All Questions
9,056 questions
4
votes
1
answer
328
views
Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 21.3k
9
votes
3
answers
4k
views
homology and cohomology of a quotient manifold
suppose $M$ is a manifold , $G$ is a lie group (may be finite) ,then let $G$ act on $M$ freely , $N=M/G$ is then a manifold ,so my question is what relations may be between the homology and ...
- 35.9k
8
votes
2
answers
844
views
Reference request for relative bordism coinciding with homology in low dimensions
It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $[H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ for $n<5$...
- 22.1k
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 ...
- 4,519
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 $M/G$?
- 57.6k
10
votes
1
answer
544
views
Ranicki symmetric L-groups of finite fields?
Can anyone tell me what the Ranicki symmetric L-groups $L^*(F)$ are when $F$ is a finite field? (and maybe provide a reference?) Thanks!
- 3,901
9
votes
2
answers
798
views
Does the category of topological symmetric spectra satisfy the monoid axiom ?
In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering ...
- 30.4k
4
votes
1
answer
250
views
Compatibility of classifying space with inner-hom?
Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category
$\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...
- 27.2k
4
votes
0
answers
495
views
Spectral sequence for cohomology of open subset
Let $X$ be a smooth compact orientable manifold (or variety) and let $j: U \subset X$ be the complement to a union $\bigcup_{i \in I} X_i$ of smooth compact orientable manifolds. Suppose that $A$ is a ...
4
votes
1
answer
497
views
Riemann Existence Theorem for Real Curve
By real curve, we mean a Riemann surface $X$ together with an anti-holomorphic involution
$\sigma : X\rightarrow X$. Let $S$ be a finite subset of $X$. For each point $x\in S$, we associate a positive ...
- 66.3k
15
votes
2
answers
973
views
Pointed Hurewicz model structure
In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves
that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and ...
- 37.8k
14
votes
3
answers
1k
views
Is intersection homology the usual homology of something else?
Let $X$ be the sort of topological space for which it makes sense to talk about the intersection homology. Fix a perversity $p$, or just take $p= 1/2$ if you like.
Is there some naturally ...
- 1,335
10
votes
1
answer
745
views
The algebro-geometric counterpart of the Dijkgraaf-Witten model
Can the Dikgraaf-Witten model for a finite gauge group $G$ [Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393] be described in ...
- 5,984
2
votes
1
answer
445
views
Homology and submanifolds...
I'm reading a paper which includes the following line, and I can't find a reference anywhere to the result the authors mention:
"Let M be a compact orientable embedded minimal hypersurface of a ...
- 20.9k
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 ...
- 20.8k
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 $\Omega_U^*?$
- 822
1
vote
1
answer
400
views
The topological realization functor reflect coequalizers ?
Reading the book of Goess-Jardine or of Gabriel-Zisman on the simplicial homotopy there are coker presentation of the boundary $\partial\Delta^n $ of the elementary simplex $\Delta^n $ or the horn $\...
- 27.2k
2
votes
1
answer
406
views
Are these systems of linear equations always solvable
Let $X$ be a (finite) simplicial complex and let $f$ be a map from the set of its $n$-Simplices to a abelian group $A$, with the property, that every cycle maps to $0$ (extending $f$ linearly).
Let $...
- 73.2k
16
votes
2
answers
1k
views
Is there a higher homotopical spinor theory?
Gabriele Vezzosi and I have been musing on the following. Consider the standard double cover $\mathop{\rm Pin}_{n} \to \mathrm{O}_{n}$, whose kernel is $\mathbb Z/2\mathbb Z$. This allows to associate ...
CommunityBot
- 1
1
vote
1
answer
304
views
good perspective in viewing manifolds of infinite dimension
Borel conjectued aspherical closed manifolds are topologically rigid.(i.e.a homotopy equivalence between two aspherical manifolds is homotopic to a homeomorphism).
now,soppuse M is a K(G,1) space,
it ...
CommunityBot
- 1
15
votes
4
answers
2k
views
Brown representability beyond CW complexes
Brown representability states that any contravariant functor from the homotopy category $CW_*$ of pointed CW complexes to the category of pointed sets is representable if it turns coproducts into ...
- 5,575
0
votes
2
answers
2k
views
wedge sum deck transformation
For any universal cover p of the wedge S1 V S1 is it true that the two actions of π1(X, x_0) on the fiber p^-1(x0) given by lifting loops at x0 and the action given by restricting deck transformations ...
- 8,803
4
votes
1
answer
430
views
Does trivial on local cohomology implies trivial on global cohomology?
I know this question is absolutely trivial, but having self-studied the subject I feel extremely unsure on the basics. Do not hesitate to downgrade the question, if you feel it deserves so.
Given a ...
- 22.6k
1
vote
1
answer
346
views
Does Thom's J-equivalence imply Whitehead's simple homotopy?
Rene Thom came up with the idea of J-equivalence:
Let $M_1$ and $M_2$ be manifolds that are oriented, compact and smooth. Then they are J-equivalent if there is a smooth manifold $X$ with boundary $...
- 55.9k
8
votes
2
answers
1k
views
fundamental group and complete invariant of irreducible 3-manifolds
I heard that,by Perelman's work,we can get that the fundamental group is a
complete invariant of irreducible 3-manifolds (except for lens spaces).
can someone help explain this.Thank you!
- 4,404
5
votes
0
answers
181
views
Are $n$-vector bundles an $(\infty,n)$-symmetric monoidal category with duals?
In Lurie's On the Classification of Topological Field Theories, one of the main characters are $(\infty,n)$-symmetric monoidal category with duals. A basic example of this should be $n$-vector spaces, ...
- 6,777
2
votes
0
answers
1k
views
Recognition principle
Hello,
The classical statement of the recognition principle (after Boardman, Vogt, Milgram and May) that I know is:
Let $X$ be a (group-like) topological space acted on by the little $n$-discs ...
- 2,521
3
votes
1
answer
890
views
$A_{\infty}$ structure of (co)homology of a space
Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$.
(1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $Homeo(X)$?
(2) Does $H_*(X)$ also ...
- 2,734
7
votes
1
answer
859
views
How commutative is Quillen's Plus-Construction?
This question is inspired by this question about the dependence of K-theory on the order of multiplication in the ring. I did not think long about it, so maybe the answer really lies on the surface; ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
9
votes
1
answer
789
views
Cohomology of a configuration space
The symmetric group $\Sigma_k$ acts on $X=F({\mathbb R}^n,k)$,
the ordered configuration space of $k$ points in
${\mathbb R}^n$.
If $n$ is odd, the cohomology $H^*(X;{\mathbb Q})$ is
a rank-one ...
- 2,537
10
votes
3
answers
2k
views
Fundamental group of R^2 minus the (ir)rationals
Let
$$E =\{(x,0) \in \mathbb{R}^2 \colon x \in \mathbb{Q} \}$$
$$F = \{(x,0) \in \mathbb{R}^2 \colon x \in \mathbb{R} \setminus \mathbb{Q}\}$$
compute the fundamental group of $\mathbb R^2\setminus E$ ...
- 7,193
8
votes
3
answers
535
views
Order of the identity map of a Moore space.
Write $M_n = S^n \cup_2 D^{n+1}$. I know, as a matter of folklore, that the identity map $\mathrm{id}_M$, considered as an element of the group $[M_n, M_n]$, has order $4$ (for $n > 3$, let's say)....
- 1,467
7
votes
1
answer
364
views
Aspherical homotopy orbit space of configurations on the 2-sphere
The group SO(3) acts naturally on $S^2$ and thus on $Conf(S^2, q)$, the configuration space of $q$ distinct points on the 2-sphere, via the diagonal action on $S^2 \times...\times S^2$. This is a ...
- 2,734
13
votes
4
answers
1k
views
How often does suspension define an action of Z/2 on a category of module spectra?
Let R be the 2-periodic complex K-theory spectrum, or any other naturally occuring 2-periodic E-infty ring spectrum. The suspend-once functor gives an autoequivalence of the category of R-module ...
- 27.2k
6
votes
1
answer
1k
views
Chern classes generating cohomology
The fact that Chern classes are Hodge classes (and are rational combinations of algebraic cycles) is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my ...
- 12.4k
2
votes
2
answers
285
views
How can I compute the full set of nodes of a surface?
The nodes of a surface are special cases of more general singularities. For example, the Cayley cubic has four nodes.
The full set of singularities of a surface can be characterized by finding all ...
1
vote
0
answers
371
views
differential form of charge for pi_4(S^3) or pi_4(S^2)
How to write a 4-form of topological charge which would correspond to non-zero element of
the homotopy group $\pi_4(S^3)$ or $\pi_4(S^2)$ (both are equal to $Z_2$) ? ?
An example of such a mapping (...
- 11
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 32.4k
13
votes
1
answer
863
views
homology of a symplectic leaf in GL(n)
I would be grateful for any info
concerning the following question.
Given an n-tuple z=(z_1,...,z_n) of
nonzero complex numbers, let X(z) denote the
GL(n,C)-conjugacy class of the
diagonal matrix ...
- 959
1
vote
1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 538
3
votes
2
answers
1k
views
presentations of the trivial group
I just came across this statement in Bowditch's notes on geometric group theory that $\langle a,b\ |\ aba^{-1}b^{-2},a^{-2}b^{-1}ab \rangle$ is a presentation of the trivial group. Does anyone know if ...
CommunityBot
- 1
11
votes
2
answers
843
views
covers of $Z^\infty$
Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...
CommunityBot
- 1
29
votes
5
answers
3k
views
Topologists loops versus algebraists loops
Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also ...
- 538
10
votes
2
answers
1k
views
Abelian groups as fundamental groups of topological groups
Hi,
It is well known that the fundamental group of a topological group is abelian, and that every group is the fundamental group of some topological space.
My question is: Does every abelian group ...
CommunityBot
- 1
6
votes
0
answers
360
views
The Space of Cellular Maps
Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
- 12.5k
22
votes
1
answer
1k
views
Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 114k
11
votes
3
answers
1k
views
Which properties of finite simplicial sets can be computed?
A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I ...
- 56.9k
3
votes
0
answers
207
views
Symmetric monoidal structure on cosimplicial spaces
Is there a monoidal structure on the category $Spc^{\Delta}$ of cosimplicial spaces such that in the adjunction
$$
\Delta^{\bullet}\otimes-\colon Spc\leftrightarrows Spc^{\Delta}\colon Tot(-)
$$
the ...
- 101
2
votes
0
answers
757
views
Leray-Hirsch for HOMOLOGY?
Let $E\to B$ be a fibre bundle. The Leray-Hirsch theorem states under suitable assumptions, the cohomology of $E$ is an $H^*(B)$-module generated by suitable cohomology classes in $E$.
Is there any ...
- 1,207