All Questions
9,056 questions
20
votes
2
answers
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views
What's the cell structure of K(Z/nZ, 1)? Does it let me sum this divergent series? What about other finite groups?
The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be ...
- 27.7k
12
votes
4
answers
1k
views
Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 63.1k
4
votes
1
answer
543
views
Can two Riemannian manifolds (dim≠4) be homeomorphic without being bi-Lipschitz homeomorphic?
Topological manifolds of dimension ≠4 have a Lipschitz structure. [Ed: Is this "well-known"? Is it obvious? Can somebody give a reference?] Does this imply the following result?
Assume M and N ...
- 32.4k
4
votes
3
answers
1k
views
De Rham cohomology and antiderivatives
A couple of recent questions about antiderivatives reminded me of the following, which I can't recall seeing tackled explicitly anywhere to my satisfaction and that I sketched to an ambitious calculus ...
- 27.5k
12
votes
2
answers
902
views
When do the Reedy and injective model category structures agree?
Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on $\...
- 13.6k
46
votes
6
answers
7k
views
Why does one think to Steenrod squares and powers?
I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ...
CommunityBot
- 1
13
votes
2
answers
1k
views
Homotopy commutativity of the cup product
Where can I find explicit formulas for the higher homotopies, which exhibit the cup product (in singular simplicial cohomology, say) as homotopy commutative on the cochain level? Same question in ...
- 4,990
17
votes
4
answers
2k
views
Applications of the Brown Representability Theorem
Probably you can "google" this question, but I can't find anything relevant. The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : ...
- 4,990
3
votes
2
answers
467
views
Euler characteristics and operator indices as exponents for Laurent polynomials
This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$\sum q^{a_i}\in\mathbb{Z}[q,q^{-1}]$$
where the $a_i$'s are either Euler characteristics ...
- 3,474
18
votes
1
answer
2k
views
Homotopy colimits/limits using model categories
A homotopy (limits and) colimit of a diagram $D$ topological spaces can be explicitly described as a geometric realization of simplicial replacement for $D$.
However, a homotopy colimit can also be ...
- 27.2k
5
votes
1
answer
580
views
Free actions of finite groups on products of even-dimensional spheres
Suppose a finite 2-group G acts freely on X = $\prod_{i=1}^k$ *S*$^{2n_i}$, a product of k even-dimensional spheres, k > 2. Is it possible for G to be non-abelian? What if we additionally assume that ...
- 47.3k
5
votes
2
answers
454
views
Burnside ring and zeroth G-equivariant stem for finite G
Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
- 4,990
9
votes
1
answer
2k
views
The Join of Simplicial Sets ~Finale~
Background
Let $X$ and $S$ be simplicial sets, i.e. presheaves on $\Delta$, the so-called topologist's simplex category, which is the category of finite nonempty ordinals with morphisms given by ...
CommunityBot
- 1
4
votes
1
answer
149
views
singular cohomological dimension
Let $X$ be a finite CW complex. Is there an integer $N,$ such that $H^i(X,F)=0$ for all $i>N$ and all abelian sheaves $F$ on $X?$ The cohomology is defined to be the derived functor of the global ...
- 47.3k
4
votes
1
answer
299
views
Changing the orientation of a Landweber exact cohomology theory
Let the ring R be a MU*-module via a ring homomorphism φ and suppose it satisfies the condition of the Landweber exact functor theorem such that we obtain a ...
- 52.6k
2
votes
0
answers
409
views
virtual bundle with compact support
A virtual bundle with compact support is a triple $E$={$E_1$,$E_2$,$a$},
where $E_1$ and $E_2$ are bundles over $M$ of the same dimension, $M$ is a closed manifold, $a$ is a bundle map from $E_1$ to $...
- 21k
2
votes
0
answers
270
views
Homotopy equivalences and cores
Hi all,
Before asking my question, I need to fix some terms and notation.
Let $M$, $M'$ be locally compact, Hausdorff spaces, and $f:M\rightarrow M'$ a homotopy equivalence with homotopy inverse $g:...
- 93
1
vote
2
answers
9k
views
How To Calculate A Winding Number?
We have a closed curve C on the plane given by parametric equations: x=x(t), y=y(t), t changes between a and b, x and y are smooth functions.
We want to calculate the winding number of this curve ...
- 9,366
28
votes
5
answers
9k
views
Why can't the Klein bottle embed in $\mathbb{R}^3$?
Using Alexander duality, you can show that the Klein bottle does not embed in $\mathbb{R}^3$. (See for example Hatcher's book Chapter 3 page 256.) Is there a more elementary proof, that say could be ...
- 28.2k
10
votes
0
answers
735
views
Adams Spectral Sequence for Equivariant Cohomology Theories
In ordinary algebraic topology the Adams spectral sequence can be applied for any cohomology theory $E$ and in good cases it converges to the stable homotopy classes of maps (of the E-nilpotent ...
- 1,273
11
votes
1
answer
1k
views
Pontrjagin numbers and exotic spheres
Hi everyone, im reading Milnor's article "On manifolds homeomorphic to the 7-sphere", in which he constructs the first example of an exotic structure, id like to know if there's a particular reason ...
- 13.2k
2
votes
2
answers
463
views
homotopy type of complement of subspace arrangement
I am studying the homotopy type of a space,and i hope it would be a $K(\pi,1)$ space.
now i have find its covering,once we can say the covering is $K(\pi,1)$,so is the space
itself.and the covering is ...
- 23.5k
3
votes
3
answers
2k
views
Homology with Coefficients
We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in ...
- 54.6k
14
votes
3
answers
991
views
Homotopy type of set of self homotopy-equivalences of a surface
Let $\Sigma$ be an oriented topological surface. For simplicity, assume that the genus of $\Sigma$ is at least $2$. There are a number of classical results on the homotopy types of various groups of ...
- 20k
7
votes
1
answer
672
views
How does this geometric description of the structure of PSL(2, Z) actually work?
There is a beautiful way to see that the congruence subgroup $\Gamma(2)$ is free on two generators: the action of $\Gamma(2)$ on $\mathbb{H}$ is free and properly discontinuous, and there is a modular ...
- 118k
14
votes
2
answers
2k
views
Sheaves over simplicial sets
Is there a good way to define a sheaf over a simplicial set - i.e. as a functor from the diagram of the simplicial set to wherever the sheaf takes its values - in a way that while defined on simplex ...
CommunityBot
- 1
4
votes
1
answer
972
views
relationship between borromean rings and hanging-a-picture-from-three-nails puzzle?
I recently heard the following puzzle: There are three nails in the wall, and you want to hang a picture by wrapping a wire attached to the picture around the nails so that if any one nail is removed ...
CommunityBot
- 1
11
votes
1
answer
2k
views
K-theory as a generalized cohomology theory
Which of the statements is wrong:
a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point
reduced complex $K$-theory $\tilde K$ and reduced real $K$...
- 15.4k
-1
votes
1
answer
1k
views
Covering maps on Euclidean spaces and spheres [closed]
Hello. I have two questions.
Does there exist an exactly 2-fold covering map
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ ?
Does there exist an exactly 2-fold covering map
$g:S^{n}\rightarrow S^{n}$ ?
...
- 24.1k
7
votes
2
answers
419
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
- 15.4k
18
votes
1
answer
853
views
Can we reconstruct positive weight invariants in algebraic topology using algebraic geometry?
I can't really say that I understand what a weight is, but the qualitative distinction between weight zero and positive weight has come up a couple times in MathOverflow questions:
The étale ...
CommunityBot
- 1
2
votes
1
answer
1k
views
Fixed points of continuous involutions of the plane
Hello. I would like to know how to prove that every continuous involution $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$
(that is, $F(F(x))=x$ $\forall x \in \mathbb{R}^{2}$ ) has a fixed point?
Thank you very ...
- 32.4k
5
votes
1
answer
3k
views
Does the category Monoid of monoids have finite coproducts?
Does the category Monoid of monoids have finite coproducts?
- 156k
-2
votes
1
answer
780
views
commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 65.4k
7
votes
1
answer
1k
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Mumford Conjecture
The Mumford Conjecture (now a theorem) says basically what is the (tautological subring)* of the rational cohomology ring of the stable moduli space of curves. Meaning that we know the ring structure (...
- 2,132
1
vote
1
answer
364
views
Analogs of left, right, inner, and Kan fibrations in CGWH
It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. However, I know next to ...
- 19.6k
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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12
votes
7
answers
1k
views
Cohomology classes annihilated by pullbacks
A friend of mine is interested in examples of the following situation:
an oriented smooth fiber bundle $\pi \colon M \to B$ with $M$ and $B$ compact
and a non-zero class $a \in H^3(B; \mathbb{Q})$
...
- 18.2k
3
votes
1
answer
4k
views
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
6
votes
1
answer
890
views
Serre spectral sequence with spectra
A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than ...
- 27.7k
4
votes
1
answer
2k
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Fiber bundle = principal bundle + fiber?
This question is heavily related to this question.
Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
CommunityBot
- 1
8
votes
2
answers
2k
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Chas-Sullivan string topology
I recently read the original paper by Chas-Sullivan on string topology, in which they introduce some operations on homology of free loopspace LM, where M is a compact oriented manifold, giving it the ...
- 252
8
votes
1
answer
2k
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homotopy associative $H$-space and $coH$-space
Let $[X, Y]_0$ denote base point preserving homotopy classes of maps $X\rightarrow Y$. A multiplication on a pointed space $Y$ is a map $\phi: Y\times Y\rightarrow Y.$ From this map, we can define a ...
- 11.1k
6
votes
1
answer
818
views
Spectra and localizations of the category of topological spaces
Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces
using some kind of localization combined with other categorical ...
- 12.5k
2
votes
2
answers
1k
views
Simple question of topological cofibration
I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
- 8,803
14
votes
2
answers
2k
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Applications of homotopy groups of spheres
The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is ...
- 44.4k
3
votes
1
answer
424
views
Principal bundle for contractible group is weak homotopy equivalence for ind schemes
This is may be obvious, but I am not comfortable with ind-schemes.
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
- 27.5k
14
votes
4
answers
5k
views
homology with compact supports
In one of the exercises in McDuff and Salamon, they mention homology with compact supports. I know how to define *co*homology with compact supports, but I can't picture the homology version. How do ...
- 4,404
5
votes
2
answers
357
views
Truncated exact sequence of homotopy groups
This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma ...
- 3,376
7
votes
2
answers
2k
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Intersection form in twisted homology (homology with local coefficients)
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
- 57.6k