces, Poincaré duality, the ability to put real-analytic structures on them sometimes which brings its own set of bells and whistles (complex and symplectic structures, Donaldson invariants etc.).

However, Euclidean spaces are not the only possible "basis" in topology. For example, one can consider spaces locally homeomorphic to Zariski spectra of rings. Hochster and others have proved some interesting things about them. What other rich classes of topological spaces can we get this way?

The questions overlaps to some extent with this one.

Improve this question
$\endgroup$
2
  • 4
    $\begingroup$ Hilbert manifolds and orbifolds are two more classes of examples $\endgroup$
    – Alessandro Codenotti
    Commented Aug 7, 2020 at 20:15
  • 3
    $\begingroup$ I'll repeat my comment from the other link: Menger compacta are a family generalizing the Menger cube are homogeneous and have interesting manifolds built on them. An $n$-manifold can be essentially uniquely replaced by an $n$-$k$-Menger manifold, keeping homotopy information below dimension $k$ and destroying information above dimension $k$. @AlessandroCodenotti "Hilbert manifold" is ambiguous between non-locally-compact objects modeled on Hilbert spaces and locally compact spaces modeled on the Hilbert cube $[0,1]^\infty$. $\endgroup$
    – Ben Wieland
    Commented Aug 7, 2020 at 23:16

0

Reset to default

You must log in to answer this question.