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Results tagged with at.algebraic-topology
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user 8032
Homotopy theory, homological algebra, algebraic treatments of manifolds.
4
votes
what is the image of $\partial( 1_{S^n})$ for the exact sequence for the fibration $X \to E ...
Any fibration $X \to E \to S^n$ up to fiber homotopy equivalence is given by the "clutching construction" applied to a map $f: S^{n-1} \to G(X)$, where $G(X)$ denotes the topological monoid of self ho …
- 18.5k
3
votes
Examples for open disc bundle which is not vector bundle
Some remarks which amount in some way to an answer.
(1) Let $\text{Diff}(D^n)$ be the diffeomorphisms of $D^n$ which restrict to the identity on
the boundary.
When $n\gg k$ is large, $\pi_k(\text{D …
- 18.5k
4
votes
Acyclic complexes for extraordinary cohomology theories
If you go to exotic cohomology with twisted coefficients, then the answer is yes.
Alternatively, one can state the condition on the level of a universal cover and then the answer is yes.
It is certa …
- 18.5k
2
votes
Fibrations with non-simply connected base and rational homology
I'm wondering the extent to which the assumptions can be tweaked. Let's assume
$B$ is connected and with basepoint. Let $F$ be the fiber over the basepoint.
However, I won't assume $F$ is homotopy …
- 18.5k
13
votes
Accepted
Reverse map of a homology equivalence.
As I was writing this answer, Oscar beat me to the punch. I will keep it posted anyway.
Let $X^3$ be the Poincare homology sphere. Let $\tilde X \to X$ be the universal cover (note: $\tilde X$ is $S^ …
- 18.5k
3
votes
Does the classifying space of monoids commute with wedge sum up to weak equivalence?
Charles has given a very good answer to the question.
The following is not meant to be an answer, but just a heuristic argument which I cannot
make into a proof.
There should be an operation, "fr …
- 18.5k
10
votes
When can you desuspend a homotopy cogroup?
Hopkins' result alluded to above gives a coordinate-free approach a lá Segal. The paper I wrote with Schwänzl and Vogt:
Comultiplication and suspension. Topology Appl. 77 (1997), no. 1, 1–18,
gives …
- 18.5k
10
votes
The fundamental group of space which has both an H and a co-H structure
Another argument which does it:
A path connected based space $X$ is a co-$H$ space if and only if the evaluation map $\Sigma \Omega X \to X$ admits a section up to homotopy. This will imply that $\p …
- 18.5k
10
votes
Accepted
Contraction of a family of loops simultaneously
Surely not.
Let $S^1 \to LS^2$ be adjoint to the map
$c: S^1 \times S^1 \to S^2$ which collapses $S^1\vee S^1$ to a point.
The latter has degree one.
Let $p\in S^1$ be any point but the basepoint.
…
- 18.5k
9
votes
Can both G and BG be finite CW complexes?
The answer is always no unless $G$ is trivial.
In fact, I can generalize your statement slightly: I only need to assume
that $G$ and $BG$ have merely the homotopy type of finite complexes.
I will ou …
- 18.5k
14
votes
How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?
Here's a different way to see it. Let $M$ and $N$ be aspherical of dimension at least 3.
Then the wedge $M \vee N$ is aspherical (but not a manifold). Let $M\sharp N$ be the connected sum. Then we get …
- 18.5k
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) = …
5
votes
Homotopy type of the plane minus a sequence with no limit points
Here is a more naive solution, as least if the sequence is countable. Let
$\Bbb N \subset \Bbb R^2$ be the embedding defined by the sequence.
Then there is an isotopy from this embedding to the stand …
- 18.5k
10
votes
Accepted
Algebraic K-theory of odd-dimensional spheres
Let $\tilde A(X)$ be the reduced functor, i.e., the homotopy fiber of the map $A(X) \to A(\ast)$. Since $A(*)$ is rationally a product of $K(Q,4j+1)$ for $j \ge 1$, we may as well study $\tilde A(X)$ …
- 18.5k
5
votes
Obstruction Theory for Vector Bundles and Connections
Correction: the definition below is wrong. It isn't true that a 1-flat reduction is the same as a flat reduction. One also needs to require that the map $Z \to BG$ factors through $BG^\delta$, where $ …
- 18.5k