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Results tagged with homotopy-theory
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user 8032
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
Accepted
Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?
I think the following can be turned into a proof, but I haven't checked the details.
By a result of Milnor, $\Omega (S^1 \vee S^1)$ coincides up to homotopy with $F(S^0 \vee S^0)$, the free group fu …
- 18.9k
1
vote
Components of a loop space, semidirect products, and multiplicativity
No in general. Yes if and only if the fibration $\tilde X \to X \to BG$ is trivializable,
where $X\to BG$ classifies the universal cover $\tilde X$. (Let me assume here that $X$ is connected.)
For if …
- 18.9k
4
votes
Accepted
Extreme rigidification of homotopy self-equivalences
This is a substantial revision of my original post. It shows that if we replace the "equivalence" Tyler is asking for by a "retract" then the answer is yes.
Given a CW space $Y$, we can take $G(Y) = …
7
votes
weak equivalence of simplicial sets
The answer is no, and there are plenty of counterexamples. Note that simplicial sets are not relevent here; one can cook up examples with spaces and take their total singular complexes.
For example, …
- 18.9k
9
votes
Accepted
Homotopy Units in $A_\infty$-spaces
For your first question:
If $X$ has the homotopy type of a CW space, then you can replace $X$ by any CW space $Y$ that
is homotopy equivalent to it (in the unbased sense).
Then $Y$ is also $A_\inft …
- 18.9k
10
votes
Homotopy Groups of Connected Sums
Here is something that's valid in the stable range.
If $M$ and $N$ are closed $n$-manifolds, there is a cofibration sequence
$$
S^{n-1} \to M_0 \vee N_0 \to M\sharp N
$$
where $M_0$ denotes the effec …
- 18.9k
7
votes
Accepted
What is the homotopy fiber of $X \to X_{hG}$, where this is a pointed homotopy orbit?
Here is a special case which gives a partial answer:
(i). Suppose $G$ acts in a homotopically trivial way on $X$. This means that there is a trivial $G$-space $Y$ and a pair of $G$-equivariant maps …
- 18.9k
5
votes
Accepted
pullback and fiber sequence
Yes. Here are some details.
The space $P$ sits in homotopy pullback diagram
$\require{AMScd}$
$$
\begin{CD}
P @>>> D \\
@VVV@VVV \\
A\times C @>>> D\times D
\end{CD}
$$
where the the right vertica …
- 18.9k
5
votes
Accepted
Models for P map in EHP sequence
Dear Dev,
You probably know this, but the map $P$ is the Whitehead product map in the metastable range in the following sense: Suppose $X = \Sigma Y$ is a suspension. Then the generalized Whitehead …
- 18.9k
8
votes
Maps with Hopf invariant zero are suspensions
The integral statement is most generally this:
For a based CW space $X$ one has a suspension map $E: X\to \Omega \Sigma X$ and
a Hopf invariant $H: \Omega \Sigma X \to \Omega \Sigma (X\wedge X)$ (con …
- 18.9k
11
votes
Homotopy classification of selfmaps of product of spheres?
No chance. For example, take the self maps of $S^3 \times S^3$. Then based maps gives
$$
\text{maps}_\ast(S^3\times S^3,S^3 \times S^3) = \text{maps}_\ast(S^3\times S^3,S^3) \times \text{maps}_\ast(S …
- 18.9k
12
votes
Accepted
Is the J homomorphism compatible with the EHP sequence?
Added 9/7/16:
I just got access to the paper:
James, I. M. On the iterated suspension. Quart. J. Math., Oxford Ser. (2) 5, (1954). 1–10
which is an explicit reference to Greg's questions on the leve …
- 18.9k
5
votes
Is the J homomorphism compatible with the EHP sequence?
In my comment to my first answer, I noted that I didn't address Greg's question (1). This answer aims to address that question. I am indebted to Bill Richter for explaining the following argument to …
- 18.9k
2
votes
Identify the sphere bundle of a complex line bundle $BD_{2n}\to BU(1)$
Here is a solution to a toy version of your question.
Namely, I will identify $S(\xi)$ in the special case
when $\mathbf n = \infty$.
In this case, I claim that after suspending once, we have
$$
\Si …
- 18.9k
2
votes
What is the homotopy fiber of a fold map?
The easiest way I think to get what you want is to use Omar Antolín-Camarena's approach.
I just want to point out that the result is actually a very special case of something that is much more general …
- 18.9k