All Questions
9,056 questions
2
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687
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How to build the principal SU(2) bundles on surfaces?
Is there a way to classify (and build) the principal SU(2) bundles over a given topological surface up to homeomorphism? In the end, I would like to examine the associated bundle whose fiber is a ...
- 4,404
19
votes
2
answers
1k
views
What manifold has $\mathbb{H}P^{odd}$ as a boundary?
This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting.
...
CommunityBot
- 1
7
votes
1
answer
871
views
Whitehead Products without Base Points?
Let $(X, x_0)$ be a pointed space. Then we can define the homotopy groups $\pi_i(X, x_0)$ for $i \geq 1$. They are abelian groups for $i \geq 2$. It is well-known that the fundamental group $\pi_1(X, ...
- 2,145
5
votes
3
answers
2k
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Classification of simply connected smooth projective varieties?
This question is related to this one.
I am wondering whether there is any sort of classification of simply connected smooth projective varieties, or any work in related directions.
The reason I am ...
CommunityBot
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7
votes
1
answer
588
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Whitehead products on manifolds
What are some good examples of simply connected manifolds with interesting Whitehead Lie algebras over R? Most of the manifolds that one thinks about if one is pretty naive are not so interesting--- ...
7
votes
3
answers
1k
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Group cohomology vs. topological cohomology in the case of non-trivial action
When A is an abelian group with trivial G-action (G being a discrete group) we get that Hn(G,A)≅Hn(BG,A). Is there a similar connection between group cohomology and topological cohomology if A ...
- 4,404
6
votes
1
answer
675
views
Some questions on the intersection theory on a Hilbert scheme of points of a surface.
If $\Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $\Sigma^{[n]} \subset X^{[n]}$. $\ \ $ Where, $X^{[n]}, \Sigma^{[n]}$ ...
- 445
7
votes
1
answer
723
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Surgery and homology: a reference request
I need a reference (or a short proof) for the following statement:
Suppose a closed manifold $N$ is the result of a surgery (along an embedded sphere) on a closed manifold $M$. Then the difference $\...
- 13.2k
41
votes
3
answers
3k
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What is the classifying space of "G-bundles with connections"
Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
- 4,519
13
votes
2
answers
713
views
How do you compute the space of lifts of an E-infinity map?
Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
- 52.6k
1
vote
1
answer
231
views
blowing up the graphs
I heard the phrase from many mathematician using in the colloquials. I heard algebraic geometer using it. I was never bother about it until one of my professor responded to one my question as follows:
...
- 121
29
votes
3
answers
5k
views
finite generated group realized as fundamental group of manifolds
This is discussed in the standard textbooks on algebraic topology.
Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$
where $g_i$ are generators and $r_j$ are ...
CommunityBot
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6
votes
1
answer
779
views
primitive of an exact differential form with special properties
We were working on a smoothing problem and ran across the apparently simple following question:
X is a triangulated smooth manifold of dimension $n$, and $\alpha$ is an exact differential form of ...
CommunityBot
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12
votes
4
answers
2k
views
triviality of fibre bundles
are there some general method in judging if a fible bundle is trivial?
At least,for vector bundles,there is a well-developed theory,that is Charicteristic Classes.the triviality of vector bundles is ...
- 23.5k
5
votes
1
answer
283
views
how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
- 6,162
4
votes
1
answer
599
views
When are two natural transformations of infinity-categories equivalent?
Suppose
C and D are two ∞-categories (quasi-categories),
$F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the ...
- 66.8k
3
votes
2
answers
1k
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What can be said about the homotopy groups of a CW-complex in terms of its (co)homology?
One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H_1 is nontrivial. What other examples are there? (The CW-complexes I ...
- 1,355
23
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9
answers
4k
views
What methods exist to prove that a finitely presented group is finite?
Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...
CommunityBot
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20
votes
1
answer
2k
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Every Manifold Cobordant to a Simply Connected Manifold
I am wondering if it is true that every compact, connected, oriented manifold is cobordant to a simply connected manifold.
I believe that some sort of surgery will do the trick. Roughly speaking, I ...
- 27.5k
11
votes
2
answers
1k
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Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?
Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of quivers)...
- 3,051
1
vote
1
answer
190
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Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
- 25.2k
20
votes
3
answers
2k
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How does $\pi_1(SO(3))$ relate exactly to the waiters trick?
I hope this is serious enough. It is a well-known fact that $\pi_1(SO(3)) = \mathbb{Z}/(2)$, so $SO(3)$ admits precisely one non trivial covering, which is 2-sheeted.
Another well known fact is that ...
- 15.4k
41
votes
7
answers
5k
views
Simplicial objects
How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
- 3,917
25
votes
4
answers
3k
views
A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
CommunityBot
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6
votes
1
answer
463
views
The (n+1)-st cohomology of K(Z/p,n).
I was looking through my notes for a homotopy theory course and found the following mysterious statement (K is of course the Eilenberg-Maclane space):
$$H^{n+1}(K(\mathbb Z_p,n);\mathbb Z_p) \cong \...
- 4,404
4
votes
2
answers
551
views
Normality of an affine semigroup
An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
- 1
3
votes
4
answers
658
views
A specific branched cover of S^2 as a subgroup of Pi_1
This is a follow-up question to: Degree 2 branched map from the torus to the sphere
This is a silly computation, but for whatever reason this is taking me much, much longer than it should. So ...
CommunityBot
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11
votes
6
answers
1k
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Computing the structure of the group completion of an abelian monoid, how hard can it be?
Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. ...
CommunityBot
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6
votes
4
answers
873
views
Interaction of topology and the Picard group of Algebraic surfaces
It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are ...
- 57.6k
7
votes
2
answers
370
views
Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.
First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
- 4,842
10
votes
1
answer
635
views
Free action of SL_2(F_p) on a sphere
Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}_2({\mathbb F}_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}...
- 3,929
3
votes
0
answers
189
views
Which local homeos to numerical space are bijective?
I am reading T. Szamuely's book on Galois groups and fundamental groups.
As preparation to the algebraic case, he recalls the topological case.
So I am wondering if a surjective local homeomorphism $f$...
- 65.4k
7
votes
5
answers
979
views
Killing the torsion in homotopy
Origin
This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the ...
- 12.5k
6
votes
2
answers
657
views
Properties of the class of topological spaces possessing a CW-structure
Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).
Is it true that for a covering map $E\stackrel{f}{\to} ...
- 29.1k
12
votes
1
answer
651
views
Does a triangulation without fixed simplex property always exist?
Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\...
- 28.9k
3
votes
1
answer
225
views
Explicit classifying spaces for crossed complexes
I'm trying to understand the topology behind a certain group which fits into a truncated crossed complex, so I've been trying to understand Brown's construction of the classifying space of a crossed ...
CommunityBot
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5
votes
3
answers
1k
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Computation of Joins of Simplicial Sets
It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and ...
- 9,627
20
votes
1
answer
8k
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Universal Covering Space of Wedge Products
Today I was studying for a qualifying exam, and I came up with the following question;
Is there a simple description in terms of the subspaces universal covers for the universal cover of a wedge ...
- 56.6k
4
votes
1
answer
1k
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references for models of stable infinity categories
There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...
- 27.5k
4
votes
2
answers
592
views
Five lemma in HoTop* and arbitrary pointed model categories
Let $\textbf{HoTop}^*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $\textbf{HoTop}^*$, i.e. pointed homotopy ...
CommunityBot
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11
votes
2
answers
2k
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Meaning of orientation/orientability over rings other than the integers
This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately.
We consider the homology definition of an orientation for a manifold, as ...
CommunityBot
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9
votes
2
answers
743
views
Cobordisms of bundles?
Is there a notion of a cobordism which is compatible with bundle structure?
That is, if I have bundles $E$ and $F$, is it the case that the manifold $W$ with $E$ and $F$ as boundary components, can ...
- 1,618
3
votes
1
answer
166
views
Upper bound on the genus of a k-page graph
Is there an upper bound on the genus of a graph that has a book embedding on say k pages, or can the genus be arbitrarily large? If not a general bound is known, what happens for k=3?
- 18.6k
7
votes
1
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457
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Reference for equivalent definitions of the genus
Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either ...
CommunityBot
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6
votes
0
answers
532
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Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?
So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...
- 44.7k
4
votes
3
answers
2k
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Homotopy groups of smooth manifolds?
For a fixed $d$, is there a relationship between the homotopy groups of smooth $d$-manifolds?
The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that the ...
- 44.4k
15
votes
3
answers
3k
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H-space structure on infinite projective spaces
Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.
Does ...
- 998
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2
answers
1k
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Are non-empty finite sets a Grothendieck test category?
A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
- 13.6k
11
votes
1
answer
1k
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Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?
There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (...
CommunityBot
- 1
23
votes
3
answers
6k
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Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 15.6k