All Questions
2,364 questions with no upvoted or accepted answers
1
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151
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Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper,
he wrote in eq.2.22 that
Atiyah Patodi Singer theorem says that the combination:
$$
\frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi}
$$
is a ...
- 447
1
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0
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119
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A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
- 1,367
1
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0
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107
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A question about the Conner Conjecture
In some sources, Conner conjecture is expressed as follows:
Theorem [Conner Conjecture] Let $G$ be a compact Lie group, and let $X$ have
the homotopy type of a finite dimensional $G$-CW complex with ...
- 1,367
1
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0
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145
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Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
- 447
1
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0
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240
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Examples of when $X$ is homotopy equivalent to $X\times X$
I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
1
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0
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90
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Cohomological dimension of the kernel of a homomorphism induced by a singular fibration
I have a very concrete question. Let $N={\Bbb C}-\{\pm 1\}$, and ${\Bbb Z}_2$ is acting on $N$ by rotation around $0$. Consider $M=\{(x,y)\in N^2\ |\ {\Bbb Z}_2x\neq {\Bbb Z}_2y\}$. Let $p:M\to N$ be ...
- 585
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155
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Lifting action of torus to torus bundle
Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...
- 205
1
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0
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161
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Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 593
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64
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Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
1
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0
answers
131
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Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
1
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0
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226
views
Cohomology in families of normal varieties
Let $f : X \to Y$ be a flat proper morphism of complex varieties whose fibers are normal varieties. Is it true that $\mathrm{dim}_{\mathbb{Q}} H^i(X_t, \mathbb{Q})$ is constant?
For non-normal fibers, ...
- 3,730
1
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181
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Non-trivial homotopy, but vanishing homology
I wonder if there are examples of 5-dimensional manifolds with vanishing integral second homology group, but non-vanishing second homotopy group? Or is it impossible by some Hurewicz theorem type of ...
- 129
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0
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57
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Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
- 1,286
1
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0
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380
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G-local systems via the classifying stack BG
First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...
user501319
1
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0
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54
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The number of $n$-cells attaching to $K^{n-1}$ in Wall's construction
Let $\phi:K\to X$ be a map, with mapping cyliner $M=X\cup_{\phi}(K\times I)$. We define $\pi_n (f)$ as $\pi_n (M,K\times 1)$. An element of $\pi_n (f)$ is represented by a pair of maps $\beta :S^{n-1}\...
- 287
1
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0
answers
37
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Unique smallest degree algebraic solution to polynomial ODE
Let's assume we are given a degree $d$ polynomial VF as a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$
$$f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a^j_{i_{1},\dots,i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$$...
- 247
1
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0
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102
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Continuity of the 1-jet prolongation map $\text{imm}(M,N)\to \text{fimm}(M,N)$
I am studying the article Immersion Theory for Homotopy theorists by Michael Weiss for my bachelor thesis. The main theorem states that the space of immersions and formal immersions between two smooth ...
- 11
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0
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146
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Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups
Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
- 813
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0
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185
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Proving a Result About Pontryagin Numbers Without Forms
I've been reading the book Geometry of Differential Forms by Shigeyuki Morita, and I came across the following theorem on page 226 the other day:
Proposition 5.53 (Pontryagin). Two cobordant closed (...
- 377
1
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0
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96
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Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic
I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic.
I have come up with ...
- 1,805
1
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0
answers
182
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Does this sequence stop?
Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
- 1,182
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131
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Definition of Cartan Model - Equivariant differential forms
I would like to ask about an equivalence between two definitions for the Cartan Model.
Let $G$ be a connected Lie group, let $\mathfrak{g}$ be its Lie algebra and let $M$ be a $G$-manifold. The Cartan ...
- 11
1
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0
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123
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Relation of branched covers and groups
I am self-studying covering spaces of topological spaces. The following question comes to my mind.
In the case of topological covering spaces, we have a nice relation between the fundamental group of ...
- 613
1
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0
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41
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How to embed an arbitrary graph into (k,d)-kautz space (like multidimensional scaling of non-normed space)? See details in the following
How to embed an arbitrary graph into (k,d)-kautz space (like multidimensional scaling of non-normed space)? See details in the following.
Given a graph $G = \{V,E\}$,
we have a distance matrix (the ...
- 11
1
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0
answers
198
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topological functor of tor functor
The framework of Quillen's model categories gives us a very general way of defining things as derived functors. For instance, in this way one can realise the singular homology as Andre-Quillen ...
- 449
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0
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119
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Equivariant cohomology with compact support of the generalized Jacobian of a nodal elliptic curve
[Question forwarded from SE for lack of interaction]
Given an geometric genus $1$ curve with $1$ node $C$ (hence with arithmetic genus $2$), its normalization is given by the elliptic curve $\tilde{C}$...
- 33
1
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0
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137
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Is $\pi_m(M) = 0$ if $\pi_m(M-X) = 0$ for a low-dimensional subset $X$?
I am doing a problem where I am stuck at this point.
Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n&...
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0
answers
126
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Blow up of simply connected isolated singularity
Let $X \subset \mathbb{C}^n$ be a simply connected complex variety with a unique isolated singularity at $x\in X$.
Let $\tilde{X}$ be the strict transform of $X$ under the blow-up $\mathrm{Bl}_x(\...
1
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0
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293
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Does Thom-Milnor theorem provide a bound for the number of connected components of a zero set of a polynomial system?
I have a system $$f_1(x_1,\dots,x_m) = 0,...,f_p(x_1,\dots,x_m) = 0,$$
where $f_1,\dots,f_p$ are real polynomials and $x_1,\dots,x_m$ are real independent variables. I'm interested in an upper bound ...
- 1,058
1
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0
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128
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Singular or cellular homology with $L^2$ coefficients
There are a few cases that $L^2$-homology (cohomology) that can be introduced. For example, for a manifold, it can be defined in the same way as de Rham cohomology using square integrable differential ...
- 161
1
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0
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84
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Equivariant spectrum with coefficients
I am curious to know whether spectra with coefficients as defined in Adams's Blue book be defined to an equivariant setting. In the non-equivariant case, for a spectrum $E$ and an abelian group $A$, ...
- 19
1
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0
answers
105
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The Dold-Lashof construction and the Milnor filtration
This question regards Section 3.5 of the book Secondary Cohomology Operations by John R. Harper.
In Subsection 3.5.4, the author considers a surjection $f: X\times Y\to X$ of topological spaces, and ...
- 935
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0
answers
190
views
the Brouwer fixed point theorem for maps rather than spaces
Is there a version for the Brouwer fixed point theorem for maps rather than spaces ?
In other words, for a family of endomorphisms, can the fixed point be chosen continuously, under some assumptions ?
...
- 513
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0
answers
87
views
Explicit form of boundary operators of topological cones
Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$.
For many problems in computational geometry, a key operation is to ...
- 840
1
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0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
1
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0
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184
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The key step in Serre's method on higher homotopy groups
Let $n \geq 2$ and $X$ be a $(n-1)$-connected simplicial complex. This means that all of the lower homotopy groups $\pi_{k}(X) = 0$ for $k \leq n-1$. My goal is to compute the higher homotopy groups ...
- 5,230
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97
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about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
1
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0
answers
114
views
Moore space over a group with infinite generator
I am not an expert on this topic. I am trying to learn about Moore spaces of type $(G,n)$. where $G$ is abelian and $n\geq 2$.
Let $M$ be a simply connected non-compact $4$-manifold with $H_2(M;\...
- 179
1
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0
answers
109
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Positive $(p,p)$-current and subvariety
Let $X$ be a complex manifold. Any dimension $p$ subvariety(or reduced analytic subscheme) will determine a real closed positive $(p,p)$-current. But the converse is not true.
My question:
Is there a ...
- 361
1
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0
answers
78
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Diffeomorphism induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{...
- 521
1
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0
answers
183
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Second homology group of a presentation complex
I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly.
Given a finite group $G$, and a presentation $P$ of ...
- 179
1
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0
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251
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Local cohomology: Polynomial ring vs Power series ring
I study algebraic topology and am currently examining the applications of local (co)homology in algebraic topology. We have the canonical inclusion of rings $\mathbb{Z}[x_1,\cdots,x_n]\subset \mathbb{...
user340793
1
vote
0
answers
98
views
Fundamental group of intersection of two codimension=2 complements
$X$ is a smooth manifold. $Y_1\subset X$ and $Y_2\subset X$ are both complements of a finite collection of real codimension=2 (transversal intersecting) submanifolds. Suppose $Y:=Y_1 \cap Y_2$ is ...
- 31
1
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0
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88
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Tensoring by a soft flat sheaf
Let $X$ be a paracompact Hausdorff space and $R$ a commutative ring. All sheaves below will be sheaves of $R$-modules on $X$. A sheaf $S$ is soft if every section of $S$ over a closed subset can be ...
- 23.5k
1
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0
answers
122
views
Finitely presented homology group
Given a finitely presented $i$th singular homology group over $\mathbb Z$ of a topological space $X$. If one knows the family of $i$th singular homology groups of $X$ over all possible fields, can one ...
1
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0
answers
186
views
How to calculate the periodic cyclic homology group of $\overline{\mathbb{Z}}/\mathbb{Z}$
$\newcommand{\ur}{\mathrm{ur}}$Fix a prime number $p$. We let $\overline{\mathbb{Z}}$ denote the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$ and $\overline{\mathbb{Z}}_p$ denote the ...
- 519
1
vote
0
answers
173
views
The geometry of a commutative ring and the topology of its ideal complex
Suppose $R$ is a commutative Noetherian ring. Let $\mathcal{P}(R)$ be the poset of ideals of $R$ ordered by inclusion, and let $\Delta(R)$ be the order complex of $\mathcal{P}(R)$. $\Delta(R)$ is a ...
- 19
1
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0
answers
82
views
Some details about a proposition of Wall
Let $X$ be a connected CW-complex. For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi}...
- 1,182
1
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0
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254
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A question on the Chow group on stacks
Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows.
Let $\...
- 565
1
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0
answers
93
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Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805