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Results tagged with reference-request
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user 8032
This tag is used if a reference is needed in a paper or textbook on a specific result.
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
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
7
votes
1
answer
564
views
Filtered homotopy colimits and singular homology
Suppose I have a functor
$$
X_\bullet: I \to \text{Spaces}
$$
where $I$ is a small filtered category.
It seems to be a "folk theorem" that the homomorphism
$$
\underset{\alpha\in I}{\text{colim }} H_ …
- 18.5k
7
votes
0
answers
132
views
Weak homotopy type of the Cech Nerve
Let ${\cal U} = \{U_i\}_{i\in J}$ be an open cover of a topological space $X$, where the indexing set $J$ is assumed to be well-ordered. Then the Cech nerve is the "simplicial space without degeneraci …
- 18.5k
16
votes
4
answers
1k
views
Multiplicativity of the Euler characteristic for fibrations
For a Serre fibration
$$
F\to E \to B ,
$$
with $F,E,B$ having the homotopy type of finite complexes, it is known that the Euler characteristic is multiplicative:
$$
\chi(E) = \chi(F)\chi(B) .
$$
Howe …
- 18.5k
1
vote
0
answers
85
views
Iterated quotients in GIT
Suppose that $G$ is a reductive group that acts algebraically on an affine variety $X$ over an algebraically closed field $k$.
Suppose also that $G$ is equipped with a normal abelian subgroup $N$ such …
- 18.5k
2
votes
What is the homotopy fiber of a fold map?
The easiest way I think to get what you want is to use Omar Antolín-Camarena's approach.
I just want to point out that the result is actually a very special case of something that is much more general …
- 18.5k
5
votes
Is the J homomorphism compatible with the EHP sequence?
In my comment to my first answer, I noted that I didn't address Greg's question (1). This answer aims to address that question. I am indebted to Bill Richter for explaining the following argument to …
- 18.5k
3
votes
Accepted
Postnikov-type tower for a map between spaces
Here is an answer to your "Part I."
The construction you outlined (and its dual) was considered in detail in the papers of Ganea. In particular:
Ganea, T.
Induced fibrations and cofibrations.
Trans …
- 18.5k
12
votes
Accepted
Is the J homomorphism compatible with the EHP sequence?
Added 9/7/16:
I just got access to the paper:
James, I. M. On the iterated suspension. Quart. J. Math., Oxford Ser. (2) 5, (1954). 1–10
which is an explicit reference to Greg's questions on the leve …
- 18.5k
7
votes
1
answer
569
views
Question about topological monoid maps
Let Mon be the category of topological monoids. I am happy to work with the model structure mentioned here:
Model Structure/Homotopy Pushouts in topological monoids?.
I'm looking for a reference f …
- 18.5k
8
votes
Accepted
Homotopy dimension of a mapping
Regarding Question 1: No, I do not think that's correct. In my opinion, the
definition should be one of the following:
The relative homotopy dimension of $f: X \to Y$ is $\le k$ if and only if there …
- 18.5k
6
votes
Accepted
$X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a co-H-Space
The proof is not hard, but more tedious than I would have thought.
There is a canonical identification
$$
X\rtimes Y = X\wedge(Y_+)
$$
where $Y_+$ is the effect of adding a disjoint base point to $ …
- 18.5k
11
votes
Accepted
Linking topological spheres
Let $C = S^3 \setminus A$. Alexander duality says that
$$
H_1(C) \cong H^1(A) \cong \Bbb Z\, .
$$
Let $\alpha: S^1 \to C$ be any map representing a generator of $H_1(C)$ (every first homology class …
- 18.5k
12
votes
1
answer
335
views
Rational homotopy invariance of algebraic $K$-theory
Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra
$$
K( …
- 18.5k