All Questions
9,056 questions
1
vote
0
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138
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Category whose morphisms are commutative monoids but not enriched
In a recent investigation, I constructed a category $\mathcal{C}$ with the following property. For objects $X,Y \in \mathcal{C}$, the morphism set $\text{Mor}(X,Y)$ is a commutative monoid with ...
- 161
42
votes
5
answers
4k
views
What are the main structure theorems on finitely generated commutative monoids?
I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...
- 22.3k
3
votes
0
answers
185
views
Equivariant cohomology of symmetric group acting on a product
Let $X$ be a finite CW-complex. The symmetric group $S_n$ acts on the product $X^{\times n}$ in the obvious way. Let $H^{\bullet}_{S_n}(X^{\times n})$ be the (Borel) equivariant cohomology of this ...
38
votes
3
answers
2k
views
If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic?
Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible,
$$
X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.
$$
Is the ...
- 5,898
44
votes
6
answers
4k
views
Does $\mathbb C\mathbb P^\infty$ have a group structure?
Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$?
$\mathbb C\...
- 8,727
4
votes
0
answers
172
views
Brouwer fixed point theorem for non-Hausdorff spaces
Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?
More particularly, is there a formulation of the Brouwer fixed point theorem
which covers both the standard case of ...
- 513
1
vote
1
answer
210
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Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?
Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the ...
- 6,962
4
votes
1
answer
266
views
Equivariant Whitney approximation
I am wondering if there is a reference for the following:
Let $G$ be a finite group, and suppose that $f\colon M\rightarrow N$ is a continuous and $G$-equivariant map. Here $M$ and $N$ are finite ...
- 55
6
votes
3
answers
411
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Problem 0.9.10 in Cohn's "Free Ideal Rings and Localization in General Rings" (CUP, 2006)
Let $S$ be a monoid. On p. xvii of P.M. Cohn's Free Ideal Rings and Localization in General Rings (CUP, 2006), one reads that
an element $u \in S$ is regular if (quote) "[...] it can be ...
- 10.5k
2
votes
0
answers
50
views
How to calculate the genus of a 3 dimensional curve (f(kx,ky,kz)) using the Newton polyhedra?
Given a plane affine curve $\sum_{i,j}a_{i,j}k_x^ik_y^j = 0$, its genus can be calculated as the number of integral points of the interior of the convex hull of $\{(i,j) \mid a_{i,j} \neq 0\}$.
How to ...
- 21
1
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0
answers
41
views
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
vote
1
answer
213
views
Multi-connected sum decomposition of $n$-manifolds
A connected sum decomposition of a closed $n$-manifold $M^n$:
$M^n = M_1^n \# M_2^n$, is to view $M^n$ as two closed $M_1^n$ and $M_2^n$, joined
by a neck $I\times S^{n-1}$.
Similarly, a $k$-connected ...
- 4,826
4
votes
1
answer
622
views
Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$?
As the title suggests, I have the following question:
Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$?
Clarification:
Denote by $b_k$ the $k$th Betti number of a compact ...
- 1,379
8
votes
1
answer
414
views
Augmentation ideal of a free group
If $F$ is a free group then it has cohomological dimension one, which implies that the augmentation ideal $IF=\operatorname{ker}(\epsilon:\mathbb{Z}G\to \mathbb{Z})$ of its group ring is a projective $...
- 35.9k
1
vote
0
answers
95
views
Eventual stabilization for repeatedly adding multiplayer games
This question is an outgrowth of a couple previous questions of mine. In order: 1,2,3. This should be fully self-contained, but those questions may help motivate this one.
To keep things readable, I'...
- 20.5k
14
votes
0
answers
341
views
Is this class of groups already in the literature or specified by standard conditions?
In recent work
Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators
Scott Balchin, Ethan MacBrough, and I ...
- 141
31
votes
6
answers
5k
views
Book recommendation for cobordism theory
I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.
The audience is familiar with ...
- 7,583
1
vote
0
answers
184
views
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
4
votes
1
answer
186
views
Alexander polynomials for a certain family of closed braids
Let $n\geq 3$ be a positive integer and $\kappa=(k_1, \dots, k_n)\in \mathbb{Z}^n$. Denote by $B_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma_\kappa:=\sigma_1^{k_1}\...
- 685
7
votes
6
answers
959
views
Intersection of two Jordan curves lying in the rectangle
Given a rectangle $ABCD$.
Let a Jordan curve $L_1$ joins the vertexes $A$ and $C$
and all points of $L_1$ belong the rectangle.
Let a Jordan curve $L_2$ joins the vertexes $B$ and $D$
and all points ...
- 97
11
votes
3
answers
1k
views
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
My question is
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
(My thoughts on this which might not be useful at all.) Since an ...
- 1,565
3
votes
1
answer
1k
views
Geometric/Algebraic intersection numbers of curves on surfaces
I have the following problem, and struggling to find some references.
Suppose I start with a homology class of a curve on a closed genus $g$ surface $$h=(a_{1},b_{1},\dots,a_{g},b_{g})\in H_{1}(\...
- 459
4
votes
1
answer
575
views
Who introduced nerves in category theory?
Who was the first to consider that categories were semi-simplicial sets (and in particular groupoids were simplicial sets)?
I think there was a concept of nerve of a covering in algebraic topology ...
- 894
8
votes
1
answer
478
views
What is $TP(\mathbb{Z}_p)$?
Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$?
(i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-...
- 81
2
votes
1
answer
545
views
Computation of cohomology of Eilenberg-Maclane spaces
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Spf{Spf}$Background:
If $E$ is a complex-oriented spectrum, then $E^*(K(\mathbb{Z}/p^k,1))$ sits inside a long exact ...
- 448
2
votes
1
answer
179
views
Can information theory characterise a (topological) space?
Consider an objective function f: $\mathbb{R}^n\rightarrow\mathbb{R}$, with a vector of variables $\theta$, i.e. $f(\theta)$, $\theta \in \mathbb{R}^n$. Depending on $f$, there can be interesting ...
8
votes
2
answers
961
views
Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?
Let $k$ be a field and $X$ a topological space.
Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...
- 1,635
3
votes
1
answer
95
views
Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy
I am looking for non-trivial examples of flat $U(2)$ connections over the complement of a torus link $\mathcal{S}^3-L$ i.e.
$\mathcal{A}:\mathcal{S}^3-L \longrightarrow \mathfrak{U}(2)$ such that $F_{\...
- 45
13
votes
3
answers
1k
views
Extending group actions to vector bundles
Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$.
Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...
- 139
11
votes
1
answer
592
views
What is the connection between Lurie's definition of shape and Čech homotopy?
It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).
For instance, Lurie [Higher topos theory] defines this one:
Definition 1.
The ...
- 15.9k
0
votes
0
answers
185
views
Interpreting the edges in the Serre spectral sequence
Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre ...
2
votes
0
answers
293
views
Monodromy action
Let us consider the following tower of (finite) ramified Galois covers
$$S \xrightarrow{p} \mathbb{P}_1 \xrightarrow{q} \mathbb{P}_1,$$
where $S$ is a Riemann surface. Denote by $R \subset \mathbb{P}...
- 73
1
vote
0
answers
274
views
Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,306
28
votes
5
answers
5k
views
Are rational varieties simply connected?
Is it true that every smooth rational variety X is simply connected? How is the proof?
Would it be still true if X has mild (for example orbifold) singularities?
11
votes
2
answers
864
views
Solving polynomial equations in spectra?
Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
- 64k
7
votes
0
answers
162
views
Relative version of Browder's theorem on H-spaces
A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology (meaning that $H_*(X)$ is finitely generated) has $\pi_2(X) = 0$. This generalizes Cartan's ...
- 19.4k
58
votes
12
answers
29k
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Homological Algebra texts
I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe).
As usual, ...
6
votes
0
answers
529
views
Infinite-dimensional "algebraic varieties"
This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
- 1,543
2
votes
1
answer
194
views
Fundamental class of topological compact surfaces
Let $S$ be a connected orientable topological compact surface of genus $g$. There are various ways to prove that the second singular homology group with integer coefficients $H_2(S, \mathbb{Z})$ is a ...
- 21
10
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0
answers
199
views
"Homotopy homomorphisms" of homeomorphisms of Euclidean space
For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...
- 8,167
14
votes
1
answer
2k
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What's the cohomology ring structure of a blow-up?
Let $X$ be a compact Kähler manifold, with $j_Z: Z\hookrightarrow X$ a submanifold of complex codimension $r$, $\tau: \widetilde{X} \to X$ the blow-up of $X$ along $Z$, with exceptional divisor $j: E \...
- 498
35
votes
5
answers
11k
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What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
- 921
10
votes
1
answer
707
views
Tensor products of $\mathbb{E}_\infty$-spaces
In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...
- 11.8k
4
votes
0
answers
333
views
Is etale sheafification of algebraic K-theory related to analytic continuation of the zeta function?
The Riemann zeta function can be recovered from algebraic K-theory and the Borel regulator. Analytic continuation is therefore a reasonable proceedure to do to Algebraic K-theory. How can we ...
- 705
14
votes
2
answers
740
views
Examples of topoi that are not ordinary spaces
In [SGA6] we find:
Mais nous lui conseillons néanmoins, de préférence, de s'assimiler le langage des topos, qui fournit un principe d'unification extrêmement commode. (DeepL translate: However, we ...
user144439
3
votes
0
answers
195
views
Is there such an isotopy for every homology sphere?
Let $n \geq 3$, and $\Sigma^{n-1} \subset \mathbf{S}^n$ be a smoothly and properly embedded, orientable, and connected submanifold of the sphere. This divides the sphere into two open sets, $U_-$ and $...
- 5,048
5
votes
1
answer
258
views
Nondegeneracy of kernel of map on homology induced by covering of surfaces
Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces and let $K$ be the kernel of the induced map $f_\ast: H_1(X) \rightarrow H_1(Y)$. Here homology has rational ...
- 53
9
votes
0
answers
195
views
Every locally presentable $\infty$-category can be presented by a proper model category
Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ?
Of course if one ...
- 42.4k
3
votes
0
answers
155
views
Posets whose homotopy type can be efficiently studied without fibrant replacement?
Let $P$ be a poset and $NP$ its nerve. In order to study the homotopy type of $NP$ via the tools of simplicial homotopy theory, we generally need to take a Kan-fibrant replacement of $NP$, e.g. by ...
- 64k
16
votes
0
answers
325
views
Rational equivalence of smooth closed manifolds
All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they ...
- 23.5k