All Questions
1,240 questions
14
votes
1
answer
3k
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Dijkgraaf-Witten TQFT vs. Representation Theory?
From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a ...
- 22.8k
14
votes
2
answers
2k
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Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-...
- 2,974
14
votes
1
answer
1k
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Are infinite simplicial complexes all manifolds?
Are infinite dimensional simplicial complexes manifolds locally modeled on $\mathbb R^\infty=\operatorname{colim}\mathbb R^n$? If they are homotopy equivalent, are they homeomorphic?
Of course not. ...
- 8,727
14
votes
3
answers
656
views
Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\...
- 5,596
14
votes
2
answers
2k
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Well-pointed space which is not locally contractible
I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
- 6,210
14
votes
1
answer
800
views
Is there a category whose isomorphisms are precisely the simple homotopy equivalences?
If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...
- 18.7k
13
votes
2
answers
3k
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Is every homology theory given by a spectrum?
Let $E$ be a spectrum. For any CW complex $X$, define $h_*=\pi_i(E\wedge X)$. Then we know that $h_*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a ...
- 1,373
13
votes
1
answer
1k
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Sheaves on Contractible Analytic Spaces
Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude,...
- 4,920
13
votes
2
answers
3k
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Generalized categories for "higher homotopy groupoids"
I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...
- 639
13
votes
3
answers
3k
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Representations of \pi_1, G-bundles, Classifying Spaces
This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first ...
- 2,684
12
votes
1
answer
961
views
Toda's book on homotopy groups of spheres
Yesterday one my friend told about recent book of Hiroshi Toda, where the computations of 3-torsion in homotopy groups of spheres are given up to a very high stem (about 75). This book is published ...
- 123
12
votes
2
answers
660
views
Vector bundle for prescribed Stiefel-Whitney classes
I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
- 2,554
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 $\...
- 25.2k
12
votes
3
answers
872
views
Can triangulations (or some related combinatorial structure) distinguish smooth structures on $RP^4$?
There are exotic versions of $RP^4$, constructed by Cappell-Shaneson, which are homeomorphic but not diffeomorphic to the standard $RP^4$. One way to distinguish them is via the $\eta$ invariant of $...
- 545
12
votes
4
answers
2k
views
The most general context of Mather's Cube Theorems
Quite simply, I'd like to know what is the broadest or most natural context in which either (or both) of Mather's cube theorems hold. If you like, this may mean any of
What properties of $Top$ or $...
- 1,987
11
votes
1
answer
2k
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Loop spaces motivation
I read that one of the main goals of utilization simplicial methods is to prove that a space is a loop space. On the other hand where lies the main importance to recognize topological spaces as loop ...
- 6,048
11
votes
1
answer
256
views
Do spaces admit a weak cogenerating set?
Let $\mathcal C$ be a category. Say that a class of objects $\mathcal S \subseteq \mathcal C$ is weakly cogenerating if the functors $Hom_{\mathcal C}(-,S)$ are jointly conservative, for $S \in \...
- 64k
11
votes
2
answers
1k
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Is the geometric realization of a level-wise weak equivalence a weak equivalence?
For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I ...
- 27.5k
11
votes
1
answer
948
views
In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?
I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also?
Given the broad scope of this question I ...
- 4,862
10
votes
4
answers
1k
views
The periodic values in Bott periodicity
After Bott periodicity is proved, one still has to compute the stable values. For the unitary group $U$, this is easy since you can get away with just $\pi_0$ and $\pi_1$. However, I'm having ...
- 103
10
votes
3
answers
3k
views
Topological dimension versus cohomological dimension
This should be really well known but I don't seem to find a statement about it nor a question in MO answering this.
Consider a Compact Hausdorff topological space $X$. The cohomological dimension of ...
- 3,928
10
votes
1
answer
707
views
Tensor products of $\mathbb{E}_\infty$-spaces
In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...
- 11.8k
10
votes
3
answers
761
views
Spin-H structures
Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
- 187
9
votes
2
answers
930
views
Is there a long exact sequence associated to a ramified covering?
A covering map $p:X\to Y$ between topological spaces can be viewed as a fiber bundle $\Sigma\to X\to Y$ with a discrete group $\Sigma=Gal(X/Y)$ as fiber. Such a fiber bundle leads to a long exact ...
- 681
9
votes
1
answer
810
views
The connective $k$-theory cohomology of Eilenberg-MacLane spectra
Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum.
Is it known what $ku^{*}(H\...
- 761
9
votes
3
answers
637
views
Group Extensions and Line Bundles on $BG$
I am sure the answer to this question is well-known, but
It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a ...
- 2,283
9
votes
0
answers
376
views
Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
- 66.3k
9
votes
2
answers
689
views
"Skew Cohomology" of a Space
Let $X$ be a space. The symmetric group $\Sigma_{n+1}$ acts on the function space
$$
X^{\Delta^n}
$$
of continuous maps from the standard $n$-simplex to $X$. The action is induced
by permuting the ...
- 18.9k
9
votes
2
answers
739
views
Correspondence between operads and monads requires tensor distribute over coproduct?
In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...
- 1,446
9
votes
4
answers
2k
views
How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?
The intuitive idea is that the sphere connected the two manifolds is not contractible, which implies the (n-1)th homotopy group is not zero. Another argument, which I am not totally understand, uses ...
- 1,598
9
votes
3
answers
3k
views
Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?
A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows. Choose any vector ...
- 54.6k
8
votes
1
answer
1k
views
Relations between Stiefel-Whitney classes
I need to know all relations between Stiefel-Whitney classes for closed manifolds of dimensions 3 and 4. Unfortunately, I found the literature on the subject quite confusing. The answer for all ...
- 1,579
8
votes
1
answer
474
views
Spin cobordism v.s. KO theory in low or in any dimensions
It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...
- 10.5k
8
votes
2
answers
1k
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rational cohomology of finite real grassmannian
Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
- 2,223
8
votes
1
answer
677
views
Equivalent fomulations of Bott periodicity
Is there an easy way to see the equivalence of the two statements of Bott periodicity.
$$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and
$$K(X)\otimes K(S^2) \cong K(X\times S^2)$$
- 953
8
votes
1
answer
617
views
Universal covers of non-prime 3-manifolds
Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...
- 601
8
votes
1
answer
458
views
Left Bousfield localization without properness, what is known?
I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...
- 42.4k
8
votes
1
answer
901
views
Is Margolis's axiomatisation conjecture still alive?
The construction of the category of finite spectra is easy, but there are different constructions of the whole homotopy category of spectra, all of which leading to the same result up to an ...
- 666
8
votes
1
answer
543
views
What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)
The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem.
Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...
- 103
7
votes
4
answers
788
views
Topological invariance of Stiefel-Whitney classes for open smooth manifolds
It is well known that Stiefel-Whitney classes are homotopy invariant for closed smooth manifolds. But in the case of open manifolds even $w_1$ is not a homotopy invariant (take just open cylinder and ...
- 141
7
votes
1
answer
768
views
More on completion/compactification of open manifolds
This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact ...
- 21.6k
7
votes
2
answers
728
views
Vector bundle over an oriented manifold with non-vanishing w_2w_3
I am looking for an example of an oriented rank 5 (or lower) real vector bundle $V$ over an oriented manifold such that the cup product $w_2(V) w_3(V)$ of Stiefel-Whitney classes does not vanish. It ...
- 1,582
7
votes
2
answers
1k
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Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?
I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the ...
- 7,127
7
votes
3
answers
1k
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What is the geometric realization of the the nerve of a fundamental groupoid of a space?
It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows:
Obj: $X \mapsto \pi_{\leq 1}(X)$, ...
- 1,215
7
votes
1
answer
565
views
Reference for base change of cohomology pull-push for clean intersections.
Let $X$ be a compact oriented manifold, and $A$ and $B$ closed oriented submanifolds intersecting cleanly. Then I've always been under the impression that pushing forward a cohomology class from $A$ ...
- 44.7k
7
votes
2
answers
629
views
Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?
Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?
- 91
7
votes
3
answers
911
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A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
- 13.6k
7
votes
2
answers
1k
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All mapping space between CW complexes is a CW complex?
Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\...
- 699
7
votes
2
answers
2k
views
categorical homotopy colimits
let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone ...
- 63.1k
6
votes
1
answer
954
views
What is a higher derived constructible sheaf
Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...
- 7,962