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Results tagged with homotopy-theory
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user 10819
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
Crossed module structure on homotopy groups
1) The book "2D homotopy and combinatorial group theory" (1993) is decidedly oriented towards applications and problem solving, and does discuss crossed modules in Chapters 2 and 4, and further mentio …
- 5,575
12
votes
Accepted
is there any fibration $\mathbb{R}^n\to \mathbb{S}^n$?
Edit: The following simplifies the original answer (which unnecessarily used singular cohomology).
If $f:\Bbb R^n\to S^n$ is a fibration, then as Mark noted, a fiber $F$ of $f$ is weak homotopy equiv …
- 5,575
16
votes
Well-pointed space which is not locally contractible
Edit of edit: I'm adding still more details and references.
The one-point compactification $W^+=W\cup\{\infty\}$ of the Whitehead manifold $W$ is locally contractible (in the sense of geometric top …
- 5,575
12
votes
Accepted
Well-pointed space which is not locally contractible
My initial answer was fully ignorant of the literature on the subject; I'm now recalling that it does exist. In 1934, Borsuk and Mazurkiewicz constructed a $2$-dimensional AR (=contractible, locally c …
- 5,575
6
votes
Accepted
Terminology question on covering spaces
"Fool's covering spaces" are very close to overlays of R. H. Fox (see this paper in the first place and also this one), which I think are still better: they retain all nice properties of "fool's cover …
- 5,575
12
votes
The role of ANR in modern topology
ANRs are (and have always been) irrelevant as long as homotopy-invariant properties of spaces homotopy equivalent to CW-complexes are concerned. But modern algebraic topologists do not seem to be real …
7
votes
1
answer
1k
views
categorifying induction in homotopy type theory
In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts.
Let $B=\sqcup_{n\in\Bbb N} BS_n$, which I'd like to think of …
- 5,575
6
votes
0
answers
312
views
homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equ...
Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with …
- 5,575