All Questions
8,725 questions
0
votes
1
answer
97
views
Extending binary operation used by homotopy classes
There is this operation you learn in algebraic topology when working with homotopy groups and loops i.e. paths on a topological space $X$, $p:[0,1]\rightarrow X$ with $p(0) = p(1)$. Basically it is ...
- 2,275
4
votes
0
answers
426
views
Pull-back and push-forward of higher local systems
This is a follow up to the following two MO questions: q1,q2
What I'm interesting in understanding is the universal property (if any) of the morphism $Sum_n:Fam_n(\mathcal{C})\to \mathcal{C}$ ...
- 6,777
1
vote
1
answer
326
views
Is there a map of spectra implementing the inverse of the Thom isomorphism?
In the top answer to the question "Is there a map of spectra implementing the Thom isomorphism?" it is explained (with a reference to Rudyaks book) that from a rank $r$ vector bundle $\mu:V\to X$, a ...
- 2,782
3
votes
0
answers
320
views
Counting smooth structures on manifolds
Kervaire and Milnor found a formula for the number of smooth structures on the $4n - 1$ sphere (see, e.g. the last part of this MO answer). It is relatively easy to compute the number of smooth ...
- 7,062
3
votes
4
answers
658
views
A specific branched cover of S^2 as a subgroup of Pi_1
This is a follow-up question to: Degree 2 branched map from the torus to the sphere
This is a silly computation, but for whatever reason this is taking me much, much longer than it should. So ...
- 1,386
3
votes
3
answers
316
views
Existence of sequence of examples of braking 'Cancellation law in homeomorphic products'
I know there are manifolds (with or without boundary) $A$ and $B$ such that $A\times C$ is homeomorphic to $B\times C$ but $A$ is NOT homeomorphic to $B$.
My question is (in the Diffeomorphism ...
- 1,101
2
votes
0
answers
1k
views
Double Torus Parametric Surface [closed]
In the process of trying to find continuous parametric surface equations for the double torus and for a pair of pants, I believe that the problem is unsolvable for some topological reason.
I have ...
1
vote
0
answers
198
views
Euler class and self-intersection number of a surface in a 4-manifold [duplicate]
In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that
For a compact oriented surface $X$ in a 4-dimensional oriented ...
- 655
11
votes
1
answer
594
views
co-$A_\infty$ spaces
A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n =...
- 18.9k
0
votes
1
answer
547
views
Continuity of a homotopy-like function
Let $X$ and $Y$ be topological spaces. Assume $Y$ is contractible (hence, path- connected).
Let $f,g: X \to Y$ be continuous maps. At any fixed $x\in X$, there is a path $P_x: [0,1]\to Y$ from $f(...
- 789
4
votes
0
answers
176
views
Are injective modules flabby on basic open sets?
In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:
Statement: If $A$ is a commutative ring and $...
- 1,064
2
votes
1
answer
312
views
local model structure on simplicial presheaves
Hello,
Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.
Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...
- 5,562
1
vote
0
answers
211
views
Toral decomposition
I have a couple of questions on the following theorem:
Theorem. (Jaco, Shalen)
Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection $\...
- 43
0
votes
1
answer
379
views
A form of Lefschetz duality
Let W be a manifold with boundary such that \partial W is a union of two compact manifold A,B attached along their boundary. Does poincare duality hold for (W,A) and (W,B)?
- 537
2
votes
1
answer
897
views
Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis
Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
4
votes
1
answer
288
views
Spaces of Finite Subsets - homeomorphism type
This is a followup to Spaces of Finite Subspaces. Just for convenience, $\exp_nX$ is the space whose underlying set is the set of nonempty subsets $S\subseteq X$ with $|S|\le n$.
As Alex Suciu ...
- 17.5k
1
vote
0
answers
3k
views
Is this a covering space? [closed]
In Hatcher's book Page 79 I was asked to provide two-sheeted covering space $Y \rightarrow X_{1}$ such that the composition
$Y \rightarrow X_{1} \rightarrow X$ of the two covering spaces is not a ...
- 799
4
votes
1
answer
363
views
Homotopy colimits of cyclic spaces
Let $\Lambda$ denote Connes's cyclic category. It is an extension of the simplex category $\Delta$ (of nonempty finite linearly ordered sets) obtained by adding an automorphism of order $n+1$ to the ...
- 6,229
3
votes
1
answer
727
views
Automorphisms of the rooted tree operad
This follows Ryan Budney's comment to the question asked here.
What is the automorphism group of the rooted tree operad?
(By the rooted tree operad, I just mean the operad with object rooted trees ...
- 2,734
3
votes
0
answers
141
views
Homology of the fixed points of the singular complex of a G-space
I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me.
Suppose $X$ is a topological space and $G$ a ...
- 1,961
11
votes
1
answer
671
views
Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?
Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...
- 118k
1
vote
0
answers
136
views
restriction to the boundary in Morse theory
Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map
$$ H^*(M) \to H^*(\partial M) $$
I'm wondering if there is a reference that:
1) constructs this map in ...
- 1,331
12
votes
0
answers
434
views
Higher holonomies for higher local systems
In Jacob Lurie's classification of tqfts, one finds a version of the cobordism hypothesis for $(X,\zeta)$-structure, where an $(X,\zeta)$ structure on a manifold $M$ is the datum of a continuous map $...
- 6,777
9
votes
0
answers
514
views
E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product
Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 Ben-...
- 3,758
5
votes
1
answer
304
views
flat maps of monoids which are not localizations
It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.
Rather, I am looking for extensions of rings which share certain properties of localizations, like ...
- 6,210
4
votes
3
answers
348
views
Cohomological dimension of a group acting on a cellular complex
Let $G$ be a group acting on $X$, $X$ a cellular complex and $cd(G)$ the cohomological dimension of $G$.
2 things:
(1) I'm looking for a reference (or proof!) of this:
Suppose $X$ is acyclic. Then $...
- 2,734
0
votes
0
answers
635
views
Do homotopic non-intersecting simple closed curves separate the surface?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...
user13946
3
votes
0
answers
383
views
small maps, extension of IC sheaves and BM homology
This is a question very much related to the one here extensions of IC sheaves
I didn't know if I should ask a new question or edit the old one which wasn't very precise...
Homology means Borel-Moore ...
- 887
15
votes
1
answer
496
views
Geometric models for classifying spaces of $GLn(Fq)$.
The title pretty much says it. In a follow-up to my question about alternating groups, does anyone know of a "geometric" model for $BGL_n(F_q)$? By "geometric" I mean "a space you would have heard ...
- 5,000
1
vote
1
answer
278
views
Homotopy type of complement of a plane algebraic curves.
Assume that $X$ is the complement of a plane algebraic curve $C$ in $\mathbb{C}^2$ and Y is the complement of the union of $C$ and a line $L$ (not contained in $C$). Assume that $Y$ is $K(\pi, 1)$. ...
- 2,444
1
vote
0
answers
692
views
The stable-homotopy-homology-theory
Hi
Is there a way to stabilise relative homotopy groups into giving the stable-homotopy-homology-functor?
The fact that the homotopy excision theorem holds for exactly the same kind of pair that ...
- 333
1
vote
0
answers
561
views
Profiniteness Condition for Hochschild-Serre Spectral Sequence?
This question may seem elementary to experts but I am quite confused about it:
According to the entry of Lyndon–Hochschild–Serre spectral sequence on wikipedia, for a group extension $1\to N\to G\to ...
- 1,108
2
votes
1
answer
530
views
Classifying space commutes with geometric realization - reference request
Let $G$ be a nice topological group, say a compact connected Liegroup.
Then one can construct a model of its classifying space as $EG/G$ where $EG$ is any contractible space with free $G$ action.
...
- 13.2k
1
vote
0
answers
459
views
deformation retraction of the complement
Let $X$ be a topological space and $V$ and $N$ are subspaces of $X$. It is known that if $V$ is homotopy equivalent to $N$ then $X-V$ need not be homotopy equivalent to $X-N$, the Alexander horned ...
- 11
4
votes
1
answer
410
views
When is a Massey product the image of a Bockstein operator?
I have a discrete group $G$ and classes $x,y\in H^1(G;\mathbb{Q})$ (group cohomology with coefficients in the rationals viewed as a trivial $G$-module) such that the Massey product $$\alpha:=\langle x,...
- 35.9k
2
votes
1
answer
503
views
Hurewicz theorem related to Galois group (or Tannakian categories)?
Is there a proof of the Hurewicz theorem $\pi_1(X)^{ab} = H_1(X, \mathbf Z)$ ($X$ a connected topological space) expressing $\pi_1(X)$ as the "Galois" group of $X$, i.e., group of deck transformations ...
- 31
3
votes
1
answer
206
views
Topological question about right-lifting property and the evaluation map
Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ ...
- 3,901
15
votes
0
answers
2k
views
Covers of $Z^k$
This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
2
votes
2
answers
285
views
How can I compute the full set of nodes of a surface?
The nodes of a surface are special cases of more general singularities. For example, the Cayley cubic has four nodes.
The full set of singularities of a surface can be characterized by finding all ...
8
votes
1
answer
652
views
How do branched coverings of complex surfaces "fit" with branched coverings of curves?
Since I'm used to working with algebraic $\pi_1$'s, which don't work well with surfaces, I find myself lacking geometric intuition when I attempt to do these types of purely geometric arguments. I'm ...
- 5,929
2
votes
1
answer
459
views
Giambelli and Porteous Formula
I was looking at the formula to compute the schubert class of a grassmanian in terms of a more elementary schubert cycles via giambelli's formula, and on the other hand Porteous formula tells us how ...
- 327
4
votes
1
answer
1k
views
Genera and the Milnor Conjecture on the Unknotting Number of a Torus Knot
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)...
2
votes
0
answers
189
views
About the Lie algebra of polyvector fields
I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
- 1,609
8
votes
0
answers
706
views
Finite generation of equivariant cohomology rings
Let $G$ be a finite group acting on a topological space $X$. (In my applications, $X$ is the classifying space of a compact Lie group.) Let $H^\ast(-)=H^\ast(-;\mathbb{Z}/2\mathbb{Z})$ denote mod 2 ...
- 35.9k
3
votes
1
answer
1k
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Orientation of a "glued"-manifold
Im wondering if there's a short way to prove that when two manifolds with diffeomorphic boundaries are glued together along the boundaries the orientations of these must be inverse to each other. That ...
- 947
1
vote
0
answers
320
views
Does the closure of a ``nice'' smooth submanifold define a homology class?
Let $M$ be a smooth compact, oriented manifold. Let
$X$ be a submanifold which is of the following type
$$X := \{ p \in M: \psi(p) =0, ~~\varphi(p) \neq 0 \} $$
where
$$ \psi: M \rightarrow V, \...
- 3,245
1
vote
0
answers
167
views
How many ways we have to prove that a topologically (or analytically) nice mapping is injective?
I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and $\...
- 1,881
4
votes
1
answer
599
views
When are two natural transformations of infinity-categories equivalent?
Suppose
C and D are two ∞-categories (quasi-categories),
$F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the ...
- 1,947
4
votes
1
answer
742
views
Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
- 2,160
7
votes
1
answer
459
views
Why do Delta-sets not allow quotients?
A $\Delta$-set is a contravariant functor from the category $\Delta'$ of order-preserving injections to the category of sets (this is essentially what Allen Hatcher calls a $\Delta$-complex).
A main ...
- 727