All Questions
8,725 questions
7
votes
0
answers
370
views
A question about a blue fan and a red fan and their common refinement
Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $...
- 24.7k
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 $\Sigma_g$?
- 471
2
votes
0
answers
338
views
Do non-ordinary Bredon cohomology theories extend?
As shown by Lewis, May, and McClure (MR0598689), the ordinary equivariant Bredon cohomology theory $H^*_G(-; M)$ extends to an $RO(G)$-graded cohomology theory precisely when the coefficient system $M$...
- 865
0
votes
1
answer
305
views
Are braid links proper links?
Are braid links proper links? Or are the concepts involved unrelated?
- 63
2
votes
0
answers
430
views
The signature of a mapping torus
Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold $B$...
- 1,582
3
votes
0
answers
228
views
Extension of homeomorphism of boundaries to a homeomorphism of a cobordism
Suppos we have a cobordism $(M, \partial_{-}M, \partial_{+}M)$, where $M$ is a oriented compact (topological) 3-manifold.
Assume we have orientation preserving homeomorphism $f_{\pm}: \Sigma \to \...
- 93
4
votes
0
answers
314
views
Combining Lefschetz numbers with Euler classes
Given an $n$-manifold $M$ (say), we can talk about its Euler characteristic
$\chi(M)$.
This can be generalized to the Euler number of any $n$-dimensional
bundle ${\mathcal V}$. Or indeed, the Euler ...
- 27.9k
2
votes
1
answer
424
views
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]\...
- 3,494
1
vote
0
answers
75
views
Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
- 111
4
votes
0
answers
383
views
Exercise concerning locally constant presheaves [closed]
Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\...
- 4,902
4
votes
0
answers
264
views
Infinity-groupoid on the etale site of a scheme.
Let $X$ be a topological space. We may naturally associate to $X$ a simplicial set which is a Kan complex. This in turn gives rise to the fundamental $\infty$-groupoid associated to $X$. From this ...
- 491
-1
votes
1
answer
263
views
Is $X$ homeomorphic to $S^1 \times Y$? [closed]
Is it true that when the first fundamental group of a topological space $X$ is isomorphic to $\mathbb{Z}$ then $X$ is homeomorphic to $S^1 \times Y$ where the first fundamental group of $Y$ is ...
- 1,671
8
votes
0
answers
370
views
Dualizing complex of the product of two locally compact spaces
Hello!
In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
- 2,756
3
votes
0
answers
423
views
Cohomologies associated to residually torsion-free nilpotent groups
This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...
- 373
2
votes
0
answers
251
views
Knots that turn around an axis [closed]
Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to ...
- 3,033
9
votes
0
answers
700
views
Is a functor which is a sheaf for open covers and finite closed covers automatically a sheaf for covers by simplices?
Suppose $F:Top^{op}\rightarrow Set$ is a functor which is a sheaf for open coverings as well as for finite closed coverings, i.e. such that, apart from the usual sheaf property we also have that for ...
- 12.3k
10
votes
0
answers
325
views
H-space structure on the Calkin algebra
By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional ...
- 7,637
7
votes
2
answers
268
views
What are the fibres of a representable simplicial sheaf (in the Nisnevich topology)
Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, ...
- 71
0
votes
0
answers
236
views
Topological K-theory of Bohr compactification of real numbers
I am interested in the
K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers.
Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ?
More generally, what do we ...
- 357
10
votes
0
answers
484
views
Applications of sheaf theory to the computation of invariants of LS-category type
I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
- 35.9k
1
vote
1
answer
245
views
Simplified Jones trace invariant for links
Jones (1985) defines a simplified trace invariant for knots by $W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W_K(i)$. Does this work for oriented links as ...
- 63
2
votes
2
answers
463
views
homotopy type of complement of subspace arrangement
I am studying the homotopy type of a space,and i hope it would be a $K(\pi,1)$ space.
now i have find its covering,once we can say the covering is $K(\pi,1)$,so is the space
itself.and the covering is ...
- 157
10
votes
0
answers
227
views
Spaces with free MU-homology
Let $E$ be a homology theory whose coefficients $E_*(pt)$ are concentrated in even dimensions. This could be complex bordism $MU$, but also complex $K$-theory, $BP\langle n \rangle$, $E(n)$, ...
Let ...
- 12.1k
5
votes
0
answers
442
views
Reference for homotopy orbits of pointed spaces
Can someone point me to a good (hopefully simple and brief) place to read about the basics
of homotopy orbits for pointed spaces?
More detail:
As I understand it, in the unpointed case,
we use the ...
- 12.5k
2
votes
1
answer
216
views
Second cohomology group with finite coefficients of the product of two varieties
This is surely well known. Let $X$ and $Y$ be smooth, projective, connected complex varieties. Then
$$H^2(X\times Y,Z/n)=H^2(X,Z/n)\oplus H^2(Y,Z/n) \oplus (H^1(X,Z/n)\otimes H^1(Y,Z/n))$$
for any $n&...
5
votes
1
answer
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 ...
- 52.4k
1
vote
0
answers
238
views
Twisted homology of free products
Let $G_1$ and $G_2$ be groups and let $M$ be a vector space equipped with actions of $G_1$ and $G_2$. The free product $G_1 \ast G_2$ thus acts on $M$. How can one compute the twisted group homology ...
- 11
1
vote
1
answer
414
views
Equivariant maps inducing isomorphism in integral cohomology
Consider the following statement.
Suppose $X$, $Y$ are finite CW-complexes with free involution
and $\mu:X\to Y$ is an equivariant map.
If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an ...
- 11
2
votes
1
answer
200
views
Homotopy type of the simplicial action groupoid
Let $X$ be a simplicial $G$-set, where $G$ is a simplicial group. What is the homotopy type of the simplicial action groupoid $X//G$?
user2529
0
votes
1
answer
162
views
3-manifolds, cubes with handles
Somebody knows where I can find some proof of the following fact:
If F is compact, connected 2-manifold with nonempty boundery why there exist n=1-X(F) pairwise disjoint properly embedded 1-cells {A1,....
3
votes
1
answer
214
views
Is there a common general setup for both Weil cohomologies and generalized cohomology theories?
My question can be simply (and loosely) stated as follows:
Is there a general (but not too general) construction that captures, as specializations, both Weil cohomologies in algebraic geometry and ...
- 23.4k
10
votes
0
answers
735
views
Adams Spectral Sequence for Equivariant Cohomology Theories
In ordinary algebraic topology the Adams spectral sequence can be applied for any cohomology theory $E$ and in good cases it converges to the stable homotopy classes of maps (of the E-nilpotent ...
- 1,273
0
votes
0
answers
77
views
Hit problem and $\left( \mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$
I try to determine the $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ as a $\mathbb{F}_2$- vector space, in which $P_6$ is polynomial algebra $\mathbb{F}_2[x_1,x_2,\dots,x_6]$ and $A$ is Steenrod ...
1
vote
0
answers
121
views
Existence of open dense subset in a Lie group
Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group
and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
- 645
11
votes
0
answers
561
views
How to get a Dehn-twist presentation of a periodic map of a Riemann surface?
Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ).
A classical result says such $f$ is ...
- 471
0
votes
0
answers
199
views
Finding a ribbon graph for a mapping class group action
Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is ...
- 1
2
votes
0
answers
337
views
simplicial deRham complex and model category structure
To every simplicial manifold is associated its simplicial deRham complex.
Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
6
votes
0
answers
367
views
Does every exact six-term sequence arise as the K-theory of a locally compact pair?
Consider six countable Abelian groups and six group homomorphims as in the following diagram
G → H → I
↑ ↓
L ← K ← J
Assume that the resulting sequence is exact ...
- 3,184
5
votes
0
answers
200
views
Whitehead products and a realization problem for graded Lie algebras
Many $\mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ we would like to study are non-degenerate in the sense that
$\dim_{\mathbb{C}} \mathfrak{g}_n < \infty \ \forall n \in \...
- 952
3
votes
1
answer
424
views
Principal bundle for contractible group is weak homotopy equivalence for ind schemes
This is may be obvious, but I am not comfortable with ind-schemes.
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
- 156k
0
votes
1
answer
141
views
Known graph/surface invariants that can be extracted from homology over different fields
The $Z_2$-homology of a surface viewed as a simplicial complex allows us to extract interesting invariants from the resulting homology groups. $\beta_0$ is the number of connected components, $\beta_1$...
- 4,462
6
votes
0
answers
510
views
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
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 ...
1
vote
0
answers
112
views
When are graphs of cohomologically complete groups cohomologically complete?
A group $G$ is cohomologically $p$- complete if the canonical map from $G$ to it's pro$-p$ completion $\hat G^p$ induces an isomorphism on cohomology $H^\ast_{cont}(\hat G^p, \mathbb{Z}_p) \rightarrow ...
- 11
5
votes
0
answers
323
views
Vector bundles of schemes and their topological realizations
Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$.
Does $R_\mathbb{R}$ send an ...
- 103
2
votes
1
answer
534
views
Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?
Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, \mathbb{Z})$,...
- 77
3
votes
1
answer
292
views
Cartesian-closed category of spaces with the Whitehead property?
I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes). ...
- 19.6k
1
vote
0
answers
170
views
Definition of the $L^2$-metric for the Determinant of Cohomology of a Vector Bundle on a Riemann surface
I start describing my setup. $X$ is a Riemann surface with a metric which can have a finite number of singularities, $E$ is a vector bundle on $X$ equipped with an Hermitian structure.
In an article (...
5
votes
0
answers
517
views
A smooth twisted tensor product of dg algebras?
I want to consider a Z/2Z dg algebra. As an algebra, it is generated over $\mathbb{Q}$ by two elements where x is even and e is odd with the relations $xe=ex$ and $e^2=1$(this makes it in particular ...
- 3,758
7
votes
0
answers
80
views
Explicit expression of WZ term for orthogonal groups
Consider the Wess Zumino term on the the space $W=I\times D$, where D is a two dimensional disk disk and $I$ is an interval, $[0,1]$, say, i.e.,
$$
\int_{I\times D} \langle(u^{-1} \, du)^3\rangle
$$
...
