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Results tagged with homotopy-theory
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user 3634
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.
40
votes
In a topological space if there exists a loop that cannot be contracted to a point does ther...
Every finite simplicial complex is weakly homotopy equivalent to a finite space. Therefore there are finite spaces with nontrivial loops; and these are obviously not embedded.
- 12.5k
38
votes
2
answers
2k
views
Finite complexes whose homotopy groups are not "finitely generated"
I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that t …
- 12.5k
27
votes
Why is it so hard to compute $\pi_n(S^n)$?
I suppose that the proof that $\pi_1(S^1) \cong \mathbb{Z}$ using covering spaces is homotopy-theoretic.
The Freudenthal Suspension Theorem (via the James construction) tells us that
$\Sigma: \pi …
- 12.5k
24
votes
Accepted
Homotopy equivalent Postnikov sections but not homotopy equivalent
This is a pretty well-known phenomenon, linked with phantom maps.
One of the first existence results was Brayton Gray's paper
Spaces of the same $n$-type, for all $n$, Topology
5 (1966) 241--243
Cla …
- 12.5k
20
votes
Accepted
Divisibility in the homotopy groups of spheres?
Yes. In fact $\bigcup_{n\geq 1} P_n $ is the set of all primes. Serre proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi_{k+(p-1)}(S^k)$, for example.
There is …
- 12.5k
20
votes
Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?
My gut reaction is always to work with CW complexes because, being a topologist, I like to work with spaces. Simplicial sets, as nice as they may be, are definitely not spaces.
19
votes
1
answer
2k
views
Slick Proof of Kudo Transgression Theorem
The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal example …
- 12.5k
19
votes
4
answers
2k
views
Difference between represented and singular cohomology?
Ordinary cohomology on CW complexes is determined by the coefficients. There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or
by defining $\ …
- 12.5k
17
votes
An abstract nonsense proof of the Hurewicz theorem
I’d argue that it boils down to the generator $S^n\to K(\mathbb{Z},n)$ being an $(n+1)$-equivalence.
More detail: If you take the represented version of homology, it is given by
$$
H_n(X;\mathbb{Z} …
- 12.5k
15
votes
2
answers
969
views
Pointed Hurewicz model structure
In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves
that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and homot …
- 12.5k
14
votes
Teaching Steenrod Operations
I like to observe that the diagonal map $X\to X\times X$
is $\mathbb{Z}/2$-equivariant, hence induces a map of homotopy colimits.
Analyzing this map with $X = K( \mathbb{Z}/2,n)$ gives the squares.
Th …
- 12.5k
14
votes
5
answers
2k
views
Good reference for homology of $K(\mathbb{Z}, 2n)$?
The homology algebra $H_*( K(\mathbb{Z},2n); \mathbb{Z})$ contains a
divided polynomial algebra on a generator $x$ of dimension $2n$.
I suppose I could read through the Cartan seminar for a proof, bu …
- 12.5k
13
votes
2
answers
2k
views
Quasifibrations and homotopy pullbacks
I'm wondering about the theoretical placement of quasifibrations.
One nice thing about "weak fibrations" (maps homotopy equivalent in the category of maps to Hurewicz fibrations) is that a pullback s …
- 12.5k
12
votes
1
answer
734
views
Open subspaces of CW complexes
I am looking at the paper
Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West
and a claim is made in the proof of their first main theorem t …
- 12.5k
12
votes
Why does homotopy behave well with respect to fibrations and homology with respect to cofibr...
Fibrations are defined as target-type concepts -- they have good formal homotopy properties when you map into them, for example, if you apply the functor $\pi_n(-) = [S^n,-]$.
Now I'll subject you to …
- 12.5k