All Questions
9,056 questions
5
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If $k[S]$ is noetherian, is S finitely generated?
Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is ...
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3
votes
3
answers
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Another group cohomology cup product question
I am wondering if there is a way to see the cup product, in some cases, without using cochain complexes. The situation I am interested in is the following:
Let $G=F/R$ be a finitely presented group ...
- 16.2k
12
votes
4
answers
832
views
$S^n \to S^m \to B$ bundle: possible?
Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with.
(1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the ...
CommunityBot
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5
votes
1
answer
492
views
Pair consisting of a compact manifold and Morse function
Consider the following situation:
Let $M$ and $M'$ be two closed manifolds and suppose $f:M\to \mathbb{R}$ and $f':M'\to \mathbb{R}$ are smooth morse functions on $M$ and $M'$ respectively. We say ...
- 68.9k
36
votes
2
answers
4k
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Timeline of cohomology (1935 to 1938)
There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific ...
- 13.2k
17
votes
1
answer
832
views
Loop spaces and infinite braids
The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$. The infinite symmetric group has interesting interactions with homotopy ...
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6
votes
1
answer
662
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Finiteness of higher homotopy groups of finite complexes
Throughout, let $X$ be a connected finite CW-complex.
Question: If $X$ is of dimension $n$. Is there some integer $n'$ (maybe depending only on $n$), such that all homotopy groups $\pi_k(X)$ for $...
- 25.5k
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 ...
- 44.7k
5
votes
3
answers
2k
views
How to show that a space has the homotopy type of wedge of spheres ?
Let me try and put the question in context. I am studying certain subsets of the tangent bundle of a sphere. I also have a regular CW complex which is a deformation retract of such a subset. Hence I ...
- 8,803
12
votes
1
answer
2k
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Monodromy with algebraic fundamental groups
Topological setting
Say we have a fiber bundle: $p: X \rightarrow B$. Let $s: B \rightarrow X$ be a section. Then $\pi_1(B,b)$ acts on $\pi_1(F_b,s(b))$ (where $F_b:=p^{-1}(b)$) by: Let $\gamma \in \...
- 56.6k
3
votes
2
answers
1k
views
Do Smash Products and Quotients Commute?
Let $X$ be a subcomplex of a CW-complex $Y$. Is $(Y/X)^{\wedge k}$ homotopy equivalent to $Y^{\wedge k}/X^{\wedge k}$, where $\wedge k$ is the $k$-fold smash product? I know it is not true for ...
- 55.9k
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 ...
CommunityBot
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votes
2
answers
2k
views
What is the Poincare dual of a symplectic form?
Every symplectic form on a manifold $M^n$ determines a De Rham cohomology class in $H^2(M)$ (often a nontrivial class), and this in turn determines a class in $H_{n-2}(M)$. What in general can be ...
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1
answer
796
views
Lie's third theorem via differential graded algebras?
Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, real)...
- 19.8k
2
votes
2
answers
438
views
Reference for the proof of this statement?
Can anyone give me the reference for this statement?:
Let $M$ be a closed oriented smooth 4-manifold. Any element $a\in H_2(M)$ can be represented by a smoothly embedded, oriented surface.
I found ...
- 10.5k
8
votes
2
answers
596
views
Infinite loop space maps into or out of BAut(F_n)
There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become ...
- 9,263
12
votes
1
answer
460
views
How to localize a model category with respect to a class of maps created by a left Quillen functor
Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean combinatorial model category. Consider a Quillen pair
$$ L: M\rightleftarrows N: R.$$
I want the following result:
There ...
- 13.6k
3
votes
2
answers
939
views
Which Groups are Infinite Loop Spaces?
At first, if a group G is an infinite loop space (all are based), then \pi_0(G) must be Abelian. Therefore, if G is discret, then it must be Abelian. In fact, any ...
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1
vote
2
answers
378
views
Is this a pre-ordered commutative semigroup?
Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
- 21
2
votes
1
answer
378
views
How can I show that the map L-->K(\pi_n(L),n) representing the fundamental class of an (n-1)-connected space is an isomorphism on \pi_n?
As an exercise, I'm trying to show that for an $(n-1)$-connected space $L$ with $\pi=\pi_n(L)$, the map $\iota_L:L\rightarrow K(\pi,n)$ associated to the fundamental class $\iota_L\in H^n(L;\pi)$ ...
- 6,069
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1
answer
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When is a TQFT the dimensional reduction of a higher dimensional TQFT?
In Lurie's framework for TQFT's, a TQFT is a symmetric mondoial functor from $Cob_n(n)$ to some symmetric monoidal $n$-category $\mathcal{C}$. One can construct an $(n-1)$-dimensional TQFT from an $n$-...
- 56.6k
8
votes
1
answer
437
views
A sphere bundle map
I think this may all be classical bundle-theory. But I'm trying to read some old papers on classifications of bundles and the following came up as questions I couldn't immediately answer:
Consider the ...
CommunityBot
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5
votes
1
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240
views
Self-linkage of the orthogonal group $O_n({\mathbb R})$.
In Exercise 153 of my list, it is proved that the connected components $SO_2({\mathbb R})$ and $O_2^-({\mathbb R})$ of the orthogonal group are linked as curves in the three-dimensional sphere defined ...
- 52.3k
7
votes
2
answers
537
views
Residually finite + torsion free + finite index = finite complex?
Suppose $G$ is a residually-finite group and $H < G$ a torsion-free subgroup of finite index.
What characterizes such $G$ such that $BH$ is homotopic to a finite complex?
I believe Serre showed ...
- 25k
3
votes
1
answer
361
views
Posets of finite sequences are highly connected
I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?
Fix $1 \leq k \leq n$. Define $X_{n,k}$ ...
- 3,571
3
votes
1
answer
806
views
a CW-complex homotopic to a manifold
I'm reading a paper and here the authors say that a connected 4-manifold with zero rational top homology has a homotopy type of 3-dimensional CW-structure. I can't figure out how it can be done.
CommunityBot
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18
votes
5
answers
6k
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Does a finite-dimensional Lie algebra always exponentiate into a universal covering group
Hi,
I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.
Consider a finite-dimensional Lie algebra, A, spanned by its d generators, X_1,.....
CommunityBot
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30
votes
1
answer
2k
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Identifying the stacks in Devinatz-Hopkins-Smith
I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what ...
- 52.6k
4
votes
1
answer
1k
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Homology is computable because it is stable under suspension
I've heard it said that the reason why the homology groups of a space are a computable invariant is because they are a stable invariant in the sense that they are stable under suspension.
I'm ...
- 24.8k
16
votes
2
answers
4k
views
How should I think about delooping?
When talking about the Eilenberg-Maclane space $K(G,n)$, we usually restrict our attention to the situation where $G$ is abelian. In that case, we get $\Omega K(G,n)=K(G,n-1)$, so we can call $K(G,n)$...
- 21k
1
vote
1
answer
444
views
"non natural" iso between homotopy and homology
Can we classify all finite CW complexes $X$ such that for each $i$ there is some isomorphism $\pi_i(X) \rightarrow H_i(X)$? Note that it is not hard to classify all complexes for which each ...
- 6,162
1
vote
1
answer
929
views
Complement of lines and wedges of spheres
Let $L=L_1 \cup ... \cup L_n$ be the union of $n$ distinct lines through the origin in $\mathbb{R}^{3}$. I'd like a convincing argument that $\mathbb{R}^{3} \setminus L$ is homotopy equivalent to a ...
- 13.6k
0
votes
1
answer
219
views
Cofibrations of differential graded commutative algebras
Let $X$ a smooth manifold. Is the pullback morphism
$\Omega^\bullet(X)\to\Omega^\bullet(X\times \mathbb{R}^n)$ an acyclic cofibration of differential graded commutative algebras? I guess so, and even ...
- 22.6k
8
votes
2
answers
5k
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The 0th homology of a path-connected space
In singular homology one of the first calculations you can make is $H_0(X)=H_0(pt)$ for path-connected $X$. This seems to be a property which does not follow from the axioms for a generalized homology ...
- 19.6k
2
votes
1
answer
360
views
James Construction for Disconnected Spaces
When I work out the James construction for a discrete pointed space, it appears that the
induced map $\pi_0 (J(X)) \to \pi_0( \Omega\Sigma X)$ is the inclusion of the free monoid on $\pi_0(X)$ into ...
- 55.9k
5
votes
1
answer
550
views
Local homology of degenerate critical points
Given a smooth function $f:M\rightarrow \mathbb R$ on a manifold, its local homology at a critical point $x$ is the group
$$ C_\star(x) := H_\star ( M_{ < c} \cup \{ x \} , M_{ < c} ) ,$$
where ...
- 55.9k
2
votes
2
answers
300
views
what conditions can one place on a finitely generated periodic semigroup that will ensure the semigroup is finite?
I am not familiar with much semigroup theory, but this question came up in my research and I've been unable to find much on it.
- 85.6k
2
votes
1
answer
424
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Cohomology groups of an intersection
Suppose that $A$ is an affine algebraic variety and $P$, and $Q$ are subvarieties. It is easy
to see that the coordinate rings $C[P]$ and $C[Q]$ of $P$ and $Q$ are modules over $C[A]$ and
$C[P]\...
- 22.6k
14
votes
1
answer
700
views
Is the model category of Complete Segal Spaces right proper?
Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories".
Has ...
- 13.6k
8
votes
2
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875
views
Is the mapping cylinder of a Serre fibration also a Serre fibration?
If we have a Serre fibration $p: E \rightarrow B$ with fiber of homotopy type $S^{k-1}$, then we can create a fibration with contractible fiber by first taking the mapping cylinder $M_p$ of $p$ to get ...
- 9,263
4
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1
answer
413
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What is the relative weight filtration of the mapping class group of a surface?
It is my understanding that Dennis Johnson defined a `relative weight filtration,' of the mapping class group of an oriented surface. My question is what is this filtration, and how does it relate to ...
- 44.8k
1
vote
1
answer
358
views
What does the weights of Lie group mean?
Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system
of the exceptional Lie group $G_2$
with $\alpha_1$ is short and $\alpha_2$ is long,
so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\...
- 43.2k
5
votes
5
answers
2k
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Loopspace of an Eilenberg Maclane space K(G,n)
I've seen the fact that the loopspace $\Omega K(G,n)$ is homotopy equivalent to $K(G,n-1)$ mentioned in some places, but I have no idea why. Can anyone offer a good explanation? Also, what happens ...
- 20k
3
votes
1
answer
727
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Automorphisms of the rooted tree operad
This follows Ryan Budney's comment to the question asked here.
What is the automorphism group of the rooted tree operad?
(By the rooted tree operad, I just mean the operad with object rooted trees ...
CommunityBot
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0
votes
1
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225
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Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?
(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \...
CommunityBot
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4
votes
3
answers
221
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Homology of bundles over a triangulated base and $A_\infty$-algebras
Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be ...
- 18.2k
6
votes
1
answer
786
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Compatibility of braids as a simplicial set and as a braided monoidal category
We can form a braided monoidal category by taking the groupoid coproduct of the Artin braid groups $B_n$. We can also make the braids into a simiplical set where the ith face operation is removing the ...
- 7,527
38
votes
3
answers
6k
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What is so "spectral" about spectral sequences?
From recent mathematical conversations, I have heard that when Leray first defined spectral sequences, he never published an official explanation of his terminology, namely what is "spectral" about a ...
- 82.7k
8
votes
1
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1k
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Beyond an intro to topological graph theory...
I'm looking to find out what active areas of research there are in topological graph theory, particularly those that interface strongly with other areas of math (say, group theory, algebraic topology, ...
- 1,180
6
votes
0
answers
510
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The Mapping Cylinder of a Pullback Square
Suppose I have a pullback square, which I think of as a map from the fibration
$q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$
from the mapping cylinder $M$ of $X\...
- 12.5k