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Results tagged with homotopy-theory
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user 58888
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.
16
votes
Why is the definition of the higher homotopy groups the "right one"?
Many answers are going to seem at least a little circular, including this one. If you agree that CW complexes are an interesting class of spaces, encompassing any spaces you may want to consider, then …
13
votes
Accepted
from a circle to higher spheres
A very general answer is given by Tony Elmendorf's paper Systems of Fixed Point Sets, http://www.ams.org/journals/tran/1983-277-01/S0002-9947-1983-0690052-0/. Very loosely speaking, it says that, if y …
- 2,062
9
votes
Generalized cohomology of CW complex is direct limit?
In general, no. Assuming, say, that the structure maps $SE_n\to E_{n+1}$ are inclusions, the correct statement is
$$
E^n(X) \cong [X,\lim_k \Omega^k E_{n+k}],
$$
and if $X$ is not compact that is not …
- 2,062
9
votes
3
answers
259
views
Looking for generalization of projective model structure
If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences …
- 2,062
9
votes
Homotopic but not equivariantly homotopic maps
$\newcommand{\RP}{\mathbb{RP}}$Connor Malin's answer is excellent. Derived from that, here is a small example: Let $G = C_2$, the group with two elements, let $X = S^1$ with antipodal action, and let …
- 2,062
5
votes
Accepted
Examples of smooth Hurewicz fibrations which are not smooth fiber bundles
Some poking around led to an example in G. Meigniez, Submersions, fibrations, and bundles, Trans. Amer. Math. Soc. 354 (2002), 3771-3787. It's Example 21 in that paper and described briefly as follows …
- 2,062
5
votes
Question regarding the paper by Atiyah, Bott and Shapiro: alternative description of K-theory
The image of the homotopy defines a subbundle of the bundle $E_n\times I$ over $X\times I$. The quotient bundle restricts to $E'_n$ over $X\times 0$ and $E''_n$ over $X\times 1$ and, in general, if yo …
- 2,062
2
votes
Accepted
Reduced Vs unreduced cohomology in the parametrized setting.
If $(X,p)$ is a space over $B$, its unreduced cohomology is the same as the reduced cohomology of $(X,p)_+ = (X\sqcup B,p,\sigma)$, $\sigma$ being the section taking $B$ to the disjoint copy of $B$.
…
- 2,062