All Questions
1,240 questions
3
votes
1
answer
987
views
Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual ...
- 315
3
votes
1
answer
691
views
Bockstein homomorphism and Square Operations: Their consistency formulas
Here are various ways to define "Bockstein homomorphism:"
Let $\beta_p:H^*(-,\mathbb{Z}_p)
\to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}...
- 10.5k
3
votes
1
answer
158
views
Cyclic polytopes whose boundary is a flag complex
A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ ...
- 1,017
3
votes
0
answers
118
views
Weak contractibility of some infinite dimensional metric spaces
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
3
votes
1
answer
467
views
Why do we need cofiltered condition on the index category in the definition of pro-categories?
Let $\mathcal{C}$ be a category. The pro-category pro-$\mathcal{C}$ is defined as (see this nLab page) follows: its objects are diagrams $F: D\to \mathcal{C}$ where $D$ is a small cofiltered category. ...
- 9,019
3
votes
1
answer
2k
views
Conditions for a parametric curve to avoid self-intersection?
Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the ...
- 151k
3
votes
1
answer
246
views
Motion planning algorithm
Consider a path connected topological space $X$, one can equip its path space $PX=\{ \gamma: [0,1] \longrightarrow X \; continuous\}$ with the compact open topology. We call a motion planning ...
- 189
3
votes
2
answers
636
views
How can I prove that Hopf fibrations are the only ones with fiber, total space and base space homeomorphic to spheres?
I know that Hopf fibrations (the four ones) are the only ones that have the form
$S^k \to S^m \to S^n$, but I never seen a proof. Could anyone link me a paper or text where this is proved, or prove it ...
- 133
3
votes
0
answers
920
views
A Künneth-Theorem version for relative singular cohomology
I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.
The Künneth-Theorem which I ...
- 1,188
3
votes
1
answer
277
views
What do you call a map of spaces which is weakly left orthogonal to all $n$-connected maps?
$\let\op=\operatorname$In $\op{Set}$, we have an $(\op{Epi},\op{Mono})$ orthogonal factorization system. Strikingly, if we reverse the roles, we get the no-less-important $(\op{Mono},\op{Epi})$ weak ...
- 64k
3
votes
1
answer
463
views
cohomology module of unit tangent vector bundles over spheres
Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow \tau(S^m)\...
- 2,223
3
votes
1
answer
909
views
Isomorphism between a mapping class group and the fundamental group of a moduli space
For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
- 1,298
3
votes
1
answer
743
views
Fiber bundle and fibration of classifying space [closed]
Let $BG$ is classifying space of $G$ topological group.
If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the
inclusion map $i:H\rightarrow G$ induces
\begin{equation*}
G/H\...
- 1,367
3
votes
0
answers
413
views
Motivation of Lawson Homology
Let $X$ be a complex projective variety. The Chow monoid of $X$, denoted by $\mathcal{C}_p(X)$, is the monoid given by $p$-dimensional algebraic cycles of $X$. Considering a fixed embedding $i\colon X\...
- 1,252
3
votes
1
answer
289
views
How many non-commensurable non-arithmetic manifolds have a quaternion algebra like this?
I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form $\Big(\frac{a,b}{F(\...
- 2,436
3
votes
0
answers
181
views
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$
Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...
- 2,147
3
votes
2
answers
503
views
Does the Gysin map in $K$-theory respect bordism?
Let $X_1$ and $X_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$.
Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X_1$ mod $2$. Let
$$f_1:X_1\to ...
- 1,913
3
votes
2
answers
798
views
Banach algebraic proof of the Borsuk Ulam theorem
I am wondering whether there exists a proof of the classical Borsuk Ulam theorem
for the Euclidean n-sphere, $n>2$ that is based only on the theory
of Banach algebras. I checked on MR but had no ...
- 687
3
votes
1
answer
124
views
Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$
Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on $[...
- 3,051
3
votes
1
answer
423
views
Is it written anywhere that open subvarieties of affine spaces have "completely impure" cohomology?
Consider complex affine space $\mathbb{C}^n$ and let $U$ be a Zariski open subset of $\mathbb{C}^n$. By a celebrated result of Deligne, the cohomology $H^i(U)$ has a canonical Hodge structure. In ...
- 44.7k
3
votes
2
answers
426
views
Acyclic complexes for extraordinary cohomology theories
Let $X$ be a CW complex such that for all extraordinary homology theories, if you plug $X$ into them you get the same value as plugging in a point. Must $X$ be contractible?
- 33
3
votes
1
answer
157
views
Structure of boundary labelling in Sperner‘s Lemma
Consider a triangulated polygon in the 2-dimensional plane, where each vertex is labelled green, blue, or orange. Sperner's Lemma asserts that a fully-colored triangle exists in the triangulation, if ...
- 6,937
3
votes
0
answers
490
views
Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$
I'll first state the question as concisely as I can and then provide some motivation.
Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
- 21.6k
3
votes
2
answers
338
views
Left determined model structure on delta-generated topological spaces
Consider the class of cofibrations of the Quillen model structure, restricted to delta-generated topological spaces (the full subcategory of topological spaces generated by the colimits of simplices). ...
- 3,901
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
3
votes
1
answer
1k
views
cohomology of the orbit space of a group action
Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...
- 1,990
3
votes
1
answer
515
views
Polynomial differential forms on $BG$
Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rahm functor on simplicial sets, where the codomain is the category of commutative differential graded algebras ...
- 1,424
2
votes
0
answers
150
views
Localization for generalized Borel cohomology
For both equivariant de Rham cohomology and equivariant K-theory (in the "naive" or Borel sense), we have localization formulae which allow us to compute this cohomology in terms of the ...
- 1,969
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
1
answer
198
views
simplicial structure on a flat fiber bundle
Edit: After helpful comments, I now know that I am concerned with flat fiber bundles up to fiber preserving homotopy.
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$...
- 404
2
votes
1
answer
589
views
Are finite correspondances flat?
In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...
- 2,871
2
votes
1
answer
403
views
Compute cohomology of flat fiber bundles - does this always work?
Edit: This has been answered in this thread Are there compact flat fiber bundles with "truly" infinite structure group?.
Setting
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber ...
- 404
2
votes
0
answers
140
views
Covering a space by cones
Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed.
I am interested in ...
- 3,173
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
184
views
combinatorical description of classifying map for principal $G$-bundle over Delta set
Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are ...
- 619
2
votes
1
answer
236
views
Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism
This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
- 77
2
votes
1
answer
1k
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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
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
2
answers
288
views
Morphisms between fundamental groups of Lie groups
Let $X$ be a compact connected manifold. Since $\mathbb T^1$ is an Eilenberg-MacLane space $K(\mathbb Z,1)$, it follows that for every morphism $\varphi\colon\pi_1(X)\to\pi_1(\mathbb T^1)$ there is a ...
2
votes
0
answers
105
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
- 1,173
2
votes
2
answers
552
views
Is the following 3-manifold irreducible?
We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $...
- 1,255
2
votes
0
answers
136
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Question about the proof of the Duistermaat-VanDerKallen-Theorem, concerning the meaning of intersecting a chain with a set
On page 228 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) a 'chain' $\Delta_\tau$ is defined as $f^{-1}([0,\tau])\cap \...
- 360
2
votes
2
answers
1k
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Simple question of topological cofibration
I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
- 367
2
votes
1
answer
308
views
Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)
every smooth manifold can be triangulated, is it true for orbifold? Is it a known result? If yes, is there any reference?
reply to the comment : G does not need to be any subgroup of Sn , any ...
- 21
2
votes
2
answers
494
views
Pontryagin numbers on a fiber bundle over $S^1$
Let $F$ be a closed manifold. What are the Pontryagin numbers on $E=F\times S^1$?
More generaly, let $E$ be a closed manifold which is a fiber bundle over $S^1$
(with fiber $F$). $E$ is also called ...
- 4,826
2
votes
0
answers
71
views
Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
- 2,147
2
votes
1
answer
308
views
The pair $(Gl(n,\mathbb{R}), O(n) )$ as a groupoid
"Is there a topological groupoid structure on the pair $(Gl(n,\mathbb{R}), O(n))$, with their standard topologies?"
This is already asked here but this linked question is a very general question, so ...
- 366
2
votes
0
answers
235
views
Volume form on real Grassmanian
There exists a unique $SO(4)-$invariant volume form on the real Grassmanian G(2,2). Is it possible to write it down algebraically in Plucker coordinates?
- 4,316
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
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