All Questions
9,056 questions
1
vote
0
answers
55
views
Schemes for conditional distributions (monads)
(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...
- 121
1
vote
0
answers
111
views
When a semigroup ideal is a determinantal ideal?
Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
- 113
1
vote
0
answers
127
views
Lift up characteristic class to chain complex
In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...
- 677
1
vote
0
answers
73
views
Filtrations of spectra related to cellular ones and singular homology
I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...
- 16.9k
1
vote
0
answers
96
views
What are the "ouverts convenables" used to prove Brieskorns lemma?
In the proof of Brieskorns lemma, see 3.3 here, Brieskorn mentions that we take "ouverts convenables" satisfying some properties, but, as far as I can tell, never specifies what these opens actually ...
- 2,788
1
vote
0
answers
163
views
Whitehead Theorem for maps
Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains ...
- 517
1
vote
0
answers
90
views
Cobordism of an annulus with a non-vanishing vector field
Let $M$ be a compact three-dimensional manifold with corners, which is a cobordism of the two-dimensional annulus. In particular, the codimension one boundary of $M$ consists of two copies of the ...
- 778
1
vote
0
answers
147
views
Example of open manifold with no free integer homology non-homeomorphic to a ball
I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...
- 333
1
vote
0
answers
427
views
On Lefschetz theorem and sum of Betti numbers as lower bounds for fixed points
Let $M$ be a closed manifold with holomorphic cell decomposition (if it is complex), or at least with only even cohomology. In particular, its Euler characteristic is equal to the sum of its Betti ...
- 1,227
1
vote
0
answers
468
views
Path lifting property of holomorphic unbranched map
Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
- 632
1
vote
0
answers
116
views
A generalized Cauchy type functional equation
Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...
- 1,209
1
vote
0
answers
49
views
Weighted cancellation norm of a word computation
A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$.
We say that two disjoint pairs $\{...
- 111
1
vote
0
answers
80
views
Is $Iso(V)$ a deformation retract of $GL(V)$ when $V$ is a finite dimensional linear normed space
Assume that $V$ is a finite dimensional real or complex normed linear space. Let $Iso(V)\subset GL(V)\subset L(V)$ be the space of linear isometric endomorphisms, invertible endomorphism and ...
- 356
1
vote
0
answers
142
views
Comparing Different Notions of Unicoherence in the Plane
Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
- 439
1
vote
0
answers
173
views
Modeling scientific theories with category theory (or, how to represent a biological system categorically)
Suppose I wanted to compare Linnaean classification, which arranges species by similarity in ranked taxa, to modern phylogenetic systematics, which appeals to descent with modification and branching ...
- 87
1
vote
0
answers
82
views
What is known about the cohomology of the matrix monoid?
When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object).
Let $M_n(R)$ be the monoid of matrices with matrix ...
- 1,726
1
vote
0
answers
75
views
partially commutative like monoids [duplicate]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\}...
- 1,269
1
vote
0
answers
256
views
partially commutative monoid [closed]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
- 1,269
1
vote
0
answers
255
views
Presentation of amalgamated sum as a quotient of the direct sum
I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...
- 65
1
vote
0
answers
408
views
Nice proof of the Reidemeister-Singer’s theorem?
Is there a nice proof (preferably with pictures) of the Reidemeister-Singer theorem? I'd prefer some classical methods, perhaps in a book or lecture notes?
I want to learn how things are done.
- 1,465
1
vote
0
answers
40
views
A exemple of a strongly-continuous contraction semigroup : how to prove the contraction?
I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is ...
- 111
1
vote
0
answers
207
views
free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$
I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.
For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 ,...
- 395
1
vote
0
answers
29
views
Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (...
- 10.5k
1
vote
0
answers
109
views
Triangulation induces morphism of Cochain Complexes
Let $X$ be a topological space, $R$ a ring, $n \in \mathbb{N}$ natural. Let $S_n(X, R) = \bigoplus_{s: \Delta_n \to X} R$ where the $s: \Delta_n \to X$ are the singular n-simplices, therefore ...
- 6,038
1
vote
0
answers
84
views
example of smoothing of quotient surface singularities with maximal Milnor number
In Wahl's paper "Smoothing of normal surface singularities", he shows that smoothing in the Artin component of a quotient surface singularity has the maximal Milnor number in the versal family. ...
- 31
1
vote
0
answers
59
views
Integer valued signature of $4n$ dimensional orbifolds
Let $M^{4n}$ be a smooth oriented $4n$-dimensional manifold without boundary. Then we have an intersection form in $H^{2n}(M^{4n},\mathbb R)$ and such a form has signature $(n_+, n_-)$.
Question. I ...
- 14.3k
1
vote
0
answers
273
views
Mayer-Vietoris for local system cohomology
We define local system cohomology of $(X,M_\rho)$, $\rho:\pi_1(X)\to Aut(M)$, as the cohomology associated to the complex
$$Hom_{\mathbb{Z}[\pi_1]}(C_0\tilde X,M_\rho)\to Hom_{\mathbb{Z}[\pi_1]}(C_1\...
- 2,788
1
vote
0
answers
92
views
How are representations of SU(2) identified with non-intersecting curves in a disk? (Rumer-Teller-Weyl)
Apparently (e.g. see bottom left of first column on p3 of this manuscript) the representations of $SU(2)$ can be identified with diagrams of non-intersecting curves (unoriented embedded $1$-manifolds) ...
- 607
1
vote
0
answers
317
views
Fixed point set of torus action is discrete and infinite?
Let $T=(\mathbb{C}^*)^k$ act holomorphically on a smooth quasi-projective complex algebraic variety $M$.
Can the fixed point set $M^T$ be discrete and infinite?
I think the answer is no because ...
- 297
1
vote
0
answers
105
views
The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11
1
vote
0
answers
112
views
Automorpshism of a free abelian group of a free product
Let $G=A_1*A_2*...*A_n*F_k$ be a free product, where each $A_i$ is free abelian of different rank (rank at least 2), and $F_k$ is a free group. Consider an arbitrary automorphism $\phi$ in $Aut(G)$, ...
- 11
1
vote
0
answers
205
views
Sheaves and isomorphisms with chain complex of singular chains (Sheaf Theory, Bredon)
Let $\Delta_{\ast}(X,A)$ (resp. $\Delta_{\ast}^c(X,A)$) be the chain complex of locally finite (resp. finite) singular chains of $X$ modulo those chains in $A$.
How to show that the homomorphism of ...
- 11
1
vote
0
answers
125
views
Transition functions and twisting functions
A principle bundle over a topological space can be given by transitions functions. On the other hand, in the simplicial world principle bundles can be given by twisting functions. Is there an explicit ...
- 2,312
1
vote
0
answers
409
views
Rational homotopy and l-adic cohomology
In rational homotopy theory there is a basic construction which, given a prime number $l$ and a $CW$-complex $X$, produces a localized space $X_l$ equipped with a map $X\rightarrow X_l$ that induces ...
user95283
1
vote
0
answers
222
views
The lattice subgroup quotient
Let $G$ a finite group and $L(G)$ it's lattice of non trivial subgroups. Is it true that the quotient space $|L(G)|/G$ is contractible, where $|L(G)|$ is the geometric simplicial complex associated to ...
- 933
1
vote
0
answers
58
views
A class of finitely generated semigroups
Let $G$ be a finitely generated group with a probability measure $\nu$. Suppose we have a finite first moment function $f:G\to \mathbb{R}$, i.e, such that $\Sigma_{g\in G}f(g)\nu(g)< \infty$. Then, ...
- 49
1
vote
0
answers
152
views
Stone cech compactification of a zero dimensional topological space
Let $X $ be a zero dimensional topological space, that is, a topological space with a basis of clopen sets. Is there any characterization for the ston cech compactification for such a space?
- 11
1
vote
0
answers
109
views
Empty regions on the second list of unstable Adams spectral sequence
Define $\phi(n) = 4n - 2$. Is there a proof that on the second list of unstable Adams spectral sequence (for all spheres) there are no elements in squares $(n, m)$ such that $m < \phi(n)$. The ...
- 1,129
1
vote
0
answers
62
views
Sampling in a polyhedral complex
Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of ...
- 2,246
1
vote
0
answers
139
views
Cohomology algebra of a fiber bundle
Given a fiber bundle $(E,B,p,F)$ with path connected base $B$ and fiber $F$, both closed smooth manifolds of finite dimensions. The standard tool to compute the cohomology of the total space $E$ is ...
- 63
1
vote
0
answers
251
views
Copylefted introduction to topology
Is there a textbook in topology with a copyleft license?
$$ $$
- 45k
1
vote
0
answers
80
views
Conical resolution with torsion in the 1st homology group
By a conical resolution I mean a resolution of singularities $\pi:X\rightarrow Y$ where $X$ and $Y$ are endowed with an action of $\mathbb G_m$ (commuting with $\pi$) such that $Y$ is contracted to a ...
- 790
1
vote
0
answers
206
views
Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary
The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a ...
- 21
1
vote
0
answers
496
views
Invariance of combinatorial/geometric euler characteristic
I am trying to read and understand the paper:
TARGET ENUMERATION VIA EULER CHARACTERISTIC INTEGRALS
by YULIY BARYSHNIKOV AND
ROBERT GHRIST.
And I am having trouble with a statement. First of all, ...
- 909
1
vote
0
answers
89
views
On the Multi-Compression theorem of Rourke and Sanderson
Suppose $M$ is a compact manifold so that it admits an embedding $f:M\to N\times\mathbb{R}^l$ with a splitting of the normal bundle as $\nu_f\simeq\nu\oplus\epsilon^l$ where $\nu$ is some $k$-...
- 3,173
1
vote
0
answers
177
views
Why is any one Wirtinger relation a consequence of the remainder? [closed]
As the question title suggests, how do I see that any one Wirtinger relation is a consequence of the remainder?
- 19
1
vote
0
answers
639
views
What is the real name of this relation and operation on a particular set of maps between cancellative monoids?
Let $A,B$ be cancellative monoids and define a transducer as a map $f\colon A \rightarrow B$ such that $f(1)=1$ and for all $a_1 ,a_2 \in A$, there exists a $b \in B$ such that $f(a_1 a_2)=f(a_1) b$. ...
1
vote
0
answers
209
views
Simplicial approximation theorem and related stuff
I would like to ask for a list of references that explain the simplicial approximation theorem, barycentric subdivisions and Kan $Ex^{\infty}$ functor in details, fully exploiting the language of ...
- 1,403
1
vote
0
answers
305
views
Steenrod's geometric view of fiber bundles
In Dan Freed's notes pp.4 beneath (12.24) he comments that, let $P \to M$ be a principal $G$-bundle and $E = (P \times F)/G = P \times_G F$ its associated fiber bundle. The fibers of $E \to M$ are ...
- 457
1
vote
0
answers
133
views
Standard proof that cyclic ordering of edges is preserved under planar graph homotopy?
I have several questions about the following theorem statement:
Thm: Let $G = (V, E)$ be a planar graph, and let $\varphi_0 : G \rightarrow \mathbb{R}^2$,
$\varphi_1 : G \rightarrow \mathbb{R}^2$ be ...
- 11