All Questions
8,725 questions
1
vote
1
answer
437
views
Induced fibration of Eilenberg-MacLane spaces
How does the inclusion $\mathbb Z\rightarrow \mathbb Q$ induce a fibration
$K(\mathbb Z,n)\rightarrow K(\mathbb Q,n)$ with fibre $\Omega K(\mathbb Q/\mathbb Z,n)$?
- 21
6
votes
0
answers
723
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
- 21.8k
5
votes
1
answer
493
views
Pair consisting of a compact manifold and Morse function
Consider the following situation:
Let $M$ and $M'$ be two closed manifolds and suppose $f:M\to \mathbb{R}$ and $f':M'\to \mathbb{R}$ are smooth morse functions on $M$ and $M'$ respectively. We say ...
- 2,299
1
vote
0
answers
365
views
Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
2
votes
1
answer
167
views
On the simply connectedness of Symmetric products and Hilbert schemes of points
My first question is whether $m$-th symmetric product of $\mathbb{C}^{n}$ is simply connected, where $n\geq 3$.
The second question is whether $Hilb^{m}(\mathbb{C}^{n})$ is simply connected, where $n\...
- 399
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
8
votes
1
answer
637
views
Cohomology map induced by the group actions on homogeneous vector bundles
Here is a topological question which seems quite elementary. The answer to this question may be useful e.g. in estimating the orders of the automorphism groups of some algebraic varieties and in ...
- 23.5k
3
votes
1
answer
613
views
Union of disjoint Vitali Sets...
We say $X$ is a Vitali set if there exists a countably dense subgroup, $\Gamma$, of the additive group $\mathbb{R}$, such that $X$ is a selector of the partition of $\mathbb{R}$ canonically associated ...
- 463
3
votes
1
answer
408
views
Morava's "Motives and cell bundles"?
Hello, do you know more about, or some exposition of Morava's talk?
- 10.8k
2
votes
1
answer
200
views
On direct limit of Stiefel mainfold
I'd like to build a model for the space $EU(n)$: the total space of universal bundle $\pi:EU(n) \rightarrow BU(n)$. $\;$ $EU(n)$ must be a conctractible space on which $U(n)$ acts freely. So I ...
- 243
3
votes
0
answers
306
views
Cohomological criterion for $G$-equivariant maps
Let $f:M\rightarrow N$ be a surjective map from a fourmanifold $M$ to a surface $N$, with connected fibers (each fiber is connected). Assume that $f$ admits a multiple section $s:N\rightarrow M$.
...
- 583
2
votes
2
answers
349
views
When does the equivariant homology of the fixed part of a $G$-space surject onto the equivariant homology of the whole space?
Let $X$ be a $G$-space. Are there examples, i.e. conditions or classes of spaces, such that the map induced by the inclusion $X^G\to X$ of the fixed part into the whole space induce a surjection of ...
user2529
12
votes
0
answers
414
views
Hilton-Eckmann dual of the Steenrod Algebra
In essence my question can be stated as follows: fill in the analogy
$$
\text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product}
$$
$$
\updownarrow \qquad\qquad \...
- 18.9k
4
votes
1
answer
320
views
a naive question about homogeneous polynomials
Suppose $p$ is a homogeneous polynomial in $n$ complex variables. Let S be the hypersurface
defined by $p(z)=0$. Then is the 1-form $dp/p$ always non-exact on the complement $C^n\setminus S$?
Any ...
- 89
1
vote
0
answers
220
views
Change the fiber of a fibration
Let $F \rightarrow E \rightarrow B$ be a (Serre) fibration of topological spaces. Given a map $F' \rightarrow F$, is there a criterion for the existence or even an explicit construction of a fibration ...
- 2,040
2
votes
0
answers
143
views
Weak notions of Kan fibrations
I have two semi-simplicial complexes $X_\bullet$ and $Y_\bullet$, along with a simplicial map $f_\bullet: X_\bullet \to Y_\bullet$. Now $f_\bullet$ is not a Kan fibration: for an $n$-simplex in $Y_\...
- 4,133
0
votes
1
answer
423
views
What Is This Quotient Space?
Let $X$ be a finite CW-complex with only even cells $x_1,\ldots, x_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. ...
- 61
1
vote
0
answers
291
views
Topological invariance of intersection number
Let $X$ and $Y$ be algebraic curves in ${\mathbb P}^2$ and suppose they have an isolated intersection at $(0,0)$. Let $\hat{X}$ and $\hat{Y}$ be another such pair, and suppose that there exist ...
- 11
1
vote
1
answer
246
views
Is the nerve of a groupoid an EM space?
Let $G$ be a connected groupoid. Is the nerve $BG$ a $K(\pi, 1)$, and if so, is there a groupoid homomorphism $f:G\to \pi$ that induces the homotopy equivalence?
- 681
1
vote
0
answers
158
views
Is the $K$-Theory of $\mathbb{P}X$ a $\lambda$-Ring?
If we let $\mathbb{P}X$ denote the free commutative algebra generated by the spectrum $X$, then we have a weak equivalence
$$
\mathbb{P}X\simeq \bigvee_r E\Sigma_{r+}\wedge_{\Sigma_r}X^{\wedge r}.
$$
...
- 535
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
0
answers
668
views
Euler Characteristic in a neighborhood of a Singularity of Complex Curve, and Deformations
Hello all.
Let's say we have a one parameter (complex) family of complex curves on a family of corresponding surfaces. i.e, think of the whole thing living inside some threefold, fibered along ...
- 1,893
1
vote
1
answer
526
views
Discrete subgroups of isometry group of proper metric space
Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.
Consider the following ...
- 13
2
votes
1
answer
144
views
Does the following categorial sum preserve weak equivalences?
In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following category....
- 681
2
votes
1
answer
359
views
Induced Map on Sp(2g,Z) is surjective
Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$ through the induced map ...
- 105
5
votes
0
answers
624
views
What is the right notion of equivariant Cech cohomology?
What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?
- 1,138
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 ...
- 569
5
votes
0
answers
187
views
Interpretation of the product $K(X)\otimes K^{-1}(X) \to K^{-1}(X)$
We can represent every element of the group $K^{-1}(X)=\tilde{K}(SX)$ by a isomorphism of trivial vector bundles $L:\, X\times \mathbb{C}^k \to X\times \mathbb{C}^k$ because $SX$ is the union of two ...
- 145
5
votes
1
answer
799
views
Exotic spheres detected in higher homotopy
Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space \...
- 2,734
3
votes
0
answers
175
views
"Cut-off" of the Adams exact couple
(This question has been asked on Math.StackExchange where it attracted a few upvotes, but - unfortunately - no answer.)
I have been reading Chapter 2. of A. Hatcher's draft of "Spectral Sequences in ...
- 2,197
0
votes
0
answers
235
views
Two-dimensional distributions in R^3
I was wondering: if $X$ is a non-vanishing smooth vector field defined on an open subset $U \subset \mathbb{R}^3$, there are two smooth vector fields $Y$ and $Z$ on $U$ such that $X(p) = Y(p) \wedge Z(...
- 73
8
votes
1
answer
469
views
Computing H^2(X, T_X(-\log D))
Let $X\subset \mathbb P^3$ be a smooth variety and $D \subset X$ be an effective, reduced, irreducible divisor. My question is the following.
If I know the defining equations of $X$ and $D$ then is ...
- 325
4
votes
3
answers
221
views
Homology of bundles over a triangulated base and $A_\infty$-algebras
Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be ...
- 2,734
1
vote
1
answer
201
views
$\hat{A}$ genus of total space of fiber bundle
Is there any formula for calculate the $\hat{A}$ genus for the total space of a fiber bundle?
- 1,101
7
votes
0
answers
191
views
Torsion in Whitehead group
Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
- 468
1
vote
1
answer
444
views
"non natural" iso between homotopy and homology
Can we classify all finite CW complexes $X$ such that for each $i$ there is some isomorphism $\pi_i(X) \rightarrow H_i(X)$? Note that it is not hard to classify all complexes for which each ...
- 2,035
4
votes
1
answer
331
views
objects in the derived category with flat homology
This is my first MO question, so please go easy on me if you think this is too vague.
Is there anything to say about the collection of chain complexes with flat homology? Is there a name for them, ...
- 499
8
votes
0
answers
549
views
Description of virtual fundamental class
For some concrete examples, is there an easy way to describe the virtual fundamental class (say, by capping off the moduli pace with an obstruction bundle ). Consider the moduli space of stable maps ...
- 147
0
votes
1
answer
170
views
Is there a Lie group which is made $S^n \cup_f ~e^m$?
Is there a Lie group G which has only two cells?
i.e. $ G = S^n \cup_f~ e^m$ where $f:S^{m-1}\to S^n$ with $m>n$
How many exists that groups? Infinitely many?
If there is no Lie group which has ...
- 699
3
votes
1
answer
215
views
number of ribbon structures (or punctured surfaces) on a graph
Suppose $G$ is connected undirected graph.
Does the calculation of the number of topologically distinct punctured surfaces that can arise from putting a ribbon structure on $G$ exist in the ...
- 47
4
votes
1
answer
1k
views
Twisting an L-infinity-morphism with "non-associated" Maurer-Cartan elements
Background
Suppose we are given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$ and an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$. Furthermore, we have a Maurer-Cartan element $\pi$ of $(g,Q)$.
One ...
- 93
3
votes
1
answer
476
views
Parallelizability of exotic structure
I'd like to discuss a little bit about the problem I asked Diarmuid Crowley that whether all smooth structures on $S^7$ are parallelizable. He first came to using semi-characteristic but then came up ...
- 1,003
9
votes
0
answers
409
views
Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?
I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner.
They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
- 6,829
1
vote
0
answers
145
views
Does compactly supported cohomology make sense for cosimplicial spaces?
As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...
- 1,162
2
votes
0
answers
201
views
Characteristic classes for general $G$ bundles, not just $G=SO(n)$ or $G=U(n)$
Is there a nice reference where I can get the information of $H^*(BG)$ with coefficients in $\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$, with $G$ not just $SO$ or $U$ but for example $G=PSU(n)$, $Pin(n)$,...
- 6,133
2
votes
1
answer
360
views
James Construction for Disconnected Spaces
When I work out the James construction for a discrete pointed space, it appears that the
induced map $\pi_0 (J(X)) \to \pi_0( \Omega\Sigma X)$ is the inclusion of the free monoid on $\pi_0(X)$ into ...
- 12.5k
2
votes
1
answer
471
views
JSJ-Decomposition - proof of finiteness in Hatcher's Notes
I'm currently trying to understand the proof of the (elementary) JSJ-Decomposition in Allen Hatcher's notes on 3-Manifolds.
Specifically, I have trouble understanding the application of proposition 1....
- 123
1
vote
0
answers
54
views
Lattice-isotopic essentialization of arrangements
I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
- 741
11
votes
0
answers
1k
views
the first cohomotopy group of a P-adic solenoid
Greetings.
I have a problem concerning the definition of the first cohomotopy group of P-adic solenoid. In [1], E. Spanier defines group operation on the set of homotopy classes of maps from X to n-...
- 219
10
votes
0
answers
458
views
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