All Questions
9,056 questions
0
votes
2
answers
295
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Does the Deligne-Mumford space module $S_{n}$ action have a fundamental chain?
Does the Deligne-Mumford space (without ordering for marked points) $\bar M_{g,n}/S_{n}$ has fundamental chain in signular simplicial chains? (because I read Costello's paper GW potential to TCFT, as ...
- 23.5k
8
votes
0
answers
340
views
Do p-compact groups have a sufficiently good notion of "flag variety" and "intersection cohomology"?
This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think.
Background
An outstanding problem ...
CommunityBot
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19
votes
2
answers
1k
views
Geometric model for classifying spaces of alternating groups
The classifying space of the nth symmetric group $S_n$ is well-known to be modeled by the space of subsets of $R^\infty$ of cardinality $n$. Various subgroups of $S_n$ have related models. For ...
- 12.6k
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
CommunityBot
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4
votes
1
answer
563
views
Why is the induced map between pullbacks (of inclusions) by a right fibration a deformation retract?
Let $X$ be a simplicial set. Let $X\to \Delta^n$ be a right fibration (has the right lifting property with respect to right horn inclusions), and let $$\Delta^{\{n-i\}}\hookrightarrow \Delta^{\{n-i,\...
- 19.6k
4
votes
2
answers
443
views
map of manifolds inducing iso on top cohomology, but not surjective on one other cohomology group
Given a map between two manifolds that induces an isomorphism on integral cohomology in the top dimension, it follows from naturality of the cup product and Poincaré duality and universal coefficient ...
- 6,162
7
votes
3
answers
444
views
An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences?
Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the plain sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are ...
- 8,717
19
votes
3
answers
5k
views
What determines a model structure?
It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal).
cofibrations and weak ...
- 2,782
7
votes
2
answers
573
views
L-S category versus Betti numbers
Is there a sequence of topological spaces $X_n$ (manifolds ideally), where the sum of the Betti numbers of $X_n$ remains bounded but the Lusternik–Schnirelmann category is unbounded, as $n \to \infty$?...
- 12.5k
0
votes
1
answer
485
views
What is a right-handed Dehn twist of a cut curve of a Riemann surface?
Let $\Sigma_g$ be a Riemann surface of genus g, and $C$ is a cut curve of $\Sigma_g$, i.e. an oriented simple close curve.
What is a right-handed Dehn twist of $C$ of $\Sigma_g$?
- 12.3k
5
votes
1
answer
765
views
Bousfield-Kan spectral sequence with local coefficients
Let $F : \mathcal{D} \to \mathbf{Top}$ be a diagram of topological spaces. A local system of coefficients $M$ on $\mathrm{colim}_\mathcal{D} F$ pulls back to a local system $M_d$ on $F(d)$ for each $d ...
- 55.9k
1
vote
0
answers
3k
views
Is this a covering space? [closed]
In Hatcher's book Page 79 I was asked to provide two-sheeted covering space $Y \rightarrow X_{1}$ such that the composition
$Y \rightarrow X_{1} \rightarrow X$ of the two covering spaces is not a ...
- 12.6k
-4
votes
2
answers
785
views
Spectral sequence [closed]
what is Koszul resolution? what is its role played in the computation of spectral sequence?
- 12.6k
14
votes
7
answers
6k
views
The Symmetry of a Soccer Ball
Let $P$ be a polyhedron which satisfies the following three conditions:
$P$ is built out of regular hexagons and regular pentagons.
Three faces meet at each vertex.
$P$ is topologically a sphere.
An ...
- 5,293
19
votes
0
answers
504
views
Other examples of computations using transfer of structure from the chains to the homology?
There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
- 3,880
1
vote
2
answers
989
views
Is the complement of a strong deformation retract of a manifold M homotopic equivalent with the boundary of M?
It seems an easy problem but I couldn't prove it.
Let $M$ be a manifold with boundary and $N\subset \mathrm{int}(M)$ is a strong deformation retract of $M$.
Then I wonder whether $M-N$ is homotopic ...
- 29.1k
9
votes
3
answers
1k
views
Integration in equivariant K-theory
Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the ...
CommunityBot
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4
votes
1
answer
3k
views
Notation for algebras
Is there standard notation for
(1) exterior algebras
(2) free graded commutative algebras
(3) divided polynomial algebras ?
I've seen (and used) $\Lambda$, $\Gamma$, $\Delta$ etc. used for ...
- 20.8k
8
votes
2
answers
2k
views
Splitting of the Universal Coefficients sequence
The really beautiful way to prove the Universal Coefficients theorem, to my taste,
is to use the fibration sequence $K(\mathbb{Z}, n) \to K(\mathbb{Z}, n) \to
K(\mathbb{Z}/k, n)$ (I'm using $\mathbb{...
- 3,526
2
votes
1
answer
831
views
Betti Cohomology of singular Kummer Surface
Let $A$ be a complex torus of (complex) dimension 2 and $X$ the associated Kummer variety $A/\sigma$, where $\sigma(x)=-x$. I would like to compute the cohomology of $X$ with $\mathbb{Z}$ coefficients....
- 22.6k
2
votes
1
answer
2k
views
Is an algebraic geometer's fibration also an algebraic topologist's fibration?
When some papers say"XXX fibration", I see it seems that it is just that the surjective map f: X ---> Y, such that the fiber is XXX,but it is really not "fibration",I didn't see it prove that it is a ...
- 13.2k
15
votes
1
answer
1k
views
Complex orientations on homotopy
I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It ...
- 3,726
6
votes
2
answers
972
views
Pointed vs. unpointed homotopy colimits
Let $C$ be a category with a zero object, i.e. an object 0 which is both initial and terminal. Then $C$ is automatically (and uniquely) enriched over the category $Set_\star$ of pointed sets with ...
- 10.9k
13
votes
3
answers
2k
views
What are normalized singular chains good for?
One of the common definitions of homology using the singular chains, i.e. maps from the simplex into your space. The free abelian group on these can be made into a chain complex and one can take the ...
- 8,717
17
votes
1
answer
3k
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Do these conditions on a semigroup define a group?
As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions ...
CommunityBot
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22
votes
1
answer
805
views
Twistings for other cohomology theories
Twistings in cohomology theories have a long history and have been used to great effect. The classical example is cohomology with local coefficients. Using this one can formulate Poincaré duality and ...
- 27.5k
5
votes
0
answers
350
views
Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,882
14
votes
5
answers
2k
views
Good reference for homology of $K(\mathbb{Z}, 2n)$?
The homology algebra $H_*( K(\mathbb{Z},2n); \mathbb{Z})$ contains a
divided polynomial algebra on a generator $x$ of dimension $2n$.
I suppose I could read through the Cartan seminar for a proof, ...
- 52.6k
5
votes
1
answer
632
views
Showing an Ext^2 element is zero
If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1_X(G,E)$, we need to show that this sequence splits. To produce a splitting ...
- 39.3k
3
votes
1
answer
2k
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Proof of the triangulation of a surface [duplicate]
Possible Duplicate:
Triangulating surfaces
Recently I the proof of an important fact (first proved by Rado), that there exist a triangulation of a compact surface, on the book Riemann Surfaces (...
CommunityBot
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1
vote
1
answer
3k
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The Simply Connected Subgroups of GLn(C)?
A friend of mine and I were trying to answer a question related to his research and he couldn't remember whether or not the special linear group over the complex numbers, SLn(C),was simply connected. (...
- 14.5k
12
votes
0
answers
440
views
K-Weil cohomology theories?
I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard ...
- 21k
8
votes
2
answers
4k
views
tangent sphere bundle over sphere
are there some general description about tangent sphere bundle over sphere?
(it is a special $S^{n-1}$bundle over $S^n$)
say for n=1,it is trivial,$S^0\times S^1$,for n=2,it is $SO(3)\cong \mathbb{R}...
- 44.4k
3
votes
1
answer
458
views
Slick verification of the model category axioms for Spaces and SSets with the q-model structure?
We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$.
Questions:
1.) Is there any sort of slick argument to verify that CGWH with the ...
- 721
3
votes
2
answers
604
views
Oriented Cobordism Rings
Hey everybody!
I was wondering if anybody had available the calculation of the Oriented cobordism groups in dimensions higher than 10? Or if anybody knew if there is another kind of torsion beside 2-...
- 1,618
17
votes
2
answers
2k
views
Is a conceptual explanation possible for why the space of 1-forms on a manifold captures all its geometry?
Let $M$-be a differentiable manifold. Then, suppose to capture the underlying geometry we apply the singular homology theory. In the singular co-chain, there is geometry in every dimension. We look at ...
- 3,230
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
8
votes
2
answers
950
views
Can we make rigorous the 'obvious' characterisation of singular homology?
It is a well known and often touted fact that the singular homology groups 'count the k- dimensional holes' in a space (see: How does singular homology H_n capture the number of n-dimensional "...
CommunityBot
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22
votes
7
answers
3k
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Essential theorems in group (co)homology
I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along the lines of
Hopf's ...
- 8,467
5
votes
1
answer
2k
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How to compute the (co)homology of a compact Riemann surface?
The situation is the following.
A finite-index subgroup $\Gamma$ of $SL_2(\mathbb Z)$ acts on the upper-half plane $\mathcal H$. It has a fundamental domain, obtained by a union of translates of the ...
- 3,699
4
votes
2
answers
705
views
(∞,1) vs Category weakly enriched over spaces
What is the difference between:
($\infty,1$) categories - in which have for two objects you have an ($\infty,0$) category of morphisms (i.e. a space of morphisms)
and
categories weakly enriched ...
- 15.5k
7
votes
1
answer
2k
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Functoriality of Poincaré duality and long exact sequences
Hi all,
Today I was playing with the cohomology of manifolds and I noticed something intriguing. Although I might just have been caught out by a couple of enticing coincidences, it feels enough like ...
- 56.6k
6
votes
1
answer
1k
views
Is geometric realization of the total singular complex of a space homotopy equivalent to the space?
Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$.
Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular ...
- 20k
4
votes
3
answers
3k
views
Homotopy Equivalence of Punctured Tori
I was told recently that if I take a 2-torus (genus 2) and remove 1 point, then this is homotopy equivalent to a torus with 3 points removed. This may be really easy but I don't see it.
Thank you!
- 47.3k
25
votes
3
answers
2k
views
Are fundamental groups of aspherical manifolds Hopfian?
A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth ...
- 68.9k
4
votes
1
answer
3k
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Quick ways to calculate cohomology of vector bundle/local system from transition functions?
Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The ...
- 8,717
13
votes
3
answers
3k
views
Are there applications of algebraic geometry into algebraic topology?
It is described in many sources that algebraic topology had been a major source of innovation for algebraic geometry. It is said that the uses of cohomology, sheaves, spectral sequences etc. in ...
- 19.8k
4
votes
0
answers
317
views
(Equivariant) Sheaves of Equivariant Spectra?
This is a very naive question but 1)given a compact Lie group G, is there a good notion of a sheaf of equivariant spectra on a G-space X analogous to the model structure that Brown develops in his ...
- 3,758
10
votes
2
answers
1k
views
Čech cohomology of compact spaces via closed covers?
Let X be a compact space.
Recall that its Čech cohomology $H^\bullet(X,\mathbb Z)$
is given by the colomit $\mathrm{colim}_U\big(H^*(C^\bullet(U;\mathbb Z),\delta)\big)$, where $U=(U_i)$ runs over ...
CommunityBot
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0
votes
2
answers
356
views
Can all induced maps be described categorically.?. (or at least as generally as possible)
Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol.
I am pretty confused about induced maps in different areas of algebraic
topology; I do know how these induced maps are ...
- 1,030