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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

4 votes
1 answer
723 views

Homotopy groups of fiber products

Let $X, Y, B$ be three smooth manifolds, and $f : X\to B$, $g : Y\to B$ submersions. Then $X\times_BY$ exists. (1) If $X, Y, B$ have the homotopy type of a finite CW complex, does $X\times_BY$? (2) W …
4 votes
0 answers
438 views

Finite good covers on smooth manifolds

Let $M$ be a connected smooth manifold that is not necessarily compact but has the homotopy type of a finite CW complex. Does $M$ admit a finite good cover? (i.e. a finite cover by contractible op …