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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.
13
votes
Do the two orientations on an orientable manifold $M$ uniquely witness lifts of $\tau_M: M \...
Up to homotopy, there is a fibration
$$
BSO_n \to BO_n \to B\mathbb Z_2.
$$
The space of orientations of $M$ is the (homotopy) fiber of the induced map of mapping spaces
$$
\text{map}(M,BSO_n) \to \te …
- 18.9k
3
votes
Accepted
Explaining some detail in Wall's paper of CW-complexes
As to (1):
If we choose a basepoint in $K$, then $\phi$ can be viewed as a map of based spaces. Let $F$ be the homotopy fiber of $\phi$. Then there is
a well-defined action $\Omega K \times F \to F$ w …
- 18.9k
3
votes
Accepted
Homotopy between sections
Not in general.
Suppose $f: S^1\times T \to S^1$ is the projection, where
$f$ is the first factor projection and $T = S^1 \times S^1$ is the torus.
Then a section amounts to a map $S^1 \to T$ and the …
- 18.9k
6
votes
Accepted
Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\i...
The question as stated probably requires clarification. If
$X$ is a space, then the S-dual $D_+(X)$ (i.e., functions from $X_+$ to the sphere) is always an $E_\infty$-ring spectrum. In particular, it …
- 18.9k
10
votes
Accepted
Whitehead product and a homotopy group of a wedge sum
Here are some details which are related to Tyler's comment.
I recommend looking at the paper "Induced Fibrations and Cofibrations" by Tudor Ganea (1967). For connected based spaces $X$ and $Y$, there …
- 18.9k
7
votes
Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) i...
An argument showing that the two models of suspension are equivalent will probably be based on something like the following:
Assertion: Suppose we are given a commutative diagram of the form
$\requir …
- 18.9k
1
vote
Spherical objects and K-theory
I believe the theorem you are trying to prove is due to Brinkmann who was an early student of Waldhausen.
What you are suggesting as the proof is only part of the story.
In addition to section 1.7 …
- 18.9k
3
votes
Is it true, the space of embeddings segments is homotopy equivalent to the subspace of all l...
Here are some remarks which may be relevant.
First of all it seems to me that the correct topology to use is the Whitney $C^\infty$-topology on the embedding space.
Let $M$ be an closed manifold. T …
- 18.9k
10
votes
When is a homotopy pushout contractible?
Let me add some additional remarks on the enumeration question:
How many such spaces $A$ are there sitting over $B\times C$ such that the homotopy pushout
$$
B \leftarrow A \to C
$$
is contractible …
- 18.9k
5
votes
Accepted
pullback and fiber sequence
Yes. Here are some details.
The space $P$ sits in homotopy pullback diagram
$\require{AMScd}$
$$
\begin{CD}
P @>>> D \\
@VVV@VVV \\
A\times C @>>> D\times D
\end{CD}
$$
where the the right vertica …
- 18.9k
2
votes
K-theory of free $G$-sets and the classifying space, and generalization
I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt–Priddy …
- 18.9k
1
vote
Existence of homotopy limits and colimits in model categories
Edit: The questioner has objected to the fact that the reference I gave to Q1 assumes functoriality.
Here is another reference which doesn't:
https://pages.uoregon.edu/ddugger/hocolim.pdf
See es …
- 18.9k
7
votes
Accepted
Turning injection of homotopy groups to an isomorphism
Your question is equivalent to the following:
Given a cellular inclusion $i : X\to Y$, when is there a retraction $r:Y \to X$?
(Being a retraction means that $r\circ i: X\to X$ is the identity.)
T …
- 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
7
votes
Homology of the universal cover
If we replace the field $k$ with the ring of integers $\Bbb Z$, then no.
There are non-trivial high dimensional knots $K: S^n \to S^{n+2}$, whose complements $X = S^{n+2}-K(S^n)$ have $\pi_1(X) \con …
- 18.9k