Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,258
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Hurwitz's construction of simple covers
What is commonly meant by Hurwitz's construction of simple covers?
- 3,717
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1
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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.
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Triangulation of moduli space.
I am recently reading the paper "Natural triangulations associated to a surface" by B.H. Bowditch and D.B.A. Epstein (http://www.sciencedirect.com/science/article/pii/0040938388900080#), where ...
- 1,703
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2
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784
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Multisimplicial geometric realization
Does anyone know a reference or proof for the following? Let $k\geq 1$ and let $X$ be a space. There is a $k$-fold multisimplicial set whose simplices in degree $n_1,\ldots,n_k$ are the maps $\Delta^...
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2
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Simplicial space whose all face/degeneracy maps are homotopy equivalences
I believe that the following is true, but I cannot find a proof. Let $X_\bullet$ be a simplicial topological space (I can add that my $X_\bullet$ comes from a bisimplicial set, so the spaces $X_n$ are ...
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Topological space with certain properties
For each integer $n$, does there exist a finite, connected $n$-dimensional CW complex $X$, that is not homotopy equivalent to an $(n-1)$-(or smaller)dimensional CW complex so that $H_0(X, \mathbb{Z}) =...
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2
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Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology)
I recently heard a talk about these topics and found them very interesting.
The talk was centered on the formal structure and didn't really focus on examples.
So my question is: what is your favorite ...
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Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
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3
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Probing a manifold with closed curves
Since fedja's excellent comment on Joseph's question on probing a manifold with geodesics remained uncommented (especially by topologists), I'd like to make a question out of it:
Conjecture: Given ...
- 9,628
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4
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Understanding four manifolds (more details inside)
I need to understand some of the theory of smooth four manifolds. Eventually I might be interested in learning about Donaldson theory. For the moment I am mainly interested in question such as: if you ...
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571
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Cohomology with $\mathbb{Z}_2$ coefficients, wedge product and integrals
I'm studing cohomology with $\mathbb{Z}_2$ coefficients in order to understand Stiefel-Whitney classes, and i have a couple of questions.Let $X$ be a n-manifold.
1) I know that $H^k_{sing}(X,\mathbb{...
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Intersections of subgroups of surface groups [closed]
Let $\mathcal{S}_g$ denote the fundamental group of an oriented surface of genus $g\ge 2$.
Does $\mathcal{S}_g$ contain subgroups $A$ and $B$ of finite index such that $A\cap B = \lbrace e\rbrace$?
- 35k
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Computing Homology via Sheaf Cohomology
Let $X$ be a finite CW complex. We can compute the cohomology groups of $X$ via sheaf cohomology of the constant sheaf. Are the homology groups of $X$ with $\mathbb{Z}$ coefficents the cohomology ...
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Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?
The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO).
On the other hand, MU and ...
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What are some triangulations of Grassmannians?
A while ago I heard that there was no known triangulation of the Grassmannian of 3-planes in 6-space.
To believe a statement like that, you have to be a little bit ungenerous about what you mean by "...
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Mayer-Vietoris sequence in homology with local coefficients
Background. I'm trying to compute some homology groups using a Mayer-Vietoris argument, but I really need local coefficients.
Question 1. What does the Mayer-Vietoris sequence look like when using ...
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252
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Cohomology with compact support and the nerve of a recouvrement
Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite ...
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2
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Are loop spaces of homotopically equivalent spaces homotopically equivalent? [closed]
Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces.
Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence?
Here, loop spaces ...
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A formal group law over oriented bordism
My question is related to the following question by Mark Grant here on math overflow:
Formal group law of unoriented cobordism
There it is stated that $MO_*$ has a formal group law $F_0$, universal ...
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euler class of the normal bundle and self intersection number [duplicate]
Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T_X|_S=T_S\oplus N_{S/X}$ where $N_{S/X}$ is the normal bundle. I have read that the euler class $e(N_{S/X})$ corresponds (via ...
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Chern numbers via Euler characteristics?
Let $X$ be a space good enough to have a fundamental class, and $E$ a complex vector bundle on $X$. Let $P$ be some polynomial expression, and say I want to evaluate $P(c_i(E)) \cap [X]$.
Is ...
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2
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A question on composites of pushforward and pullback
Let a finite group $G$ acts on an orientable manifold $X$ freely. Denote $\pi:X\rightarrow Y=X/G$ be the quotient map. This covering map defines two maps between cohomology groups $\pi^*=H^\ast(\pi):H^...
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FIltered colimits of truncated objects in $\infty$-topoi
The bare question:
Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT 6....
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When is the projective model structure cartesian? When is the internal hom invariant?
If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...
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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 ...
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Connections between topos theory and topology
What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the ...
2
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3
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585
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Explicit model of BSU(2) in terms of singular complex
What is the explicit model of BSU(2) in terms of singular complex, up to 5 dimensions,
so that one can compute $\pi_5(BSU(2))=\mathbb{Z}_2$ explicitly?
This question is related to another question of ...
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Branched Regular Cover over 4-times punctured sphere
This is probably trivial but has been bothering me all day.
Suppose $f:\Sigma_g\to \mathbb{S}^2$ is a $g+1$ fold branched conformal map with $\Sigma_g$ a connected genus $g$ surface and $f$ having $4$...
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The role of ANR in modern topology
Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
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$\pi$-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 ...
- 4,736
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Are there non-compact, non-smoothable manifolds?
There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact.
Are there any non-compact, non-smoothable manifolds?
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When do reflexive coequalizers preserve weak equivalences?
In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is levelwise a weak ...
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Simplicial path and loop spaces
I am trying to understand the relationship between the simplicial path space and loop space with the path space of a topological space, and the loop space of a topological space.
I have understood ...
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When is this braiding not a symmetry?
Given a topological space $X$ instead of forming the fundamental groupoid $\pi(X)$ which is the category whose objects are the points and morphisms the homotopy classes of paths one can also form the ...
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Mayer-Vietoris Sequence for Arbitrary Bicartesian Square of Spectra
Can anyone tell me if there is a Mayer-Vietoris sequence for an arbitrary homotopy pushout (hence homotopy pullback) of spectra and an arbitrary (co)homology theory. If this comes from some easy way ...
- 10.1k
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Homotopy Equivalences and Induced Correspondences between Fibre Bundles
Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...
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589
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When is a substack closed?
For this question we will consider the Zariski site of affine schemes and a stack $\mathcal{M}$ over it. I don't know what a substack is, but I have a guess. The stack $\mathcal{M}$ has an underlying ...
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When can an automorphism of the fundamental group be written as the induced isomorphism of some self-homeomorphism?
It is a common exercise to show that both automorphisms of $\pi_1(S^1)$ can be realized as induced isomorphisms of self-homeomorphisms of $S^1$. It is natural to ask if this is the case for spaces ...
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"Mathai-Quillen-type" form on $M\times M$?
Let $(M,g)$ be a compact, oriented, $(2n)$-dimensional Riemannian manifold. I'm wondering whether there is a "canonical" construction of a $(2n)$-form $\eta_g$ on $M\times M$, such that
$\eta_g$ is ...
- 3,192
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k3 surface as ramified double cover of $\mathbb{P}^2$
I read that one example of k3 surface is a double cover of $\mathbb{P}^2\mathbb{C}$ ramified over a sextic. My question is why a sextic? i believe that the sextic is isomorphic to the ramification ...
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No fixed components in the linear system of the line bundle generating $Pic(X)$
Let $X$ be a K3 surface. I know that $Pic(X)\simeq H^{1,1}(X,\mathbb{Z}):=H^{1,1}(X)\cap H^2(X,\mathbb{Z})$. Let's suppose that $H^{1,1}(X,\mathbb{Z})$ is one dimentional and generated by $\omega=c_1(...
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On the stable splitting of loops on a suspension
Let $X$ be a connected, based CW complex. Then the James splitting
of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra
$$
\Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad ...
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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(...
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de rham model for relative cohomology
In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M.
And in the page:
Relative De Rham cohomologies,
I got to know that there is another model for relative ...
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Cech cohomology as a colimit over maps to a CW complex
Given a topological space $X$, we consider the following category $\mathsf{CW}_{X\to}$. The objects are finite CW complexes $Y$ equipped with a continuous map $X\to Y$. The morphisms are continuous ...
- 18.3k
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2
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636
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Cohomology of configuration space of a compact manifold
There is a reference or a methode which by it we can calculate the cohomology of a configuration space of a compact manifold simply connected? It is possible to find a spectral sequence converging to ...
2
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1
answer
271
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Non-commutativity of certain Hopf spaces
How do one prove (or disprove) that $\Omega S^{2}$ and $\Omega(S^{2} \vee S^{2})$ are non commutative Hopf spaces?
I thought this is a question for math.stackexchange, but not many people even viewed ...
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votes
1
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Triangulation of the surface determined by sampling two of its cross-sections
I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...
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388
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Non-$\Sigma$ $E_n$ algebras?
Any symmetric operad can be considered as an non-$\Sigma$ operad by throwing away permutations. Does anyone know what sort of structure one gets for algebras over $C_n$ the little n-cubes operad, or ...
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votes
1
answer
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Effect of crepant resolution on the torsion in homology of a complete intersection 3-fold
Suppose I have a (2,4) complete intersection 3-fold $X\subset\mathbb P_{\mathbb C}^5$ with 118 nodal ($A_1$) singularities (if it simplifies things then assume it's the intersection of the Grassmanian ...
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