All Questions
9,056 questions
0
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307
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A modified version of K-theory for manifolds ?
If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
- 505
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
0
votes
0
answers
292
views
Is $GL_{S^1}(H)$ (for $H$ Hilbert space ) path-connected?
Due to the negative answer to my last question I want to know if at least the following is true:
Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let $\text{Gl}_{S^...
- 1,282
0
votes
0
answers
185
views
the boundary homomorphism $[\Sigma S^{n-1},X]\to[S^n,X]$ is identity?
Given a Puppe sequence $\cdots \to S^{n-1} \to Y \to S^n(\simeq Y/S^{n-1}) \to \Sigma S^{n-1} \to \cdots$, where $Y=S^{n-1}\cup_{2\iota} e^n$ where $\iota:S^{n-1}\to S^{n-1}$ is identity,
we have a ...
- 699
0
votes
0
answers
179
views
semigroup actions of groups on regular rooted trees
If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we ...
- 125
0
votes
0
answers
850
views
Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
- 1
0
votes
0
answers
142
views
Homomorphism between the set of n-flats in $R^m$ to some manifold
I am unaware of what the formal definition of "limit" is for a sequence of flats, but for the purpose of this question her is the definition of limit that I am using:
Consider a sequence $s_1, s_2, ...
- 435
0
votes
0
answers
2k
views
Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
- 24.1k
-1
votes
2
answers
814
views
Definition for fundamental group (higher homotopy groups) for a category?
How to define homotopy groups in categories as in Quillen's definition for Higher algebraic K-theory: K_i(M)=\pi_{i+1}(BQM, 0), where M is a small category and BQM is the classifying space of QM. ...
- 39
-1
votes
1
answer
209
views
A closed manifold with a subset with the same ring cohomology
Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question $...
- 356
-1
votes
1
answer
178
views
If $H_i(U_j)=0$ for infinitely many $j$ then $H_i(X)=0$ [closed]
Let $X$ be a topological space and $U_i$ open subsets. If $U_i\subset U_{i+1}$ and $\bigcup^{\infty}_{i=1}U_i=X$. How can I prove that if for infinitely many $j$, the $i$-th homology vanishes $H_i(U_j)...
- 27
-1
votes
1
answer
506
views
loop homology product for oriented compact manifolds with boundary
This is my first steeps in string topology and please forgive the basic level of my questions: I reformulate my question
Chas and Sullivan define the loop homology product for closed (=compact with ...
- 189
-1
votes
1
answer
307
views
Are there results about the group of homeomorphisms of $(T^2-\{*,*\})$ up to isotopy?
I am studying a fiber bundle over circle with fiber $T^2-\{*,*\}$.
Since this is a mapping torus, the group $Homeo(T^2-\{*,*\})/isotopy$ plays an important role.
Are there some existing theorems on ...
- 157
-1
votes
2
answers
1k
views
Regarding understanding differential geometry [closed]
I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics ...
user39066
-1
votes
1
answer
163
views
Alternate property of H^2(T, Z) [closed]
Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
- 563
-1
votes
1
answer
1k
views
Fibre bundles and flat connections [closed]
If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat ...
- 2,818
-1
votes
1
answer
1k
views
Covering maps on Euclidean spaces and spheres [closed]
Hello. I have two questions.
Does there exist an exactly 2-fold covering map
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ ?
Does there exist an exactly 2-fold covering map
$g:S^{n}\rightarrow S^{n}$ ?
...
-1
votes
2
answers
260
views
Function space and contractibility
$\DeclareMathOperator\map{map}$I have the following question:
Let $X$ and $Y$ be topological spaces. Let $\map(X,Y)$ denote the space of non-constant continuous functions from $X$ to $Y$. Suppose ...
- 315
-1
votes
2
answers
527
views
Directed colimit and homology
I am looking for a reference or a proof of the following fact:
Let $X_{1}\subset X_{2}\subset\dots $ be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each $X_{i}\...
- 511
-1
votes
1
answer
263
views
Is $X$ homeomorphic to $S^1 \times Y$? [closed]
Is it true that when the first fundamental group of a topological space $X$ is isomorphic to $\mathbb{Z}$ then $X$ is homeomorphic to $S^1 \times Y$ where the first fundamental group of $Y$ is ...
- 1,661
-1
votes
1
answer
861
views
Essential simple closed curves in a torus [closed]
Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$.
We will usually identify a closed curve with its image in $S$. A closed curve
is called essential if it is not ...
- 119
-1
votes
1
answer
137
views
Manifold for which you need to specify the action on cohomology in each degree
Let $M$ be a connected closed manifold of dimension $n\geq 2$.
Can it happen that for any $I\subsetneq I_n=\{1, \dots, n\}$ there are continuous maps $f, g:M\to M$ such that $f^*=g^*|_{\oplus_{i\in I}...
- 21
-1
votes
1
answer
339
views
A condition for Artinian topological spaces [closed]
A topological space $X$ is called Artinian if the descending chain condition holds for open subsets of $X$. If the descending chain condition holds for open basis subsets of a Hausdorff space $X$ with ...
- 13
-1
votes
1
answer
262
views
Question related to Galois covering of Projective line over rational numbers
Suppose that we have Galois covering $$X\longrightarrow P^{1}_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group....
- 591
-1
votes
1
answer
110
views
Variety of commutative semi group [closed]
V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.
- 155
-1
votes
1
answer
216
views
Poset complex of reverse ordering [closed]
This might be too easy but I cannot proof it easily. Any reference or hint will be great.
Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. ...
- 1,713
-1
votes
1
answer
720
views
What are Fake Weighted Projective Spaces? [closed]
What is the origin of and motivation for the notion fake weighted projective spaces? Could you please compare this notion to that of genuine weighted projective spaces, giving significant examples? Is ...
- 1,422
-1
votes
1
answer
787
views
How to combine correlated signals !? [closed]
Hi everybody
There are 11 signals:
S_main : The original signal
S1 ~ S10 : 10 signals that are correlated to S_main with different correlation coefficients (coeff1 ~ coeff10)
Now here's the ...
- 1
-1
votes
1
answer
499
views
How does a chain map induce another chain map on an isomorphic chain complex?
I have 2 ways of defining a chain complex on a manifold, one of which is the cellular chain complex, $C^{CW}_*$. I know that a cellular map $f: X^n \rightarrow Y^n $ such that $ f(X^n) \subset Y^n $ ...
- 1
-1
votes
1
answer
421
views
How does the discrete group act on simplicial set level by level
Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set?
For example, if ...
- 681
-2
votes
1
answer
516
views
no classification of nilpotent lie groups
there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix:
$$ \left(
\...
- 22.8k
-2
votes
1
answer
314
views
configuration space and iterated loop space
Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...
- 1,990
-2
votes
1
answer
438
views
Relation between $\mathbb{Z}\pi_1 (X)$-module $\pi_n (X)$ and $\mathbb{Z}$-module $H_n (X)$ or $\mathbb{Z}\pi_1 (X)$-module $H_n (\tilde{X})$
Let $X$ be a finite CW-complex of $n$.
For $i\geq 2$, $\pi_i (X)$ is a $\mathbb{Z}\pi_1 (X)$-module.
for $i\geq 2$, $H_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi_1 (X)$-module, where $\...
- 1,182
-2
votes
1
answer
271
views
Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$
I recently came across the following statement,
The Galois coverings of $\mathbb{P}^1_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$$ where $L$ is a number field.
How ...
- 591
-2
votes
1
answer
189
views
Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
- 521
-2
votes
1
answer
780
views
commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 1
-2
votes
1
answer
1k
views
Component and quasi-component
Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$...
- 1
-2
votes
1
answer
1k
views
Degree of a rational function [closed]
I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
- 2,091
-2
votes
1
answer
89
views
Alternating property of H_2(T, Z)
Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...
- 563
-2
votes
1
answer
215
views
Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]
I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...
- 591
-2
votes
1
answer
413
views
Covering space theory, category theory [closed]
Requiring covering spaces of a well-behaved connected topological space $X$ to be connected, let $\mathcal{Cov}(X)$ be the category of covering spaces of $X$ and maps over $X$ and maps over $X$. Can ...
-2
votes
1
answer
513
views
Sheaf cohomology and double covers
Let $\pi:X\rightarrow Y$ be a double cover of complex varieties and take $L$ holomorphic line bundle on $Y$.
I read that there are the isomorphisms
1) $H^p(X,\mathcal{O}_X)\simeq H^p(Y,\pi_*\mathcal{...
- 175
-3
votes
1
answer
215
views
Topology of the moduli space of a 2-dim closed surface
Consider the moduli space $\cal{M}_{\Sigma_g}$ of a 2-dim closed surface $\Sigma_g$ of genus $g$. What is the topology of such a moduli space $\cal{M}_{\Sigma_g}$?
For example, what is $\pi_n ( \cal{M}...
- 4,826
-3
votes
1
answer
1k
views
Continuous surjection $S^n\to S^m$, $n<m$ [closed]
I had an exam last week on algebraic topology.
"Suppose $f:S^n\to S^m$ is continuous, where $n<m$. Prove that $f$ is homotopic to a constant mapping. The fact that $S^n$ minus one point is ...
- 1
-3
votes
1
answer
330
views
Loop space of manifold [closed]
Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...
- 189
-3
votes
1
answer
1k
views
relation between smash product and suspension
let $S$ be the d-sphere.
we know that $S \wedge X = \Sigma^d X$ the $d$-fold suspension of $X$.
what can we say about
$(S\times S) \wedge (S\times S)$ in terms of suspension?
- 1
-3
votes
1
answer
234
views
A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
- 41.9k
-4
votes
1
answer
441
views
Are spectra determined by their homotopy groups?
A famous theorem of Whitehead essentially states that spaces are determined by their homotopy groups. Is this true for spectra too?, i.e,
$$
\text{question: is a spectrum $E$ determined by its ...
- 705
-4
votes
2
answers
785
views
Spectral sequence [closed]
what is Koszul resolution? what is its role played in the computation of spectral sequence?
- 1
-4
votes
1
answer
710
views
Lie algebraic Grassmannian
Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...
- 356