All Questions
Tagged with examples at.algebraic-topology
37 questions
1
vote
1
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125
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Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME
I apologize for repeating the same question from ME, but it seems more subtle than I expected.
Let me fix the notations here first:
\begin{equation}
C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...
- 3,477
21
votes
2
answers
2k
views
A simplicial complex which is not collapsible, but whose barycentric subdivision is
Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?
Every collapsible complex is necessarily contractible, and subdivision preserves the ...
CommunityBot
- 1
7
votes
0
answers
218
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Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 6,367
6
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1
answer
231
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Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
- 30.3k
3
votes
1
answer
459
views
When is the Freudenthal compactification an ANR?
Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What are ...
- 1,446
46
votes
5
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3k
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‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$
[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg–MacLane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (Or if not a manifold, say some ...
- 5,429
11
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4
answers
4k
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A map inducing isomorphisms on homology but not on homotopy
As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem:
A weak homotopy equivalence induces isomorphisms of the corresponding integral ...
- 2,339
27
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6
answers
4k
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Failure of smoothing theory for topological 4-manifolds
Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
- 1,446
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
- 2,821
8
votes
2
answers
574
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Non-trivial examples of Stably diffeomorphic 4-manifolds
I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp_n (S^2\times S^2) \cong N \sharp_r (S^2\times S^2)$$ for $r,n$ not ...
38
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4
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4k
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What manifolds are bounded by RP^odd?
Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...
- 11.2k
5
votes
1
answer
145
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Example request: seriously deficient homogeneous spaces
In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
- 309
41
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10
answers
4k
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Phenomena of gerbes
What is your favourite example of Gerbes?
I would like to know Where do we find Gerbes in "nature"?
The examples could vary from String theory to Galois theory. For example my favourite examples of ...
- 63.9k
12
votes
1
answer
904
views
Manifolds with nonwhere vanishing closed one forms
I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $N\times S^1$.
- 9,580
32
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5
answers
4k
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Some intuition behind the five lemma?
Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category)
$$\require{AMScd}
\begin{CD}
A_1 @>>> A_2 @>>> A_3 @>>> A_4 @...
- 41
45
votes
13
answers
9k
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Motivating the de Rham theorem
In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book Foundations of Differentiable Manifolds and Lie Groups. One thing ...
- 6,025
25
votes
1
answer
5k
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Example of fiber bundle that is not a fibration
It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I ...
- 79
7
votes
0
answers
455
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Is there a list of examples of orthogonal spectra?
Schwede's symmetric spectra book project provides point-set models of many important spectra as symmetric spectra, including (in §I.1) the sphere spectrum, Eilenberg-Mac Lane spectra, several Thom ...
- 6,881
10
votes
3
answers
2k
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Need examples of homotopy orbit and fixed points
I am no expert in equivariant homotopy theory. Let's say, I am planing to give a talk on homotopy fixed points and orbits. My audience will be graduate students who are doing algebraic topology or ...
- 12.3k
28
votes
1
answer
2k
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Example of 4-manifold with $\pi_1=\mathbb Q$
This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.
CommunityBot
- 1
112
votes
6
answers
10k
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Counterexamples in algebraic topology?
In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...
CommunityBot
- 1
16
votes
5
answers
2k
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What are examples when the equality of some invariants is good enough in algebraic topology?
As far as my understanding goes, most of the tools of algebraic topology (homotopy groups, homology groups, cup product, cohomology operations, Hopf invariant, signature, characteristic classes, knot ...
CommunityBot
- 1
3
votes
2
answers
445
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Example s.t. the unbased loop-space is not $\Omega X \times X$
For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...
- 12.1k
37
votes
3
answers
3k
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Are there pairs of highly connected finite CW-complexes with the same homotopy groups?
Fix an integer n. Can you find two finite CW-complexes X and Y which
* are both n connected,
* are not homotopy equivalent, yet
* $\pi_q X \approx \pi_q Y$ for all $q$.
In Are there two non-...
CommunityBot
- 1
1
vote
1
answer
356
views
Examples of nontrivial local systems in Decomposition Theorem
There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a IC_{\bar{Y_a}}(L_a)[shifts]$...
- 9,870
16
votes
6
answers
6k
views
Fundamental group of the line with the double origin.
In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole".
This ...
CommunityBot
- 1
2
votes
1
answer
412
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Vanishing Cech cohomology
Let $X$ be a manifold such that $dim(X)=n$. It is well-know that if $\mathcal{F}$ is a coherent sheaf $H^m(X,\mathcal{F})=0$ for all $m >n$ (where I denote with $H(-)$ Cech cohomology). But is ...
CommunityBot
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13
votes
2
answers
2k
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Request: A Serre fibration that is not a Dold fibration
A Serre fibration has the homotopy lifting property with respect to the maps $[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration $E \to B$ has the weak covering homotopy property: lifts with ...
- 27.5k
31
votes
3
answers
4k
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Algebras over the little disks operad
Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like" algebra over the ...
CommunityBot
- 1
6
votes
1
answer
462
views
A counter example in obstruction theory
Let $K$ denote a simplicial complex and $Y$ a path-connected topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation or a ...
- 56.9k
19
votes
1
answer
1k
views
What are "good" examples of string manifolds?
Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle ...
CommunityBot
- 1
6
votes
2
answers
926
views
Simple examples of equivariant homology and bordism
I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.
I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant ...
- 11.1k
1
vote
1
answer
716
views
An example of a space which is locally relatively contractible but not contractible?
A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ...
- 52.6k
5
votes
2
answers
878
views
What is an example of a non-regular, totally path-disconnected Hausdorff space?
I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...
- 7,193
1
vote
0
answers
1k
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Again about Bing's house with two rooms [duplicate]
Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question clearly, when "hollowing ...
CommunityBot
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3
votes
1
answer
4k
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How to show that the "bing's house with two rooms" is contractible? [closed]
I can't image this, Someone can give a clear illustration?
- 19.6k
3
votes
1
answer
928
views
Simple applications of Atiyah-Bott localization
I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
- 156k