Questions tagged [homology]
Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
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non zero differential in a spectral sequence
This is the situation:
Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
5
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0answers
60 views
Alexander-Whitney for cyclic objects
What is known about the extension of the AW map from simplicial to cyclic Abelian groups? Homological perturbation theory implies there is an A infinity-like sequence of maps, but is it known ...
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0answers
77 views
Cohomology of a chain complex over a polynomial ring
I asked this on SE but I did not get any answer; I got some progress but I hope here I can find some help to finish the problem out.
Let $R = F[x_1, \ldots, x_n]$ be a polynomial ring over a field $...
2
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1answer
272 views
Is there a theorem showing that de Rham homology is isomorphic to singular homology?
The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology.
Now ...
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67 views
Homology of SL(2,R) with finite coefficients
Consider the third homology group of a real special linear group
$H_3 (SL(2,\mathbb{R}),\mathbb{F}_p)$. It is known$[1]$ that for $p=2$ the third homology group of $SL(2,\mathbb{R})$ vanishes.
...
5
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1answer
254 views
Topology of connected subsets of the $3$-torus
Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded.
I am ...
14
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1answer
423 views
Spectra with “finite” homology and homotopy
As known, any non-trivial finite spectrum $X$ can not have non-zero homotopy groups $\pi_i(X)$ only for finite number of $i$. As I understand, the same is true for any spectrum $X$ with finitely ...
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71 views
Is there a knot invariant robust to hiding one part of the diagram behind another?
Is there a well-known knot invariant that can be computed solely by inspecting an arbitrary projection of the knot into the plane (with a marking of each crossing as "over" or "under")?
The reason ...
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0answers
115 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.
...
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0answers
67 views
Reference request for Leibniz rule and spectral sequences
Suppose $A_*$,$B_*$, and $C_*$ are chain complexes equipped with filtrations and a map $m:A_* \otimes B_* \to C_*$ respecting these filtrations. I am looking for a reference for the fact that the map $...
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2answers
532 views
Classification of closed 3-manifolds with finite first homology group?
I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$.
Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ...
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2answers
291 views
On spaces with finite homological dimension
Let $X$ be a connected $CW$-complex, such $\pi_1(X)$ is torsion-free and $H_k(X,\mathbb Z) = 0$ for all $k \geq N$ and some $N \in \mathbb N$. Then
$(1)$ Does it follow that $X$ is homotopy-...
2
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1answer
71 views
On the entries of a matrix representation for a boundary operator of a persistence module
In equation 6 of Computing Persistent Homology (page 8), the authors put forward the following identity:
$$\deg \hat{e_i}+\deg M_k (i,j)=\deg e_j$$
Where $\hat{e_i}$ and $e_j$ are elements of ...
6
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2answers
266 views
Homology spectral sequence for function space
The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\...
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1answer
95 views
Proof of $\det\partial_2^t∂_2 =m^2 ·k(G)$ for G, finite connected graph with reduced homology being 0
Let $G$ be a finite connected graph. Let $K$ be a 2-dimensional complex such that $K^{(1)} = G$, $\tilde{H}_2(K)=0$ and $\tilde{H}_1(K)=\Bbb Z_m$. Show that $\det\partial_2^t∂_2 =m^2 ·k(G).$
Over the ...
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0answers
146 views
Hochschild homology of a Hopf algebra
Let $A$ be a Hopf algebra over the complex numbers.
Denote by $\mathcal{M}$ the dg-category of dg-$A$-modules.
The Hochschild homology of $\mathcal{M}$ is not going to be the Hochschild homology of $A$...
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0answers
132 views
Eilenberg-Steenrod Axioms for Lawson Homology
Let $X\subset\mathbb{P}^N:=\mathbb{P}^N(\mathbb{C})$ be a projective variety and denote by $X(p)$ the set of $p$-dimensional subvariety of $X$. The free abelian group generated by $X(p)$ is the space ...
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0answers
161 views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
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3answers
835 views
When does homology represent an embedded sphere?
If we have a triangulation of a manifold $M$ of dimension $i$ and we have simplicial homology $H_i(M)=\mathbb{Z}$, what is the condition than there exists an embedded sphere $S^i$ that generates the ...
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1answer
113 views
Generalised homology of a split fibration
Let $E, X$ be path-connected and suppose I have a fibration $p\colon E\to X$ which admits a section $s$.
For a generalised homology theory $\mathcal{E}_\ast$, there is a splitting $\mathcal{E}_\ast (...
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1answer
251 views
Vanishing of homology for hyperelliptic locus
It is a theorem due to Harer that $H_k(M_{g,n},\mathbb{Q})=0$ for $k>C(g,n)$, where $C(0,n)=n-3, C(g,0)=4g-5$ for $g>0$, and $C(g,n)=4g-4+n$ for $g,n>0$. Here $M_{g,n}$ denotes the coarse ...
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1answer
264 views
A weak version of the Whitehead Theorems
Let $f:X\longrightarrow Y$ be a map between CW-complexes $X$ and $Y$. By the Whitehead Theorems, if one of the conditions:
1- (homotopy version) $\pi_n (f):\pi_n (X)\longrightarrow \pi_n (Y)$ is an ...
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2answers
112 views
Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?
Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $...
4
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1answer
222 views
Base change for Borel-Moore homology
For a seperated scheme of finite type $X$ over $\mathbf{C}$, let $H_*(X)$ denote its Borel-Moore homology, which is defined by
$$
H_k(X) = R^{-k}\Gamma(X, \omega_X)
$$
where $\omega\in D_c(X, \mathbf{...
10
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1answer
304 views
Smallest volume representatives of homology
Given a Riemannian manifold, I have a notion of volume for each of my chains, so it makes sense to ask for a representative of a homology class with the smallest volume. Are there conditions for when ...
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1answer
240 views
Clarification of “death event” in persistent homology
Before I ask my question let me clarify some notation:
$f^{i,j}_r$, where $i < j$, refers to the inclusion map $f: H_r(X_i) \hookrightarrow H_r(X_j)$. $X_i$ and $X_j$ are subcomplexes of a filtered ...
10
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1answer
435 views
Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms
I asked this question a while ago on MSE, got no answer, put a bounty on it, still got no answer, was advised to ask here instead, hesitated, forgot about the question for a while and now remembered ...
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0answers
123 views
What is the relationship between the Khovanov-Rozansky homology of a digraph and that of a link?
Motivation: I'm reading this preprint, which takes a digraph $G = (V, E)$ and then builds a projective algebraic set $P(G)$ by assigning a variable to each edge and then defining certain polynomial "...
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0answers
97 views
Quillen homology of a morphism
I’m interested in definition of a homology of a map in model category $C$, as an example let’s take $C = \mathrm{sGrp}$.
Let $\Gamma$ be a discrete group, its Quillen homology groups defined as $H_n \...
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2answers
336 views
Integer homology of double loop space of odd-dimensional sphere
I have checked everything "homology of loop spaces"-like, but was not able to find what is $H_*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?
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1answer
468 views
Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $
Question 1: What is a complete classification of all positive integers $m,n$ with the following property:
There is a continuous map $f:S^n \to \mathbb{C}P^m$ such that $f$ maps ...
1
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1answer
96 views
On graph imbedding genus clarification
Given a graph the minimum genus $g$ is the minimum number of handles needed so that there an imbedding of the graph on the surface with no edge crossings.
If the graph is of genus $g$ then is there ...
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56 views
show that the set of functions from $ \mathbb{Z}[X^{n}] $ to $ A $ is generated by the characteristic functions
Let $ X $ be a quandle and $ A $ be an abelian group (for simplicity assume
that $ A $ is a finite cyclic group $ \mathbb{Z}_{n} $ or the infinite cyclic group $ \mathbb{Z} $). I need to show that the ...
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4answers
556 views
Complements of Simply Connected Subsets of the Plane
this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...
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0answers
108 views
Do homological holes with unit coefficients correspond to polyhedra?
(Originally posted at m.se without answers.)
Let $T$ be a set of triangles in an abstract simplicial complex, with orientation of the triangles chosen such that
$$\partial \left( \sum \limits_{t \in ...
4
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1answer
199 views
Invariance of Khovanov homology under first Reidemester move
I am studying Khovanov homology from five lectures on Khovanov homology
and I want to try to show Khovanov homology is invariant under first Reidemester move but I cannot understand how we can write
...
2
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1answer
142 views
Cobordism/bordism group based on orbifolds with corners
We define a geometric homology group of a topological space $X$ as follows: the chain complex $C_{\bullet}$is freely generated by the maps $f$ from a compact oriented orbifold with corners $P$ to $X$, ...
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2answers
375 views
Homology $H_*(TOP, \mathbb{Z}_2)$ of the stable homeomorphism space
Let $TOP$ be the stable homeomorphism space, with $TOP(n) = Homeomorphisms(\mathbb{R}^n)$. What is known about its $\mathbb{Z}_2$-homology $H_*(TOP, \mathbb{Z}_2)$? In particular I am interested in ...
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0answers
113 views
torsion part of homology of simplicial complexes [duplicate]
Let $n$ be a fixed positive integer and let $K$ be a simplicial complex with $N$ vertices. Suppose the $n$-th integral homology group of $K$ is
$$
H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \...
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1answer
195 views
Homology groups of compact subset of $\mathbb{R}^2$
I am working over the paper: Target Enumeration via Euler Characteristic Integrals and in order to follow a proof I need to prove:
If $A$ is compact nonempty subset of $\mathbb{R}^2$, then the ...
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1answer
360 views
Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$
Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$
$$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
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1answer
229 views
Homology of the product of spaces with integer coefficients and the Massey products
Consider
$H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence
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0answers
549 views
Is this “Homology” useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
10
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1answer
586 views
Understanding Homology Operations and how to compute them
I stumbled upon this Lemma:
Let $X$ be a spectrum and $H_p(X;\Omega_q^{Spin})\Rightarrow MSpin_{p+q}(X)$ the Atiyah-Hirzebruch spectral:
The differential $d_2\colon H_p(X;\Omega_1^{Spin})\...
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0answers
212 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, ...
2
votes
1answer
160 views
index of the subgroup of the mapping class group acting trivially on Z/3Z homology
Let $S=S_g$ be the closed orientable surface of genus $g$ and let $\Gamma_3(S)$ be the subgroup of the mapping class group, $Mod(S)$, which acts trivially on $H_1(S;\mathbb{Z/3\mathbb{Z}})$. Define $\...
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0answers
88 views
A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
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0answers
89 views
computing homology of subvarieties of Euclidean spaces by persistent homology
Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$.
Suppose the ...
4
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0answers
195 views
Applications of cosheaf homology?
What are some applications of cosheaf homology within mathematics?
Some ones I've heard of Sheaves (not cosheaves) are computing global sections and the Picard Group with a sheaf on projective space.
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3answers
743 views
Which paths in a graph are orthogonal to all cycles?
Start with some standard stuff. Suppose we have a directed graph $\Gamma$. I'll write $e : v \to w \,$ when $e$ is an edge going from the vertex $v$ to the vertex $w$. We get a vector space of 0-...
