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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
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
2
votes
Accepted
Relating two notions of geometric realization
Regarding
What is the relation between $|K|$ and $X$, more specifically between $|\sigma|$ and $X_\sigma$ for each simplex $\sigma \in K$?
It seems to me that one can build an intermediate space …
- 18.5k
5
votes
What do Atiyah and Segal mean by $K_G^*(X)$?
Although not given in the Atiyah-Segal paper, I suspect the definition is as follows:
The functor $K_G$ is defined on pairs of $G$-spaces $(X,A)$, since it is a cohomology theory (I guess it's the Gr …
- 18.5k
28
votes
0
answers
1k
views
On the (derived) dual to the James construction.
Background
If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by
$$
X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots
$$
in …
7
votes
2
answers
544
views
Why does the map $BG\to A(*)$ fail to split?
There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is
Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus construct …
- 18.5k
6
votes
1
answer
256
views
An operad-like structure, is there a name for it?
Here is an example which I'd like to have a name for.
Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary.
Define $E(k,P)$ to be the space of smooth (codimension …
- 18.5k
12
votes
Accepted
"C choose k" where C is topological space
I presume you are asking is whether one can make sense of $\binom{X}{k}$ as a space in such a way that it relates to the formula you are asking about.
The answer is yes.
First let $l = 1$. For a sp …
- 18.5k
5
votes
Accepted
Models for P map in EHP sequence
Dear Dev,
You probably know this, but the map $P$ is the Whitehead product map in the metastable range in the following sense: Suppose $X = \Sigma Y$ is a suspension. Then the generalized Whitehead …
- 18.5k
9
votes
2
answers
649
views
"Skew Cohomology" of a Space
Let $X$ be a space. The symmetric group $\Sigma_{n+1}$ acts on the function space
$$
X^{\Delta^n}
$$
of continuous maps from the standard $n$-simplex to $X$. The action is induced
by permuting the ve …
- 18.5k
12
votes
1
answer
844
views
Equivariant homotopy theory: some history questions
I have sometimes wondered about the following:
(1) Who was the first articulate that in dealing with $G$-equivariant cohomology theories ($G$ a finite group or a compact Lie group), it is best to wor …
- 18.5k
10
votes
3
answers
588
views
On the naturality of the bar construction
Let $X$ be a based space. Then the Moore loop space $MX$ is defined to be the topological monoid whose points are based loops $[0,a] \to X$ where $a \ge 0$ is allowed to vary. Composition is gotten b …
- 18.5k
15
votes
2
answers
1k
views
"Strøm-type" model structure on chain complexes?
Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by t …
- 18.5k
16
votes
1
answer
2k
views
On the wikipedia entry for Borel-Moore homology
The wikipedia page on Borel-Moore homology claims to give several definitions of it,
all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and …
- 18.5k
4
votes
1
answer
486
views
Existence of homotopy inverses for co-H spaces
Suppose $c: X \to X\vee X$ is a co-$H$ structure on a based CW complex $X$.
Question: Under what circumstances can one find left (right) homotopy inverses for $c$?
Remarks: If $X$ is $1$-connect …
- 18.5k
8
votes
1
answer
507
views
Weak Vector Bundles
The following notion has arisen in a paper I'm writing.
Definition. A map $p: E\to B$ of spaces
is said to be weak vector bundle if for all compact subspaces $K \subset B$
the restriction of $p$ to …
- 18.5k
3
votes
Homotopy type of smooth manifolds with boundary
I know of two proofs in the compact case. Let $M$ be a compact smooth $m$-manifold with boundary $\partial M$.
1) Morse theory (Sketch). For this I think we need $m \ge 4$.
There is a Morse functio …
- 18.5k
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 …
- 18.5k
26
votes
Accepted
Groupoid actions on spaces
Perhaps the most natural example is given by universal covers?
Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X_x$ be the universal covering of
$X$ taken at $x$ (the fiber at $y \in X$ …
- 18.5k
2
votes
Multiplicativity of the Euler characteristic for fibrations
Here is an argument that the
Euler characteristic is multiplicative for fibrations
$$
F\to E \to B
$$
where $F$ and $B$ are finitely dominated and $B$ is connected.
Without loss in generality, we may …
- 18.5k
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 …
- 18.5k
7
votes
Multiplicativity of the Euler characteristic for fibrations
Note Added March 1, 2022:
I now think there is a gap in deducing multiplicativity of the Euler characteristic from the Pedersen-Taylor result on the finiteness obstruction. I think the argument I giv …
- 18.5k