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Results tagged with at.algebraic-topology
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user 8032
Homotopy theory, homological algebra, algebraic treatments of manifolds.
16
votes
Choice of base point in a Waldhausen category
Although it might seem surprising to some readers, Waldhausen was not going for abstraction merely for the sake of itself in making his definitions: he had concrete applications in mind (most importan …
- 18.5k
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 …
- 18.5k
16
votes
What does actually being a CW-complex provide in algebraic topology?
One should not forget that certain invariants, such as Whitehead torsion, are defined
using a choice of CW structure.
- 18.5k
8
votes
what does BG classify? i.e. what is a principal fibration?
When $G$ is discrete, another sort of answer is provided by the paper of Michael Weiss:
What does the classifying space of a category classify? Homology, Homotopy and Applications 7 (2005), 185–195.
…
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
1
vote
$G$-equivariant intersection theory using differential topology?
You may want to take a look at
Klein, J.R., Williams, B.
Homotopical intersection theory, II: equivariance.
Math. Z. 264(2010),849–880.
An arXiv version appears here:
https://arxiv.org/abs/0803.0017
I …
- 18.5k
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, …
- 18.5k
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 …
- 18.5k
1
vote
Regular homotopy invariance of Wall's self-intersection form.
Even more is true in the context of surgery theory: let
$q \ge 3$ with $q$ odd, assume $M$ is a $1$-connected closed smooth manifold of dimension $2q$. Let $$I^{\text{fr}}_q(M)$$ denote the space of i …
- 18.5k
1
vote
On a special case of Alexander duality
The proof that $S^n \setminus K$ is path connected follows directly from general position if $K$ is, say, a finite simplicial complex of codimension two (I don't know what "codimension two" means in y …
- 18.5k
1
vote
Tubular neighborhoods of chains
Here's an approach which might work (I'm not sure about the correctness of this.)
1) Assume $M$ is closed. Choose a triangulation $T$ of $M$.
If the support of $c$ is contained inside the $p$-skelet …
- 18.5k
15
votes
Computational complexity of computing homotopy groups of spheres
Here's a very useless algorithm due to Kan: Let $G(S^n)$ be the simplicial group that is
Kan loop group of the $n$-sphere. In each simplicial degree, it is a free group.
(This simplicial group has the …
- 18.5k
8
votes
What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?
See
Brown, Edward, The cohomology of $B\text{SO}_n$ and $B\text{O}_n$ with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288
for an answer to your questions. (As for applications: …
- 18.5k
3
votes
Why does the map $BG\to A(*)$ fail to split?
I just realized how the argument for Question 1 might go (I hope this isn't self-indulgence on my part):
The composite $BO \to BG \to A(*)$ factors through $Q(S^0)$. So it suffices to show there is …
- 18.5k