All Questions
9,056 questions
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346
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base change for singular cohomology?
Let $X,Y,Z$ be compact complex manifolds or smooth complex projective varieties, do we have the following commutative diagram similar to flat/proper/smooth base change for quasi-coherent sheaves?
$$\...
anon
0
votes
1
answer
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
0
votes
1
answer
100
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Connectedness of the set having a fixed distance from a closed set 2
This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $\mathbb{R}^n$ ($n>1$). Suppose the complement ...
- 411
0
votes
1
answer
215
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Computation of the groups $K(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z})$
Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$
I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact ...
- 709
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1
answer
116
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On constructing free action of the cyclic group $\Bbb Z/p \Bbb Z$ on $\prod_i S^n$($n$ is odd) which is not conjugate to the usual action.
Let $ p$ be an odd prime. Can we construct a free action of the cyclic group $\Bbb Z/p\Bbb Z$ on $S^n \times \cdots \times S^n$($n$ is odd), which is not conjugate to the free action given by ...
- 61
0
votes
1
answer
183
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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
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1
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220
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Homotopy equivalence of nerves [closed]
Where I can find a proof of the following statement:
,,We have two categories: $C$ and $C'$. If they are equivalent, then geometric realizations of its nerves are homotopy equivalent"?
0
votes
1
answer
103
views
Continuous orthogonal preserving maps between projective space
Is there a continuous map $f:\mathbb{C}P^n \to \mathbb{C}P^m$, for some $n>m$
which preserve orthogonality?Namely $x\perp y \implies f(x) \perp f(y) $?
If yes, are there two non homotopic ...
- 356
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1
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668
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A possible proof of the Borsuk Ulam theorem without "Homology-Cohomology"
Assume that $n>1$.
The configuration space of $S^n$ is defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$
We have two questions:
1.Is there a continuous function $f:M_n ...
- 356
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1
answer
206
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Definition of relative Whitehead product
I can not find a definition of relative Whitehead product. Could someone explain this product to me?
- 41
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1
answer
147
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Closure of non-closed subset in Ring theory
We say that the ring $R$ is topological, when we are given neighbourhoods $\{O_{\lambda};\lambda \in \Lambda\}$ of $0_R$.
We say ${\cal I}$ is a closed ideal of $R$ if and only if for any sub-index ...
- 781
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1
answer
227
views
Example of bundle-mapping over $S^4$ with singularity $S^2$
Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E_0,E_1$ over $S^4$ and a bundle mapping
$$0\to E_0\overset{v}{\to}E_1\to0$$
such ...
- 1,915
0
votes
1
answer
192
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cohomology algebra of unordered configuration space on Euclidean space
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents):
Line 2: ... is used to compute the precise algebra ...
- 1,990
0
votes
1
answer
148
views
Unseparability of two linked rings in higher dimensions [closed]
I am not familiar with topology. We know that in $R^3$, we cannot separate two "rings": two copies of $S^1$, if they are "linked".
I wonder that is there any similar results for two copies of $S^1\...
- 9
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1
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626
views
The Jordan-Brouwer Separation Theorem
Theorem $S^{n-1}$ disconnects $S^n$ into two open connected components, which have $S^{n-1}$ as frontier.
In $R^3$, if we replace sphere of standard torus with genus $g\geq1$, we may have "The ...
- 424
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votes
2
answers
539
views
Morse matching with 0-cells and (n-1)-cells
Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.
If $P$ is connected ...
- 9
0
votes
2
answers
335
views
What does a singular simplex with real coefficient mean [closed]
For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental ...
- 689
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265
views
Are period domains ever contractible
Which simply-connected period domains are contractible?
Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety?
Are these contractible?
- 19
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1
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562
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Helped needed with some characteristic class / number questions
Suppose M is a $2n$-complex dimensional complex manifold.
a) Why is Pontryagin class independent of orientation of the bundle?
...
- 365
0
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1
answer
197
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Hurwitz's construction of simple covers
What is commonly meant by Hurwitz's construction of simple covers?
- 3,779
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1
answer
232
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Kernel elements for the Grothendieck group map of a commutative monoid
This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
- 8,512
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1
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239
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Real Pfaffian representations of real cubic surfaces
Consider the following classical construction (which is called Pfaffian representation as Sasha indicates):
Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space
of ...
- 14.3k
0
votes
1
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364
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Existence of a chain map lifting the identity; Alexander-Whitney/Eilenberg-Zilber maps
Some preliminary definitions:
Let $\Pi = \langle \alpha | \alpha^2 = 1\rangle$ be the cyclic group of order $2$ and let $\mathbb{Z}\Pi$ denote the group ring of $\Pi$ over $\mathbb{Z}$. Embed $\Pi$ ...
- 13
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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
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1
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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
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votes
2
answers
2k
views
wedge sum deck transformation
For any universal cover p of the wedge S1 V S1 is it true that the two actions of π1(X, x_0) on the fiber p^-1(x0) given by lifting loops at x0 and the action given by restricting deck transformations ...
- 11
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314
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Homology of symmetric groups
Let $S_n$ denote the symmetric group on $n$ letters, and let $S_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H_i(S_n(p))$ in $H_i(S_n)$ the $p$-primary part of $H_i(S_n)$?
- 803
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1
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219
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Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 2,837
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1
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135
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Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
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155
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Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
- 3,307
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votes
1
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94
views
Homology of independence complex after removing a vertex
Let $G$ be a chordal graph, $I(G)$ be its independence complex, and $v \in V(G)$ be a simplicial vertex (that is, $v$'s neighborhood is a clique).
Is there a way to relate the homology of $I(G)$ and ...
- 105
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1
answer
201
views
Are these two natural cohomology classes of a manifold constructed from a 1-cochain and a group extension equal?
Let $X$ be a manifold, $G$ and $A$ finite abelian groups and $\epsilon \in H^2(G,A)$ a group cohomology class (for the moment I am assuming there is no action of $G$ on $A$). Given $\alpha \in H^1(X,G)...
- 813
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1
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215
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Can this space be homotopy dominated by $\mathbb{RP}^n\times \mathbb{RP}^m$?
A space $X$ is homotopy dominated by a space $Y$ if there are continuous maps $f:X\to Y$ and $g:Y\to X$ such that $gf\simeq id_X$.
Let $P$ be a finite polyhedron such that $\pi_1 (P)= \mathbb{Z}_2$ ...
- 1,182
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1
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225
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Does contractible imply homologically locally connected?
Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?
Definition of homologically locally ...
0
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1
answer
206
views
Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?
Let $X_1$ be the suspension of $\mathbb{R}P^2$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$.
Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...
- 1,182
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1
answer
1k
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What is definition of branched covering?
What is definition of branched covering in the page 10 of following paper ?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...
- 119
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1
answer
175
views
Fourier transform for constructible sheaves on spheres
Let $S_1 = S_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p_i : S_1 \times S_2 \to S_i$ be the projection, $j : U \to S_1 \times S_2$ the inclusion of the complement of the diagonal and $...
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1
answer
309
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About Alexander method in mapping class group
The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups"
For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
- 119
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1
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154
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Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 563
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1
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49
views
More vocabulary for periodic elements in monoids
Let $M$ be a monoid, and let $x\in M$. One says that $x$ is periodic if
$$x^{i+j}=x^j$$
for some integers $i\geq 1$ and $j\geq 0$.
An easy division algorithm argument shows that if $m$ is the ...
- 18.7k
0
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1
answer
234
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Retractions, homology and multiplication
Suppose that I have 2 CW-complexes $A\subset B $ such chat thé inclusions is a retract.
Let $H_{\ast}$ be the ordinary homology with integral coefficients.
Let
$$\mu: B\times B \rightarrow B $$
be ...
- 855
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1
answer
554
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Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or ...
- 4,195
0
votes
1
answer
655
views
How to show two semigroups are isomorphic?
I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the ...
- 233
0
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1
answer
153
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C*-algebra of free monogenic inverse semigroup
Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...
- 493
0
votes
1
answer
351
views
spectral sequence for etale cohomology [closed]
I'm reading Etale cohomology. so I need to learn elementary spectral sequence for Leray spectral sequence. Please give me a reference so that I can quickly go to the main subject.
0
votes
1
answer
395
views
Topology of manifolds and transition functions
let me start by describing some examples which may well demonstrate the motivation:
A manifold is obtained by glueing together Euclidean spaces, and there is a transition function on the overlap of ...
- 1
0
votes
2
answers
219
views
If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?
Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...
- 3,751
0
votes
1
answer
262
views
Homotopy class of maps from torus to BG for a group G [closed]
I find a statement that the set of pairs of commuting elements in a group G is bijective to the set of homotopy classes of maps from torus to BG, the classifying space in the paper Elliptic cohomology ...
- 281
0
votes
1
answer
239
views
What is this type of knot algebra called within knot theory?
https://youtu.be/co78AEqsv3s?t=1901
Within this video at 31:41, Professor Ronald Brown handles knots algebraically, I am confused to how he deals with the crossing indices. He substitutes and ...
0
votes
1
answer
232
views
Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$
I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...
- 591