All Questions
9,056 questions
1
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355
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Induced model structure?
Let $H$ be a set with a binary operation $\cdot _H$ on it. To show that it is a group, one has to show that $\cdot _H$ is associative, find an identity element in $H$, and so forth; it might take ...
- 583
1
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0
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222
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Loops in CW-complexes and the 2-skeleton
Let $X$ be a path-connected CW-complex. If $\omega: [0,1] \to X$ is a loop in $X$ and $\partial$ the boundary operator in simplicial homology, then $\partial(\omega)=\omega(1)-\omega(0)=0$, i.e. $\...
- 16.2k
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365
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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 $...
1
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0
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153
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Pushout of the skeleton of homotopy colimit of diagrams
First of all let me define what homotopy colimit i'm talking about. Let F be a functor from a small category to the category of simplicial set, the homotopy colimit is the simplicial realization of ...
- 111
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0
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241
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Two-point desuspension for augmented chain complexes?
Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \...
- 19.6k
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380
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Topological definition of intersection multiplicities of algebraic varieties
I posted this question in Stack Exchange and was recommended the appendix of Fulton's Young Tableaux. While I think it's good, it'd be nice to have some books which explain this subject in more detail....
- 1,169
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170
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Definition of the $L^2$-metric for the Determinant of Cohomology of a Vector Bundle on a Riemann surface
I start describing my setup. $X$ is a Riemann surface with a metric which can have a finite number of singularities, $E$ is a vector bundle on $X$ equipped with an Hermitian structure.
In an article (...
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0
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561
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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
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0
answers
238
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Twisted homology of free products
Let $G_1$ and $G_2$ be groups and let $M$ be a vector space equipped with actions of $G_1$ and $G_2$. The free product $G_1 \ast G_2$ thus acts on $M$. How can one compute the twisted group homology ...
- 11
1
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1
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526
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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
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0
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692
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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
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0
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358
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Fundamental group of the complement of a conic-line arrangement
This is a problem concerning a lemma in Oka's paper "On the fundamental group of the complement of a reduced curve in $\mathbb{P}^2$". Let $C$ be a curve in $\mathbb{P}^2$ and $L$ be a general line ...
- 2,444
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1
answer
364
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Decomposition of simplicial G-set?
Let $G$ be a simplicial group. Let $X$ be a simplicial $G$-set,i.e. for each level, $X_n$ is a $G_n$-set.
Can $X$ be written as a union of finite (or finite type) simplicial $G$-subsets? Here "finit ...
- 681
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1k
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Does the nerve of a category have a right adjoint?
Taking the nerve of a groupoid gives a simplicial set. This is functorial $N:{\mathbf{Grpd}}\to {\mathbf{sSet}}$. NLab tells me that, in general, nerve has a left adjoint, which is geometric ...
user2529
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99
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n-reduced Eilenberg subcomplex
It is somewhat of a standard construction, that, given a simplicial set K, we can form $E_n K$, by choosing a basepoint, and picking all simplices that map their $(n-1)$-skeleton to the basepoint. If ...
- 41
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79
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Subresultants of primitive polynomials
Given two primitive polynomials $f,g\in D[x]$ over some domain $D$, is there anything we can say about the primitiveness of their $i$-th subresultant polynomials $Sres_i(f,g)$? I.e. is there a simple ...
1
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0
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267
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subset embedding gives trefoil knot [closed]
Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$.
It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that
the embedding $S^1\...
- 11
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0
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193
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How to get countably many generators for $K_{j}^{G}(\beta G)$ ??
Hey
I am trying to find out how the Baum-Connes conjecture works over $GL(1)$ over local fiels.
I am just wondering if anybody knows how to get a countable many generators for in the L.H.S of the ...
- 85
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371
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differential form of charge for pi_4(S^3) or pi_4(S^2)
How to write a 4-form of topological charge which would correspond to non-zero element of
the homotopy group $\pi_4(S^3)$ or $\pi_4(S^2)$ (both are equal to $Z_2$) ? ?
An example of such a mapping (...
- 11
1
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0
answers
3k
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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
1
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1
answer
170
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Does there exists a (possibly homological) characterization of the Jordan curve property in all dimensions?
More precisely, let $M$ be a subspace $\mathbb R^n$ with the following properties:
$M$ is a topological manifold of dimension $n-1$.
M is compact.
Does there exist a homological characterization of ...
- 3,699
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0
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1k
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Again about Bing's house with two rooms [duplicate]
Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question clearly, when "hollowing ...
- 187
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1
answer
327
views
what is this simple topological space?
Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$...
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1
answer
801
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Can every 3-manifold be triangulated? [closed]
One of my classmates was telling me that it is an open question whether every 3-manifold can be triangulated. This was rather surprising. He said that the question as far as he remember is settled ...
- 87
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2
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146
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Space with maps detected by homotopy groups in infinitely many degrees
Is there a pointed space $(X, p)$ such that for infinitely many integers $n\geq 1$ there is a map $(X, p)\to (X,p)$ inducing an automorphism other than $\mathrm{id}$ on $\pi_n(X, p)$?
In particular $\...
- 1
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votes
3
answers
548
views
Compact Hypersurfaces Bounding Compact Domains
The following statement seems to be taken as given in papers I'm reading:
Let $\mathcal{M}^n$ be a compact, embedded hypersurface in $\mathbb{R}^{n+1}$. Then $\mathcal{M}$ is the boundary of some ...
- 1,123
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1
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470
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Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
- 1,783
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1
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201
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Ambient isotopy of the diagonal submanifold in product space
Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$...
- 73
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2
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1k
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The Schottky group and the fundamental group of a compact Riemann surface
I am quoting the following description from a paper,
"...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
- 1,893
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2
answers
295
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Does the Deligne-Mumford space module $S_{n}$ action have a fundamental chain?
Does the Deligne-Mumford space (without ordering for marked points) $\bar M_{g,n}/S_{n}$ has fundamental chain in signular simplicial chains? (because I read Costello's paper GW potential to TCFT, as ...
- 285
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2
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545
views
Is $S^1=\mathbb{R}/\mathbb{Z}$ a ring? [closed]
Is there a ring structure on $S^1=\mathbb{R}/\mathbb{Z}$?
- 705
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1
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731
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Generalizing the Madsen-Weiss Theorem via the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$
The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H_*( \mathscr{C}_{\infty})\cong H_*(\Omega_0^{\infty}AG^+_{\infty,2})$ where $\Omega_0^{\infty}AG^+_{\infty,2}$ ...
- 239
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3
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450
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Cohomologies of double covers
Let $\pi:X \rightarrow Y$ be a double cover between compact manifolds $X$, $Y$ and $\theta$ be the deck transformation. Let $H^2(X, \mathbb Z)^\theta$ be a group of $\theta^*$-invariant elements in $H^...
- 1,841
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1
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364
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Proper actions and diffeomorphism groups
Since the diffeomorphism group is not locally compact; is it true that there is no proper action of an infinite-dimensional diffeomorphism group on a finite-dimensional smooth manifold?
Edit: The ...
- 111
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votes
3
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2k
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Identifying the orientation bundle uniquely
A nonorientable surface $S$ is homeomorphic to the $k$-th connected sum
$\mathbb{R}P^2 \sharp \ldots \sharp \mathbb{R}P^2$.
For each nonorientable surface $S$ there exists an oriented $2$-fold ...
- 9
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2
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1k
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Question about the fundamental group and homotopy equivalence
Let T be a two-dimensional torus and Y be the one point
compactification of a two dimensional sphere ($S^2$) minus three points.
I have to prove:
1)they have the same fundamental group
2)they are ...
- 1,727
0
votes
1
answer
327
views
Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
0
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2
answers
333
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Technical but elementary homotopy question
Not really research level but here goes anyway: Suppose I have a topological space $X$ with a closed subset $K$ and a continuous map $f : X \times [0,1) \to X$ such that:
0) $f(x,0) = x$.
1) For all ...
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1
answer
152
views
Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 481
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1
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377
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How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]
Questions.
EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...
- 6,051
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votes
1
answer
337
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Decomposition of 3-dimensional mapping torus into connected sum
In the following article
https://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf
Hatcher mentioned that there is only one prime
closed 3-manifold with infinite cyclic fundamental group, which is
$S^...
- 4,826
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votes
2
answers
240
views
Using group presentation for its corresponding semigroup?
Somewhere Colin M. Campbell noted:
If $A$ is a semigroup defined as $$A=Sg(\pi)=\langle a_1,\cdots, a_d\mid u_1=v_1,\cdots,u_e=v_e\rangle $$ then the same generators with the same relations can ...
- 233
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1
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782
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Misprint in Switzer's algebraic topology?
I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can ...
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2
answers
353
views
Realisability cohomological class as product or as immersed sphere
Let's consider closed simply connected manifold $M^n$ and a $a\in H^k(M)$ and $a*\in H^{n-k}(M)$ is the dual to $a$.
Is it true that dual to $a$ is realisable as a immersed sphere or $ a*=bc $ for ...
- 5,063
0
votes
1
answer
360
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Triviality of finite fiber bundles [closed]
Hello,
I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...
0
votes
1
answer
274
views
Universal covering of symmetric product
Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
- 153
0
votes
3
answers
400
views
Are two different definitions for Čech cohomology equivalent?
In Spanier's book Algebraic Topology (Chapter 6 section 7) he defines Čech cohomology in terms of the nerves of open coverings.
I wish to know if this is equivalent, for a topological space A closed ...
0
votes
1
answer
286
views
Künneth formula and induced map in homologies
Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H_{*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus_{p+q=*}(H_{p}(X; \...
- 369
0
votes
1
answer
165
views
How many n-dimensional closed submanifolds of $R^n$ have Euler characteristic 1?
It is well-known that the closed $n$-ball has Euler characteristic $1$. Is it true that every closed (i.e., compact), connected $n$-dimensional submanifold (with boundary) of $\mathbb R^n$ having ...
0
votes
1
answer
509
views
How do Schubert classes form a basis for $H^{*}(Gr(k, n))$?
I've gone through many texts in algebraic geometry, specifically, Schubert calculus. They all claim that the Schubert classes $[\Omega_{\lambda}]$ form a basis for the cohomology ring of the complex ...
- 349