All Questions
9,056 questions
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75
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Homomorphic image of $B_{\lambda}^o(S)$ is the Brandt $\lambda^o$-extension of some monoid with zero
Let $S$ be a monoid with zero and $I_{\lambda}$ be an indexed set, then $B_{\lambda}(S) = \{ (\alpha, s , \beta ) : \alpha , \beta \in I_{\lambda}, s\in S \} \cup \{0\}$ is a semigroup and $J = \{ (\...
- 143
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101
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group actions preserve the cup product
Let $X$ be an oriented compact manifold of dimension $2k$. Suppose that a compact Lie group $G$ acts differentiably on $X$ in an orientation-preserving way. Let $B$ be the ${\mathbb R}$-bilinear form ...
- 371
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50
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Intersection of ideals corresponding to simplicial complexes at different points?
Suppose I have two simplicial complexies $\triangle_1$ and $\triangle_2$. Consider their Stanley-Reisner ideals $I(\triangle_1)$ and $I(\triangle_2)$. I want to get their intersections when they meet ...
- 143
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125
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Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual ...
- 10.5k
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472
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Exterior product in relative cohomology
Let us consider two pair of spaces $(X, A)$ and $(Y, B)$. We set $(X, A) \wedge (Y, B) := (X \times Y, (X \times B) \cup (A \times Y))$. Given a cohomology theory $h^{\bullet}$, we can define a ...
- 1,242
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134
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when is "fibering" preserved under homotopy equivalence
Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...
- 259
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139
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Homology of $S^n/G_x$
I tried to find the homology groups of the quotient of the unit sphere $S^{n-1}$ by an action of a finite subgroup $G$ of $SO(n)$. I'm especially concerned with
$$H_i(S^{n-1}/G),\quad 1\leq i\leq n-2.$...
- 303
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331
views
$G$-CW complex structure of certain G-space
Let $G$ be a finite group and $H$ be a subgroup of $G$ . Let us denote $X= G/H \ast G/H \ast \cdots \ast G/H $($k$ times).Where $\ast$ denotes the topological join operation. My question are as ...
- 35
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289
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Chern classes, vanishing of smooth sections or vanishing of holomorphic?
I have seen both definitions and this is getting me more and more confused.
Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic?
They can't be the same thing, can they?
- 363
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votes
1
answer
124
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How does $H_1$ change after projection
Suppose $X\subset \mathbb{CP}^N$ is a $n$ dimensional projective manifold (and complex dimension $n>1$), take a general projection $p\colon X\to\mathbb{CP}^{n+1}$. Suppose $H_1(X)$ is nontravial. ...
user39380
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answers
148
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There is no quasiregular diffeomorphism from punctured ball into ring (on the plane)
The idea is to use l2 cohomology as a quasiregular map invariant.
It is easy to see that there are closed 1-forms on the ring which are not exact, but it occures that every closed l2-form
$f_1(x,y)dx +...
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294
views
conjugate operation on vector bundle
Is the conjugate operation on $\overset{\sim}{K}(\mathbb{C}\mathbb{P}^n)$ known? If so, can I get the full formula at least in terms of the basis $\eta^i$? Here $\overset{\sim}{K}(X)$ denotes the ...
- 141
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139
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Relating overlapping simplicial complexes
Let $X$ be a simplicial complex and let $A,B\subset X$ be
subcomplexes such that $C=A\cap B$ is a non-empty simplicial
complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$,
$C_{\cdot}...
- 547
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answers
177
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Does the polynomial De Rham functor preserves finite cartesian products?
Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rham functor on simplicial sets.
I have the following questions
1) When we have a quasi-isomorphism between $\...
- 1,424
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502
views
Self homotopy equivalence
Given $X$, a simply connected CW-complex of finite type, ${\rm aut}(X)$ denotes the set of its self homotopy equivalences, that are maps $f: X\rightarrow X$ which admits a homotopy inverse (i.e., ${\...
- 189
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140
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Degree of Map between Pseudomanifold
There are two different ways to define a degree of map.
Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
- 424
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1
answer
284
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Creating topological spaces with portals [closed]
I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
- 19
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235
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Goodwillie tower of $\Omega^n$?
What are layers of the Goodwillie tower of the functor "n-th iterated loop space" from based spaces to based spaces? I know the answer for n=0.
- 743
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answers
373
views
Understanding a program for computing Khovanov homology
I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...
- 803
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369
views
K-theory of $\mathbb{RP}^\infty$
can anyone give some reference of K-theory and K-homology of $\mathbb{RP}^\infty$, both $K_0$ and $K_1$.
PS: also posted in stackexchange
- 17
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145
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Zero Dimension Intersection
Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class $...
- 105
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318
views
Divisibility in homology/homotopy
I have a simply-connected CW-complex $F$ of finite-type, and I know that the imprimitivity of its particular integral homology is divisible by an odd prime $p$; that is,
$$ \forall n,\exists \delta, \...
- 731
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answers
136
views
Monoid action on an uncountably infinite set
The action of a monoid on a finite set is equivalent to a finite state machine, however I would like a categorical way to think about an uncountably infinite state machine (a state transition system?)....
- 101
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votes
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answers
166
views
Where are there defined objects between gerbes and bundle gerbes?
Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism.
Does this exist in the literature?
- 3,880
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votes
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answers
187
views
Symplectic submanifolds in $\mathbb{R}^4$
Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...
- 251
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72
views
Decomposition results for locally commutative semigroups
Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
- 798
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answers
109
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Characterising singular homology among a more general class of cosimplicial spaces
Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
- 1,105
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1
answer
330
views
Rational formal and coformal space
In rational homotopy theory there are two concepts,formal and coformal spaces.I want to know example of a space which is
1)not formal and coformal,
2)rationally hyperbolic and not coformal.
- 141
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0
answers
77
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Hit problem and $\left( \mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$
I try to determine the $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ as a $\mathbb{F}_2$- vector space, in which $P_6$ is polynomial algebra $\mathbb{F}_2[x_1,x_2,\dots,x_6]$ and $A$ is Steenrod ...
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answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
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218
views
is this a simplicial model category?
A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
- 761
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148
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Extending a 2-frame field - manifolds with boundary
If $M$ is an orientable compact 3-manifold with boundary such that there is defined $s\in\Gamma(\partial M,V_2(TM))$ a section of the 2-frame bundle of $TM$, then $s$ extends to $\tilde s\in\Gamma(M,...
- 261
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0
answers
60
views
Relative homology of interlevel set
Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
H_{\...
- 181
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votes
0
answers
157
views
Postnikov system for a tree
The Postnikov system of a path-connected space X is a tower of spaces $...\longrightarrow X_{n+1}\longrightarrow X_{n}\longrightarrow ...\longrightarrow X_{1}\longrightarrow X_{0}$ with the following ...
- 933
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0
answers
498
views
Hodge isometries between K3 surfaces
I'm starting to study K3 surfaces and i have seen many examples of them as Kummer surfaces, smooth quartic in $\mathbb{P}^3$, double covering of $\mathbb{P}^2$ ramified over a smooth sextic...
But no ...
- 1
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0
answers
158
views
symbol map in algebraic K theory
I have a smooth projective morphism $X \to S$ or relative dimension 1 (i.e.
a family of smooth curves over base $S$). There should be a map $H^2(X, K_2) \to H^1(S, K_1) = Pic(S)$ given by integration ...
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votes
0
answers
148
views
Only finitely many fundamental groups in $M(n,k,v,D)$?
Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
- 689
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473
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Cohomology of bundles and local systems
I'm wondering if there are general techniques to calculate the singular cohomology groups of a fiber space (specifically, a non-smooth elliptic fibration) using methods of algebraic topology. More ...
- 9
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235
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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(...
- 73
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answers
801
views
About first Chern class and Poincaré duality in case of an ample divisor
Led $D$ be a very ample divisor in $X$ projective variety.
I can't understand why the first Chern class $c_1(\mathscr{O}_X(D))$ equals the Poincaré dual of $D$, $\mathscr{P}(D)$
- 107
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253
views
on variable and primitive cohomology of a hypersurface in a projective space
I have a smooth hypersurface D in $\mathbb{P}^n$: in many books about Hodge theory (as the ones of Voisin and Carlson) they take for granted that the primitive cohomology of D is equal the variable ...
- 107
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218
views
Does the group completion theorem apply to the James construction?
In other words, is the natural map $M \to \Omega B M$, for $M=JX$ the James construction on a space, a group completion? (By "group completion" I mean at the level of homology, I am aware of the space ...
- 213
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answers
85
views
Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius
I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
- 101
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109
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finiteness of the dimensions of cohomologies of open subsets of a compact manifold
Let $M$ be a compact differentiable manifold which can be covered by two open subsets $U$ and $V$. Then $H_{\text{dR}}^n(M)$ is finite-dimensional for all $n$. But how about $U$, $V$ and $U\cap V$? ...
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answers
82
views
Degree of sequence of mappings
If $f_n$ is a sequence of smooth orientation preserning mappings of degree one between open annuli $A(1,r):=\{x: 1< |x| < r \}$ and $A(1,r_n)$, $r>1$ and $r_n>1$, on the Euclidean space $\...
- 95
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0
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381
views
Is the cap product bilinear?
This is probably a stupid question, so I apologize in advance.
On p. 239 of Hatcher's book, he defines the cap product $C_k(X;R)\times C^l(X;R)\to C_{k-l}(X;R)$ for $k\geq l$, which he claims is $R$-...
- 1
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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
0
votes
0
answers
127
views
How to compute the Betti numbers of S-D for a surface S and a divisor D?
Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper $b_i(S-D)=...
- 1
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votes
0
answers
199
views
Finding a ribbon graph for a mapping class group action
Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is ...
- 1
0
votes
0
answers
236
views
Topological K-theory of Bohr compactification of real numbers
I am interested in the
K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers.
Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ?
More generally, what do we ...
- 357