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Results tagged with at.algebraic-topology
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user 8032
Homotopy theory, homological algebra, algebraic treatments of manifolds.
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