All Questions
8,725 questions
10
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458
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is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
4
votes
0
answers
362
views
Weil Kostant Integrality Result as Stated by Brylisnki
I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued 2-...
- 1,611
6
votes
1
answer
516
views
Poincare conjecture and the graph of triangulations
This was an update to this question, but I decided to make it a separate question. The definition of the graph of triangulations can be found in the previous question.
Question. I was told a few ...
user6976
10
votes
0
answers
281
views
Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$
Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...
- 28.9k
3
votes
0
answers
127
views
Additive basis for the cohomology of real flag manifolds
By Borel's description the mod 2 cohomology algebra of the flag manifold is the polynomial algebra on the Stiefel-Whitney classes of canonical vector bundles modulo ideal generated by the dual classes....
- 101
4
votes
0
answers
219
views
In the cohomology of Thom spectrum over LoopS^{2} and p-adic characteristic classes
Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map
$1+3: \Omega S^{2} \to BG_{3}$
where $1 +3$ is a unit in $3$-adics.
Here $G_{3}$ is the unit component of $\Omega^{\infty}S_{...
- 2,023
4
votes
0
answers
121
views
Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin
Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth ...
- 4,920
4
votes
1
answer
439
views
Contractible space of maps between Eilenberg-Mac Lane spaces
Suppose $A$ is an abelian torsion group, with no elements of order $p$, and let
$P$ be an abelian $p$-group (i.e., the order of each element is a power of $p$).
It sure seems to me that
$$
\mathrm{...
- 12.5k
1
vote
0
answers
289
views
The homotopy colimit of a tower of triangles
Set the framework to be a triangulated category with all set indexed coproducts.
In "Relative Homological Algebra and Purity in Triangulated Categories", J. of Algebra 227, (2000), pp. 268- 361, (...
- 666
1
vote
0
answers
380
views
Topological definition of intersection multiplicities of algebraic varieties
I posted this question in Stack Exchange and was recommended the appendix of Fulton's Young Tableaux. While I think it's good, it'd be nice to have some books which explain this subject in more detail....
- 1,169
3
votes
0
answers
199
views
Extending simplicial complex to a manifold
Let $Y$ be a simplicial complex contained in a simplex $S\subseteq \mathbb{R}^n$. Can I find a simplicial complex $X\supseteq Y$ s.t. $|X|=|S|$? Moreover, can I do it without introducing new vertices, ...
- 972
2
votes
1
answer
510
views
Are the C(S^n, S^n)'s homeomorphic ?
Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...
- 4,060
0
votes
1
answer
351
views
Understanding a proof in Adams' Stable Homotopy and Gen. Coh
[Question cross posted on stack-exchange]
I'm slowly working through Part III of the book, and I'm scratching my head a bit while reading the proof of Lemma 3.2 (here reproduced):
Let $X, A$ be a ...
- 398
2
votes
1
answer
295
views
Sg: How to Show this Sequence is Exact?
Hi,All:
I am seeing a result in which the following sequence, in the context of the genus-g surface Sg, is described as being exact:
1-->Tg-->$M^{(2)}g$-->$Sp^{(2)}(2g,\mathbb Z)$-->1
Where :
i)Tg ...
- 105
4
votes
1
answer
805
views
Homology dimension of the mapping class group of a surface with boundary
There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except $(...
- 1,499
2
votes
2
answers
369
views
Classifying space of a crossed complex
Brown defines the classifying space of a crossed complex in the following way.
Given a filtration X* of a space X, define the fundamental crossed complex by:
C_0 = X_0, C_1=\pi(X_1,X_0) (the ...
- 1,422
1
vote
0
answers
137
views
free action on mod p cohomology sphere
It is well known that the group $G=\mathbb Z_2\oplus\mathbb Z_2$ cannot act freely on mod 2 cohomology n-sphere.
Is it also true that this group $G$ cannot act freely on any mod p cohomology n-...
- 119
9
votes
0
answers
462
views
Two constructions for BU×Z
Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$:
1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...
- 7,637
2
votes
1
answer
291
views
Hopf Algebras/Rings, A Question of Terminology
I'm reading $\textit{The Hopf Ring for Complex Cobordism}$ by Ravenel and Wilson where they discuss the notion of group and ring objects over a category. They say that a hopf algebra over a ring in ...
- 822
4
votes
0
answers
166
views
Homotopy-theoretic measure of operations on sheaves failing to be sheaves
Here's something I've been wondering about for a few weeks:
Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ ...
user1437
1
vote
0
answers
190
views
Homotopy colimits over a certain subset category.
Hi!
Let $I$ denote all the finite subsets of some set (infinite or finite) S. For each n, let $I_n$ be the subset consisting of objects of cardinality n, so that there are no morphisms between the ...
- 1,071
1
vote
0
answers
430
views
Professional skills advising for math jobs [closed]
Hi,
I am a postdoc at the University of Nottingham (UK) and I am beginning to apply for Assistant Professor positions in US.
I would like to receive a feedback on the material that I am sending (...
2
votes
2
answers
601
views
Must a Strong deformation retractible 3-manifold be homeomorphic to $\mathbb{R}^3$?
Assume $M$ is an open 3-manifold which can be deformation retracted to a point. Is it necessarily homeomorphic to $\mathbb R^3$?
(I know Whitehead had an example which is contractible and not ...
- 1,101
1
vote
1
answer
206
views
A p-form taking discrete values on p-chains must be 0.
I want to show that if $w$ is a $p$-form such that its induced cochain on $p$-chains:
$w(\gamma)= \int_{\gamma} w \in S$
takes values in a discrete set $S \subset \mathbb{R}$ then $w$ must be zero.
...
- 1,611
3
votes
1
answer
361
views
Is the coproduct of fibrant spectra fibrant again?
Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...
- 51
3
votes
2
answers
344
views
Principle when limits level by level don't commute with simplicial structure
Are there general principles when a simplicial object is a (co)limit of other simplicial objects level by level, but is not a (co)limit when considering the entire simplicial structure?
Objects can ...
user2529
7
votes
0
answers
433
views
Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem
I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...
- 25.6k
5
votes
0
answers
167
views
In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal?
Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). ...
- 54.6k
0
votes
1
answer
386
views
Reference request for equivariant cohomology of G [duplicate]
Possible Duplicate:
What is the equivariant cohomology of a group acting on itself by conjugation?
Let $G$ be a compact Lie group. Where can one read about the equivariant cohomology $H_G^*(G)$, ...
- 559
3
votes
1
answer
335
views
Reference for monkeying with the topology of a mapping cylinder
In "Construction of Universal Bundles, II", Milnor has to replace the standard topology on the join with what he calls the "strong topology" which is the smallest topology such that certain maps are ...
- 12.5k
0
votes
0
answers
148
views
Extending a 2-frame field - manifolds with boundary
If $M$ is an orientable compact 3-manifold with boundary such that there is defined $s\in\Gamma(\partial M,V_2(TM))$ a section of the 2-frame bundle of $TM$, then $s$ extends to $\tilde s\in\Gamma(M,...
- 261
0
votes
1
answer
427
views
what does the coefficients ring of generalized cohomology defined by the unitary Thom spectrum like?
Let $MU$ be the unitary Thom spectrum, then it gives a generalized cohomology,
so what is the coefficients $MU^*(point)$ like?
Is it just the complex cobordism ring $\Omega_U^*?$
- 139
9
votes
0
answers
1k
views
Weight filtration over the integers
This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...
- 23.5k
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 ...
- 1,101
2
votes
1
answer
689
views
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 ...
- 22.8k
2
votes
1
answer
116
views
Complementary Sets in $\mathbb{C}P^2$
Let $U,V\subset \mathbb{C}P^2$ be complementary sets (ie $U\simeq V^c$) with $M=U\cap V$ a $3$-manifold. Assume further $M$ is orientable.
My question is, must it be the case that one of $i_*:H_2(U;\...
- 23
6
votes
0
answers
616
views
Fiber sequences in proper model categories
I am confused about the notion of a fiber sequence (or dually a cofiber sequence) in a general pointed and proper model category $\mathcal{C}$.
Following Hovey, we can define, like in topology, a map ...
- 61
2
votes
0
answers
153
views
Can class in $H^4(BT)$ be realized as the second Chern class of a principal SU(2) bundle?
The question in the title, to which I add some clarification. Can every class in $H^4(B\mathbb{T})$ be realized as the second Chern class of a principal $SU(2)$ bundle?
$B\mathbb{T}$ is the ...
- 1,331
3
votes
1
answer
200
views
Three dimensional spherical space form
How can we deduce the three dimensional spherical space form conjecture from the Poincare conjecture? More precisely, how can we deduce using the Poincare conjecture that every free action of a finite ...
- 127
7
votes
0
answers
199
views
Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
- 1,040
2
votes
0
answers
80
views
Codimension $k$ homeomorphism extensions
Let $f:D \to D$ homeomorphism of $k$ codimension manifold (closed, compact, without boundary) to itself. ($D \subset \mathbb{R}^{n+k}$). For which $f$ does homeorophism $g: \mathbb{R}^{n+k} \to \...
- 1,307
7
votes
2
answers
559
views
Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?
Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
- 8,167
5
votes
1
answer
230
views
Nonuniqueness of maps of representing spaces
In Rudyak's On Thom Spectra, Orientability, and Cobordism, two variants of Brown's representability theorem are presented: given a natural transformation $f^*: E^* \to F^*$ of cohomology theories, ...
- 1,049
0
votes
1
answer
163
views
Pencil P of quartics surface
I'm reading an article of mumford and I want to know what it means or where I can find: the pencil has no fixed components, and the pencil is fixed components.
Thanks
- 1
-1
votes
1
answer
421
views
How does the discrete group act on simplicial set level by level
Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set?
For example, if ...
- 681
0
votes
1
answer
252
views
the homotopy type of a product of some spaces
let S be the n-sphere.
how can we see that
(SxS) smash S
has the homotopy type of a wedge of spheres?
- 11
9
votes
0
answers
236
views
H-spaces without rational homology
Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space,
and whose rational homology groups vanish in positive degrees?
My space $M$ is in fact homotopy equivalent ...
0
votes
0
answers
157
views
Postnikov system for a tree
The Postnikov system of a path-connected space X is a tower of spaces $...\longrightarrow X_{n+1}\longrightarrow X_{n}\longrightarrow ...\longrightarrow X_{1}\longrightarrow X_{0}$ with the following ...
- 933
1
vote
2
answers
530
views
Why is this a local constant sheaf
If a group $G$ acts on a topological space $M$, and a representation of $G$ on a vector space $V$, why $M \times_G V$ is a local constant sheaf over $M/G$?
- 1,499
3
votes
0
answers
171
views
minimal model of $A_\infty$ structure
Hi all,
I am reading about minimal model of $A_\infty$ structure. So far, I find two different ways of the construction, given by Kadeishvili and Kontsevich, Soibelman.
1) The construction of these ...
- 583