All Questions
1,239 questions
2
votes
2
answers
212
views
Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step ...
- 7,799
2
votes
1
answer
761
views
Characteristic class for a fiber bundle over $S^1$
For an oriented manifold, we have Pontryagin classes. For a manifold with complex structure, we have Chern classes. For a orientable fiber bundle over $S^1$ (ie for orientable mapping tori), do we ...
- 4,826
2
votes
2
answers
1k
views
Uniqueness on square root of complex Line Bundle
Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
user21574
2
votes
1
answer
129
views
cohomology algebra of submanifold in euclidean space
If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline}
F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, x_{3k+1},x_{3k+2},x_{3k+3})\in\...
- 1,990
2
votes
1
answer
1k
views
Moduli space of flat connections over a torus
Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...
- 2,818
2
votes
0
answers
92
views
Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
- 527
2
votes
0
answers
246
views
A possible generalization of the Borsuk Ulam theorem via action of symmetric groups
The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...
- 366
2
votes
1
answer
131
views
Approximate Jordan-Brouwer theorem
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is ...
- 5,529
2
votes
1
answer
1k
views
Alexander duality theorem
Is the following true?
Let $\Sigma$ be a compact orientable hypersurface without boundary in $R^n$. Then $R^n\setminus\Sigma$ has at least two connected components.
- 23
1
vote
1
answer
996
views
Torsion in the (co-)homology of a smooth projective variety - what is known in general?
There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge ...
- 3,058
1
vote
1
answer
203
views
Moving of sphere embedding and its interior defined by Jordan-Brouwer separation theorem
Let $f_1:\mathbb S^{n-1}\rightarrow \mathbb R^n$ be a continuous embedding, where $\mathbb S^{n-1}$ is the unit sphere of dimension $n-1$, and a point $x$ in the interior of $f_1(\mathbb S^{n-1})$ ...
- 435
1
vote
1
answer
479
views
Possible homotopy-theoretical approach to Gauss-Bonnet
Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
- 2,129
1
vote
0
answers
150
views
Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
1
vote
2
answers
402
views
Homotopy problem for infinite dimensional topological space
Let $X$ be an infinite dimensional topological space such that :
$ \forall n \in \mathbb{N}$, $ \exists X_{n} \subset X$, $n$-dimensional subspaces verifying :
$\forall r<n$, the homotopy ...
1
vote
0
answers
138
views
Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$
Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
- 10.5k
1
vote
1
answer
153
views
For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
- 1,367
1
vote
1
answer
346
views
$\pi$-cohomology class -- a variant of cohomology class
Let $X$ be a topological space with a triangulation. The triangulation defines a
chain complex in $X$. Let $\mu_d$ be a cochain and $M^d$ be a chain. We use $<
\mu_d, M^d > \in M$ to denote ...
- 4,826
1
vote
0
answers
143
views
End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
1
vote
2
answers
482
views
(Lower) homotopy groups from triangulations
Both cohomology and homotopy groups capture global topological information of a manifold $X$. It is interesting to ask if they can be computed from local data. A triangulation $T$ is a natural ...
- 5,230
1
vote
0
answers
146
views
Do all $\mathbb{E}_{k}$-comonoids in $\mathcal{C}_*$ come from “freely-pointed” $\mathbb{E}_{k}$-comonoids on $\mathcal{C}$?
In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):
Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint ...
- 11.8k
1
vote
1
answer
1k
views
On compact, orientable 3-manifolds with non-empty boundary
I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave:
Let $M$ be a compact, orientable 3 manifold with non-...
- 1,255
1
vote
2
answers
464
views
Homotopy domination of mapping cylinders with a special condition
Suppose that there exist continuous maps $f:Y\longrightarrow Z$ and $g:Z\longrightarrow Y$ so that $g\circ f\simeq 1_Y$. Also, let $h :X\longrightarrow Y$ be a continuous map which induces an ...
- 139
1
vote
0
answers
184
views
The key step in Serre's method on higher homotopy groups
Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups ...
- 5,230
1
vote
1
answer
379
views
Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$
Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...
user21574
1
vote
0
answers
133
views
Contractibility of a $K_0^{\oplus}$ presheaf
Let's assume $X$ is a smooth projective variety over a field. Let $\Delta^{\bullet}$ be the cosimplicial scheme over the same field, where at level $n$ is just the $n$-th algebraic simplex. We can ...
- 5,901
1
vote
0
answers
53
views
Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,529
1
vote
1
answer
218
views
stable sheaves in characteristic $0$
Let $K$ be a non-algebraically closed field of characteristic $0$ and $X_K$ a smooth, projective, geometrically connected curve defined over $K$. If $F$ is a stable locally free sheaf on $X_K$, is it ...
- 2,323
0
votes
0
answers
88
views
Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? [duplicate]
Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not ...
- 10.5k
0
votes
1
answer
143
views
Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755
0
votes
1
answer
417
views
fiber, homotopy fiber of spaces
Suppose we have a pullback of topological spaces (CW-complexes) $B\rightarrow A\leftarrow C$ which I will denote by $D$.
Assumptions
The induced map $D\rightarrow C$ is a trivial fibration
The map $...
- 1,974
0
votes
1
answer
206
views
Definition of relative Whitehead product
I can not find a definition of relative Whitehead product. Could someone explain this product to me?
- 41
0
votes
1
answer
183
views
detecting weak equivalences in a simplicial model category
Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. ...
- 511
0
votes
1
answer
860
views
Sierpinski Triangle and the Chaos Game
The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
- 87
0
votes
1
answer
642
views
fibre bundle as a boundary of a fibre bundle
Let $M_{n+1}$ be a fibre bundle with $S_1$ as the base and $n$-dimensional CW complex $F_n$ as the fibre.
Assume $M_{n+1}$ is oriented.
(1) Can one show that $M_{n+1}$ is always a boundary of a CW ...
- 4,826
0
votes
1
answer
376
views
Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
- 447
-1
votes
1
answer
507
views
loop homology product for oriented compact manifolds with boundary
This is my first steeps in string topology and please forgive the basic level of my questions: I reformulate my question
Chas and Sullivan define the loop homology product for closed (=compact with ...
- 189
-1
votes
1
answer
163
views
Alternate property of H^2(T, Z) [closed]
Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
- 563
-2
votes
1
answer
215
views
Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]
I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...
- 601
-4
votes
1
answer
412
views
A topological groupoid structure on a pair $(X,A)$
Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.
Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding ...
- 366