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Results tagged with homotopy-theory
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user 61824
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
6
votes
1
answer
360
views
Immersion in $\mathbb R^3$ of a Klein bottle with Morse-Bott height function without centers
Can the Klein bottle be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers?
That is, the height function would have only Bott-type extrema and …
2
votes
0
answers
90
views
First Betti number of a Reeb graph is not greater than that of the space?
(I have asked this question at math stackexchange, it was upvoted but got no answers; maybe you can help.)
It is well-known that $\beta_1(R(f))\le\beta_1(X)$, where $\beta_1$ is the first Betti numbe …
4
votes
1
answer
490
views
Immersion of non-orientable surface in $\mathbb R^3$ with conditions on the height function
EDIT: The answer is trivially positive; the question arose from my misunderstanding of the figure below.
Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the assoc …