All Questions
9,056 questions
1
vote
1
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414
views
Equivariant maps inducing isomorphism in integral cohomology
Consider the following statement.
Suppose $X$, $Y$ are finite CW-complexes with free involution
and $\mu:X\to Y$ is an equivariant map.
If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an ...
- 11
2
votes
1
answer
215
views
Second cohomology group with finite coefficients of the product of two varieties
This is surely well known. Let $X$ and $Y$ be smooth, projective, connected complex varieties. Then
$$H^2(X\times Y,Z/n)=H^2(X,Z/n)\oplus H^2(Y,Z/n) \oplus (H^1(X,Z/n)\otimes H^1(Y,Z/n))$$
for any $n&...
3
votes
1
answer
214
views
Is there a common general setup for both Weil cohomologies and generalized cohomology theories?
My question can be simply (and loosely) stated as follows:
Is there a general (but not too general) construction that captures, as specializations, both Weil cohomologies in algebraic geometry and ...
- 23.3k
0
votes
1
answer
162
views
3-manifolds, cubes with handles
Somebody knows where I can find some proof of the following fact:
If F is compact, connected 2-manifold with nonempty boundery why there exist n=1-X(F) pairwise disjoint properly embedded 1-cells {A1,....
1
vote
0
answers
238
views
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
10
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0
answers
735
views
Adams Spectral Sequence for Equivariant Cohomology Theories
In ordinary algebraic topology the Adams spectral sequence can be applied for any cohomology theory $E$ and in good cases it converges to the stable homotopy classes of maps (of the E-nilpotent ...
- 1,273
2
votes
1
answer
200
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Homotopy type of the simplicial action groupoid
Let $X$ be a simplicial $G$-set, where $G$ is a simplicial group. What is the homotopy type of the simplicial action groupoid $X//G$?
user2529
0
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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 ...
4
votes
0
answers
495
views
Spectral sequence for cohomology of open subset
Let $X$ be a smooth compact orientable manifold (or variety) and let $j: U \subset X$ be the complement to a union $\bigcup_{i \in I} X_i$ of smooth compact orientable manifolds. Suppose that $A$ is a ...
1
vote
0
answers
121
views
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
11
votes
0
answers
561
views
How to get a Dehn-twist presentation of a periodic map of a Riemann surface?
Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ).
A classical result says such $f$ is ...
- 471
0
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
2
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0
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337
views
simplicial deRham complex and model category structure
To every simplicial manifold is associated its simplicial deRham complex.
Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
6
votes
0
answers
367
views
Does every exact six-term sequence arise as the K-theory of a locally compact pair?
Consider six countable Abelian groups and six group homomorphims as in the following diagram
G → H → I
↑ ↓
L ← K ← J
Assume that the resulting sequence is exact ...
- 3,184
5
votes
0
answers
200
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Whitehead products and a realization problem for graded Lie algebras
Many $\mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ we would like to study are non-degenerate in the sense that
$\dim_{\mathbb{C}} \mathfrak{g}_n < \infty \ \forall n \in \...
- 952
3
votes
1
answer
424
views
Principal bundle for contractible group is weak homotopy equivalence for ind schemes
This is may be obvious, but I am not comfortable with ind-schemes.
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
- 156k
6
votes
0
answers
510
views
The Mapping Cylinder of a Pullback Square
Suppose I have a pullback square, which I think of as a map from the fibration
$q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$
from the mapping cylinder $M$ of $X\...
- 12.5k
0
votes
1
answer
141
views
Known graph/surface invariants that can be extracted from homology over different fields
The $Z_2$-homology of a surface viewed as a simplicial complex allows us to extract interesting invariants from the resulting homology groups. $\beta_0$ is the number of connected components, $\beta_1$...
- 4,462
5
votes
0
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323
views
Vector bundles of schemes and their topological realizations
Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$.
Does $R_\mathbb{R}$ send an ...
- 103
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
- 1,882
1
vote
0
answers
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
1
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0
answers
170
views
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 (...
5
votes
0
answers
517
views
A smooth twisted tensor product of dg algebras?
I want to consider a Z/2Z dg algebra. As an algebra, it is generated over $\mathbb{Q}$ by two elements where x is even and e is odd with the relations $xe=ex$ and $e^2=1$(this makes it in particular ...
- 3,758
2
votes
1
answer
533
views
Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?
Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, \mathbb{Z})$,...
- 77
7
votes
0
answers
80
views
Explicit expression of WZ term for orthogonal groups
Consider the Wess Zumino term on the the space $W=I\times D$, where D is a two dimensional disk disk and $I$ is an interval, $[0,1]$, say, i.e.,
$$
\int_{I\times D} \langle(u^{-1} \, du)^3\rangle
$$
...
2
votes
1
answer
198
views
Reference for an automorphism in a paper of Toda
In Selick's very pretty paper "Odd primary torsion in $\pi_*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".
Unfortunately, Selick ...
- 12.5k
3
votes
1
answer
292
views
Cartesian-closed category of spaces with the Whitehead property?
I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes). ...
- 19.6k
2
votes
0
answers
512
views
Lifting criteria of covering space by using homology condition
Let $\pi\colon\tilde{X}\to X$ be a p-fold (regular) cyclic covering(p:prime) and $\mathcal{A} = \mathrm{Im}(\pi_* )$, where $\pi_* \colon H_1(\tilde{X};\mathbb{Z}_p) \to H_1(X;\mathbb{Z}_p)$ is ...
- 698
7
votes
1
answer
282
views
Can you construct a mapping space from local data? (looking for reference)
I'd to know if/where there is a reference for the following construction.
Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
- 12.8k
1
vote
1
answer
156
views
Sufficient Conditions for Free Indecomposability
An interesting fact was relayed to me in another question of mine that
If $M$ is any closed manifold with universal cover homeomorphic to $R^n$ for $n>1$ then $\pi_1(M)$ is freely ...
- 726
10
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0
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362
views
Question about $A_\infty$ maps
Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping ...
- 18.8k
1
vote
1
answer
72
views
Transformation terminology question
Given a transformation $t$ from the transformation semigroup $T_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a ...
1
vote
0
answers
153
views
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
2
votes
1
answer
378
views
How can I show that the map L-->K(\pi_n(L),n) representing the fundamental class of an (n-1)-connected space is an isomorphism on \pi_n?
As an exercise, I'm trying to show that for an $(n-1)$-connected space $L$ with $\pi=\pi_n(L)$, the map $\iota_L:L\rightarrow K(\pi,n)$ associated to the fundamental class $\iota_L\in H^n(L;\pi)$ ...
- 6,069
1
vote
1
answer
304
views
good perspective in viewing manifolds of infinite dimension
Borel conjectued aspherical closed manifolds are topologically rigid.(i.e.a homotopy equivalence between two aspherical manifolds is homotopic to a homeomorphism).
now,soppuse M is a K(G,1) space,
it ...
- 179
2
votes
0
answers
486
views
Casson Gordon paper - Cobordism of classical knots
It is given in Progress in mathematics 62, Guillou and Marin book. In the proof of Lemma 4, They choose $\alpha$ and $r\in \mathbb{N}$ such that $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfies $h^r_*(...
19
votes
0
answers
504
views
Other examples of computations using transfer of structure from the chains to the homology?
There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
- 3,880
0
votes
0
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
1
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0
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267
views
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
2
votes
0
answers
152
views
Does the ordinary cokernel respect weak homotopy equivalences?
Let $M_1\subset M_2$ and $K_1\subset K_2$ be inclusions of simplicial monoids. If there is a weak equivalence $f:M_2\to K_2$ that restricts to a weak equivalence $f|_{M_1}:M_1\to K_1$, does this ...
- 21
2
votes
1
answer
349
views
Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces
Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.
Question: How do you prove that the following diagram of homotopy groups commutes?:
$\pi_n(Y) \to \pi_{n-1}(\...
- 155
3
votes
0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
1
vote
0
answers
371
views
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
3
votes
1
answer
265
views
Equivariant Surgery problem
I have a question about surgery.
Let $G= \mathbb{Z}_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.(...
- 698
0
votes
0
answers
198
views
Euler characteristic of a subset of cartesian product induced by a group action
let $X$ be a CW-complex on which a finite group $G$ acts.
define
$$F=\{ (x,gx)\;|\; x\in X ,g \in G \}$$
i want to compute the Euler characteristic of $F$. I wrote $$F=\cup_{g\in G}{F_g}\;\;,\;\; ...
- 1
5
votes
1
answer
283
views
how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
- 44.7k
6
votes
0
answers
360
views
The Space of Cellular Maps
Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
- 12.5k
0
votes
1
answer
314
views
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
12
votes
0
answers
440
views
K-Weil cohomology theories?
I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard ...
- 21k
6
votes
0
answers
312
views
homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence
Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
- 5,575