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Results tagged with at.algebraic-topology

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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 …

John Klein

18.5k

answered Apr 9, 2012 at 21:11

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 …

John Klein

18.5k

answered Nov 8, 2011 at 21:19

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.

John Klein

18.5k

answered Sep 11, 2011 at 15:03

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. …

John Klein

18.5k

answered Dec 9, 2011 at 19:22

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 …

John Klein

18.5k

answered Jul 12, 2016 at 4:26

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 …

John Klein

18.5k

answered Jul 23, 2013 at 21:55

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 …

John Klein

18.5k

answered Dec 24, 2020 at 1:32

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, …

John Klein

18.5k

answered Apr 27, 2011 at 19:54

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 …

John Klein

18.5k

answered Apr 6, 2012 at 20:06

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 …

John Klein

18.5k

answered Apr 14, 2011 at 15:16

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 …

John Klein

18.5k

answered Aug 19, 2011 at 13:25

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 …

John Klein

18.5k

answered Jan 18, 2013 at 3:58

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 …

John Klein

18.5k

answered May 9, 2012 at 1:32

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: …

John Klein

18.5k

answered Feb 28, 2011 at 22:57

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 …

John Klein

18.5k

answered Jan 23, 2011 at 5:16

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