All Questions
9,056 questions
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103
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Lower periodic subsets of groups and semigroups
Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower $B$-...
- 995
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0
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137
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(The Homotopy type of the) lifting of homeomorphism of Grassmanian
For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space ...
- 356
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275
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Explicitly showing that a free group is LERF [closed]
Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...
- 11.3k
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372
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Role of determinant of the matrix corresponding to $i$-th Homology group.
I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
- 63
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256
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Is this a "new" terminology in homology/cohomology theory?
I have the following question. For our research purpose, we have introduced the following concept:
Let $f:X\to Y$ be a continuous, disrecte and open mapping between two locally compact metric spaces. ...
- 1,881
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143
views
on reductive monoids which are gorenstein
Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore http://www.cmat.edu.uy/cmat/docentes/alvaro/...
- 3,472
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304
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generalisation of the universal coefficient spectral sequence
Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective modules....
- 71
1
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136
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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
1
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0
answers
403
views
Functors with Mayer-Vietoris Sequences
Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...
- 9,853
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197
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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
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0
answers
45
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Equivalence between the definitions of Reidemeister Coincidence number
Given two applications $f$ and $g$, denote by $R (f, g)$ the set of Reidemeister classes determined by $f$ and $g$ (according to the algebraic definition, on the induced on fundamental groups). And $\...
- 29
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99
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PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
- 1,373
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75
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Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
- 111
1
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1
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151
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A formula for isotropy group $\pi_1(G_a)$
Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...
user21574
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200
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When is equivariant cohomology generated by equivariant Euler classes?
Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map $$H_T^*(...
- 4,920
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193
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Homology of a higher-dimensional full torus with a certain disk removed
Let $X=S^2\times B^5$ and let us consider an embedding of an
open $5$-disk in $X$ with its boundary
glued by a map of degree $2$ to some $\{
s_0\} \times S^{4}\subseteq \partial X$ . The embedding ...
- 19
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0
answers
284
views
Metalinear frame bundle on sphere or $\mathbb{C}P^n$
Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map $f:...
user21574
1
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0
answers
46
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spherical map of fixed points?
Let $B = \{\, x \in \Re^m : \|x\| \le 2 \,\}$, and let $f : B \to B$ be a continuous function whose set of fixed points is $S^k = \{\, x \in B : \|x\| = 1, x_{k+2} = \cdots = x_m = 0 \,\}$. Can it ...
1
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0
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187
views
Gysin sequence for $\mathbb S^3$ bundle
Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following ...
- 119
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158
views
Is the $K$-Theory of $\mathbb{P}X$ a $\lambda$-Ring?
If we let $\mathbb{P}X$ denote the free commutative algebra generated by the spectrum $X$, then we have a weak equivalence
$$
\mathbb{P}X\simeq \bigvee_r E\Sigma_{r+}\wedge_{\Sigma_r}X^{\wedge r}.
$$
...
- 535
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0
answers
230
views
How do people deal with different cohomology theories [closed]
After some time spent on it I now have some understanding of de Rham cohomology and can actually calculate some cohomology groups. However I have now and then encountered many other cohomology ...
- 639
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0
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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
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266
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What invariants are of great concern in the field of 3-manifolds and why? How much do we know about them? [closed]
I am curious about 3-manifolds though I know little.
Here I am trying to know what invariants people in this field are interested in.
The following are what I have known and what I particularly want ...
- 121
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0
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145
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Does compactly supported cohomology make sense for cosimplicial spaces?
As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...
- 1,162
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answers
137
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free action on mod p cohomology sphere
It is well known that the group $G=\mathbb Z_2\oplus\mathbb Z_2$ cannot act freely on mod 2 cohomology n-sphere.
Is it also true that this group $G$ cannot act freely on any mod p cohomology n-...
- 119
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211
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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
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447
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Lefschetz duality for manifolds with boundary / stratified spaces
Let $M$ be a manifold with corners. Let $F_p$ denote the union of all the codimension $i \geq p$ faces of $M$.
Then I have read that there is a form of Lefschetz duality that says that there is an ...
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0
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135
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Number of generators of the fundamental group of a Riemannian manifold with Ricci curvature bounded below
Is there a constant $C(n,D)$ such that for any closed Riemannian manifold $M$ with $Ric \ge -(n - 1)$ and $\mathrm{diam} \le D$, the fundamental group $\pi_1(M)$ is generated by at most $C(n,D)$ ...
- 689
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155
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Krull dimension in equivariant cohomology
Let a compact Lie group $G$ act on a manifold $X$. Let $\mathbb Q$ be the field of rational
numbers and assume that cohomology $H^{\ast} (X)$ with $\mathbb Q$-coeffiecients is
finite-dimensional. ...
1
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0
answers
133
views
equivariant singular homology
Let $M$ be a smooth $G-$manifold, where $G$ is a compact Lie group. From the result of S.Illman that $M$ admits an equivariant triangulation. Moreover, we can construct the equivariant singular ...
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0
answers
81
views
Reference request: Compact metrizable semilattices with small connected semilattices are absolute retracts
Let $X$ be a compact metrizable topological semilattice with a neighbourhood base of sub-semilattices that are path-connected. I need a reference for a proof that $X$ is an absolute retract.
Here is ...
- 900
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837
views
State of the game : cohomology of principal bundles
I would like to know what has been done in terms of specific calculations for the cohomology of principal bundle. For instance, it is known (see e.g. Greub, Halperin and Vanstone's "Connection, ...
- 502
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0
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357
views
Homology of the dg-nerve vs Hochschild homology of the dg-category
Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the homology of the ...
- 187
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0
answers
220
views
Change the fiber of a fibration
Let $F \rightarrow E \rightarrow B$ be a (Serre) fibration of topological spaces. Given a map $F' \rightarrow F$, is there a criterion for the existence or even an explicit construction of a fibration ...
- 2,040
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0
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367
views
Applications of Global Torelli theorem for K3 surfaces
Hi, i've just studied the weak global Torelli theorem for K3, which states that, given two K3 surfaces $X$ and $Y$, they are isomorphic if and only there is an Hodge isometry between $H^2(X,\mathbb{Z})...
- 11
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0
answers
786
views
Pontryagin Product on the Homology of $CP^{\infty}$
Is there an explicit description of the Pontryagin product on the homology of $CP^{\infty}$? Also, what is the homology of the classifying spaces $BU(n)$?
- 935
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0
answers
138
views
G-graphs and Cayley graphs
with which kinds of group we can make a G-graph(Bretto 2011) which are hamiltonian Cayley graph?
1
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663
views
Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space
Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$?
I know that such a bundle must ...
- 827
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515
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Two technical questions about cohomology theories
1) Milnor proved that singular cohomology is the unique additive cohomology theory satisfying the dimension axiom on the category CW of pairs (X,A) such that X and A have the homotopy type of a CW-...
- 1,242
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0
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190
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Homotopy colimits over a certain subset category.
Hi!
Let $I$ denote all the finite subsets of some set (infinite or finite) S. For each n, let $I_n$ be the subset consisting of objects of cardinality n, so that there are no morphisms between the ...
- 1,071
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0
answers
256
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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 ...
- 933
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290
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Topological invariance of intersection number
Let $X$ and $Y$ be algebraic curves in ${\mathbb P}^2$ and suppose they have an isolated intersection at $(0,0)$. Let $\hat{X}$ and $\hat{Y}$ be another such pair, and suppose that there exist ...
- 11
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320
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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
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112
views
When are graphs of cohomologically complete groups cohomologically complete?
A group $G$ is cohomologically $p$- complete if the canonical map from $G$ to it's pro$-p$ completion $\hat G^p$ induces an isomorphism on cohomology $H^\ast_{cont}(\hat G^p, \mathbb{Z}_p) \rightarrow ...
- 11
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297
views
Simple proof of an isomorphism theorem
I haven't references about a proof of this theorem:
Let $p: Y \rightarrow X$ be a fiber bundle. If for all $x \in X$ $p^{-1}(x)$ satisfies $H^{\ast}(p^{-1}(x)) \simeq \mathbb{R} $, then $p$ induces an ...
- 21
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0
answers
121
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Existence of open dense subset in a Lie group
Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group
and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
- 645
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0
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101
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How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
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289
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The homotopy colimit of a tower of triangles
Set the framework to be a triangulated category with all set indexed coproducts.
In "Relative Homological Algebra and Purity in Triangulated Categories", J. of Algebra 227, (2000), pp. 268- 361, (...
- 666
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0
answers
366
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Question on Steenrod realizability problem
René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact ...
- 257
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0
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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