All Questions
8,725 questions
1
vote
2
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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
6
votes
0
answers
388
views
Cohomology of a space with coefficients in a Lie group?
The following is a very naive construction, and I am almost embarrassed to ask questions about it.
Suppose I have a connected simple graph $\Gamma = (\Gamma_0,\Gamma_1)$ (i.e., vertex set, edge set) ...
- 18.9k
2
votes
1
answer
277
views
computing homotopy type
I'm trying to construct a computer algorithm that decides, whethear a map is homotopically trivial and came to the following question: Let U by a connected closed hypersurface of dimension n-1 that is ...
- 972
0
votes
1
answer
97
views
Connecting two hypersurfaces in R^{n+1} by embedded curves
Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$.
Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$.
We assume that $\mathbb R^{n+1}\backslash D$ is ...
- 285
7
votes
1
answer
826
views
Weight filtration for smooth analytic manifolds
In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to ...
- 23.5k
1
vote
0
answers
99
views
PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
- 1,373
3
votes
1
answer
299
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disagreement between two definitions of the singular boundary map
Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his '...
- 105
5
votes
0
answers
192
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Extensions of discrete groups by spectra
If $G$ is a discrete group, recall that a (naive) $G$-spectrum consists of based $G$-spaces $E_n$ together with based $G$-maps $\Sigma E_n \to E_{n+1}$, where we give the suspension coordinate the ...
- 18.9k
5
votes
0
answers
477
views
Alternative approaches to the universal coefficient theorem
Let $A$ be a chain complex of free $R$-modules over a PID $R$, and let's assume $A$ has finite cohomological type, by which I mean $H^\ast(A)$ is finitely generated in each dimension and $0$ for large ...
- 5,544
2
votes
0
answers
72
views
d-refining covering of normal space
If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset U_i\...
- 189
6
votes
1
answer
675
views
Some questions on the intersection theory on a Hilbert scheme of points of a surface.
If $\Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $\Sigma^{[n]} \subset X^{[n]}$. $\ \ $ Where, $X^{[n]}, \Sigma^{[n]}$ ...
- 445
4
votes
0
answers
330
views
good covers and simplicial maps
Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all ...
- 7,637
3
votes
1
answer
419
views
Question on coverings and and their classifying spaces [closed]
Hello. I have a question on covering spaces. Please let each space be paracompact, path-connected, locally path-connected and semi-locally simply connected.
Let $E\to B$ denote a normal covering ...
- 31
1
vote
0
answers
155
views
Krull dimension in equivariant cohomology
Let a compact Lie group $G$ act on a manifold $X$. Let $\mathbb Q$ be the field of rational
numbers and assume that cohomology $H^{\ast} (X)$ with $\mathbb Q$-coeffiecients is
finite-dimensional. ...
1
vote
1
answer
219
views
bounded cohomology of subgroups of groups
Let $G$ be a discrete group and K a subgroup of G . denote by $(\hat{H_K})^i$ the bounded cohomology groups of $K$ , and by $(\hat{H_G})^i$ the bounded cohomology groups of G.
then $(\hat{H_K})^i$ ...
student
0
votes
0
answers
158
views
symbol map in algebraic K theory
I have a smooth projective morphism $X \to S$ or relative dimension 1 (i.e.
a family of smooth curves over base $S$). There should be a map $H^2(X, K_2) \to H^1(S, K_1) = Pic(S)$ given by integration ...
0
votes
0
answers
253
views
on variable and primitive cohomology of a hypersurface in a projective space
I have a smooth hypersurface D in $\mathbb{P}^n$: in many books about Hodge theory (as the ones of Voisin and Carlson) they take for granted that the primitive cohomology of D is equal the variable ...
- 107
2
votes
0
answers
177
views
Geometric interpretation of higher simpicial homotopy groupoids.
As a geometric interpretation of the simplicial fundamental groupoid, resp. its equivalent relation I thought about something like arcwise differentiable (or arcwise linear) paths modulo "we can go ...
- 21
0
votes
0
answers
109
views
Characterising singular homology among a more general class of cosimplicial spaces
Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
- 1,105
5
votes
2
answers
357
views
Truncated exact sequence of homotopy groups
This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma ...
- 28.9k
1
vote
1
answer
364
views
Decomposition of simplicial G-set?
Let $G$ be a simplicial group. Let $X$ be a simplicial $G$-set,i.e. for each level, $X_n$ is a $G_n$-set.
Can $X$ be written as a union of finite (or finite type) simplicial $G$-subsets? Here "finit ...
- 681
4
votes
1
answer
299
views
Changing the orientation of a Landweber exact cohomology theory
Let the ring R be a MU*-module via a ring homomorphism φ and suppose it satisfies the condition of the Landweber exact functor theorem such that we obtain a ...
- 43
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 $\...
- 4,165
2
votes
0
answers
271
views
Fixed point indices of simplicial maps
Let $K$ be a simplicial complex and let $f:K \to K$ be a simplicial map.
Assume the existence of an embedding $\iota: K \hookrightarrow \mathbb{R}^n$ of this complex into Euclidean space and note ...
- 103
21
votes
0
answers
1k
views
What is the current knowledge of equivariant cohomology operations?
In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in $RO(\mathbb{Z}/p)$-...
- 865
6
votes
0
answers
284
views
Reference request: splittings in rational homotopy theory
It is well known that for simply-connected rational spaces,
every suspension splits as a wedge of rational spheres and
every loop space splits as a product of rational Eilenberg-Mac Lane spaces.
...
- 12.5k
0
votes
1
answer
118
views
Homeomorphism between base of conifolds and spheres
Hello
Call $Y^4$ a conifold which satisfies the following condition:
$\mathfrak{Y}(z):=\sum_{\alpha=1}^{3}(z_{\alpha})^{2}=0,$
where $z_\alpha \in \mathbb{C}$. Now intersect $Y^4$ with $S^5$ to ...
- 77
1
vote
0
answers
51
views
Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR
Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$?
...
- 11
1
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0
answers
358
views
Fundamental group of the complement of a conic-line arrangement
This is a problem concerning a lemma in Oka's paper "On the fundamental group of the complement of a reduced curve in $\mathbb{P}^2$". Let $C$ be a curve in $\mathbb{P}^2$ and $L$ be a general line ...
- 2,444
1
vote
0
answers
135
views
Number of generators of the fundamental group of a Riemannian manifold with Ricci curvature bounded below
Is there a constant $C(n,D)$ such that for any closed Riemannian manifold $M$ with $Ric \ge -(n - 1)$ and $\mathrm{diam} \le D$, the fundamental group $\pi_1(M)$ is generated by at most $C(n,D)$ ...
- 689
6
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0
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391
views
Smooth manifold with non-trivial inertia group? (wrt homotopy spheres)
Let $\Theta_n$ be the set of orientation-preserving diffeomorphism classes of homotopy spheres, with abelian group structure given by #. Then for any smooth manifold $M^n$ one defines the "inertia ...
- 732
2
votes
1
answer
169
views
Is the total space of Fiber bundle bordant to 0 if the fiber is null bordant?
Assume F is null bordant. Does it imply that the total space of fiber bundle
$F\hookrightarrow E\to M$
is null bordant?
in particular what if $F$ is sphere?
- 1,101
2
votes
0
answers
138
views
Topology of Asymmetric Symmetric Products
Let $X_1,...,X_m$ be connected, simply-connected CW sub-complexes of a CW complex $X$. Let the symmetric group on $m$ letters, $S_m$, act on $P:=X_1\times\cdots\times X_m$ in $X^m$ by permuting ...
- 8,529
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votes
0
answers
2k
views
Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
- 24.1k
0
votes
0
answers
149
views
Only finitely many fundamental groups in $M(n,k,v,D)$?
Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
- 689
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0
answers
133
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equivariant singular homology
Let $M$ be a smooth $G-$manifold, where $G$ is a compact Lie group. From the result of S.Illman that $M$ admits an equivariant triangulation. Moreover, we can construct the equivariant singular ...
1
vote
0
answers
1k
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Again about Bing's house with two rooms [duplicate]
Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question clearly, when "hollowing ...
- 187
1
vote
1
answer
258
views
Homology of abelian groups and their finite-index subgroups
Fix some $1 \leq k \leq n$. I'm looking for finite-dimensional vector spaces $M_{n,k}$ over $\mathbb{Q}$ on which $\mathbb{Z}^n$ acts such that the natural map $H_k(\ell \mathbb{Z}^n,M_{n,k}) \...
- 13
2
votes
1
answer
219
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pairs of matrices up to similiarity and vector bundles over punctured torus
I would like to construct 2D vector bundles over the punctured torus, but I don't know a lot of K-theory. Over the square, there can only be the trivial bundle, but now since $\pi_1(\mathbb{T}^2\...
- 22.8k
1
vote
1
answer
115
views
Is function from topological group to metric space Borel?
Let $G$ be a pseudometrizable compact abelian topological group, $X$ a compact
metric space and $f:X\rightarrow G$ a continuous bijective function.
Suppose there exists $g\in G$ such that if $d_{G}(...
- 307
2
votes
0
answers
252
views
Invariant Ideals in Split Hopf Algebroids
Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:
An ideal $J\subset S$ is invariant under the action of the group $\mathrm{...
- 10.5k
3
votes
0
answers
187
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Topological self-maps of smooth complex hypersufaces in complex projective spaces
This questions if related to a cute article of Beauville where he proves in particular the following theorem:
http://math1.unice.fr/~beauvill/pubs/endo.pdf
Theorem.− A smooth complex projective ...
- 14.3k
2
votes
1
answer
244
views
Increased connectiviry of cross-effect functors on simplicial modules
I'm trying to write an expository manuscript on Bousfield-Kan's Fiber lemma and the relevant constructions. The proof of one needed theorem in the way seems to claim the following:
if $F$ is a functor ...
0
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0
answers
218
views
Does the group completion theorem apply to the James construction?
In other words, is the natural map $M \to \Omega B M$, for $M=JX$ the James construction on a space, a group completion? (By "group completion" I mean at the level of homology, I am aware of the space ...
- 213
1
vote
1
answer
414
views
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
Given a fiber bundle $p:E\to B$ and a point $x\in B$, is the evaluation map $\varepsilon:\Gamma^0(E)\to p^{-1}(x)$ defined by $\varepsilon(\sigma):=\sigma(x)$ a weak homotopy equivalence when $\Gamma^...
- 143
3
votes
0
answers
556
views
When a quasifibration is a Hurewicz fibration?
In studying quasifibration I have a question.
When a quasifibration $F\to E\to B$ is a Hurewicz fibration?
If $F,E$ and $B$ are CW-complex, it is right?
- 699
2
votes
0
answers
263
views
k-theory of $\mathbb{Z}$
I have a doubt.
Borel computed the rank of the higher algebraic k-theory of $\mathbb{Z}$:
$rank(K_n)(\mathbb{Z})= 1$ if $n\equiv1 mod4$, otherwise this rank is equal to 0.
On the other hand Bjorn ...
- 235
1
vote
1
answer
101
views
ball in universal cover belongs to the union of actions on a section?
M is an n-dim manifold. $\pi :\tilde M \to M$ the universal cover of M. $\tilde p \in \tilde M$ a lift of p. We choose a measurable section $j:{B_1}\left( p \right) \to {B_1}\left( {\tilde p} \right)$,...
- 689
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
3
votes
1
answer
361
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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 $X_{n,k}$ ...
- 44.8k