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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.
19
votes
Fibrations and Cofibrations of spectra are "the same"
The following might help answer the last part of your post:
In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For exam …
- 18.9k
19
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dime...
In the simply connected case, the answer is yes.
In the general case, the theory was worked out in complete detail by Wall in the paper:
Wall, C. T. C. Finiteness conditions for CW-complexes. Ann. of …
- 18.9k
16
votes
Is an A-infinity thing the same the same as strict thing viewed through a homotopy equivalence?
The answer to your question about the loop space is a conditional "yes." The conditions are:
i) X should have the homotopy type of a CW complex, and
ii) When we say topological group, we mean with …
- 18.9k
14
votes
do spectra have diagonal maps?
The existence of an $E_\infty$-diagonal is an obstruction for equipping a spectrum $E$ with the structure of a suspension spectrum. Conversely, in
Klein, J.R.: Moduli of suspension spectra. Trans. A …
- 18.9k
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
$$
\text{map}(M,BSO_n) \to \te …
- 18.9k
12
votes
Accepted
Is the J homomorphism compatible with the EHP sequence?
Added 9/7/16:
I just got access to the paper:
James, I. M. On the iterated suspension. Quart. J. Math., Oxford Ser. (2) 5, (1954). 1–10
which is an explicit reference to Greg's questions on the leve …
- 18.9k
11
votes
Homotopy classification of selfmaps of product of spheres?
No chance. For example, take the self maps of $S^3 \times S^3$. Then based maps gives
$$
\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
11
votes
Does the Borel functor take equivariant fibrations to fibrations?
I believe what you are asking is true whenever $X\to B$, considered as an unequivariant map,
is a Serre fibration.
First some definitions:
Call a map of $G$-spaces $E \to B$ a $G$-Serre fibration …
- 18.9k
11
votes
Accepted
Connectivity of suspension-loop adjunction
If the spectrum $X$ is $r$-connected, then the map $\Sigma^\infty\Omega^\infty X \to X$
is $(2r+2)$-connected.
Here's a sketch: apply the functor $\Omega^\infty$ to get the map of spaces
$$
Q(\Omega^ …
- 18.9k
10
votes
classifying space of orthogonal groups
$BO$ can be defined as the colimit over $(k,n)$ of Grassmanians $G_k(\Bbb R^n)$ of $k$-dimensional linear subspaces of $\Bbb R^n$ (the limit over $n$ is defined by standard inclusions $\Bbb R^n \subse …
- 18.9k
10
votes
Homotopy Groups of Connected Sums
Here is something that's valid in the stable range.
If $M$ and $N$ are closed $n$-manifolds, there is a cofibration sequence
$$
S^{n-1} \to M_0 \vee N_0 \to M\sharp N
$$
where $M_0$ denotes the effec …
- 18.9k
10
votes
Accepted
Non-zero homotopy/homology in diffeomorphism groups
Here is a very naive approach: choose a basepoint in the manifold (call it $M$). Then evaluation at the basepoint gives a map
$$
\text{Diff}(M) \to M
$$
and so cohomology classes on $M$ pull back to o …
- 18.9k
10
votes
When is a homotopy pushout contractible?
Let me add some additional remarks on the enumeration question:
How many such spaces $A$ are there sitting over $B\times C$ such that the homotopy pushout
$$
B \leftarrow A \to C
$$
is contractible …
- 18.9k
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
9
votes
Accepted
$\Omega X$-action on spectral $X$-bundles
I hope I've understood your question.
Any connected based space $X$ is weak homotopy equivalent to the classifying space of a topological group $G$ in a functorial way (the group is the realization o …
- 18.9k