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.
2
votes
1answer
256 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 ...
1
vote
0answers
66 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
votes
1answer
248 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 ...
0
votes
0answers
99 views
is the homology of a free complex again free?
Let $k$ be a field and $R$ be a finitely generated graded $k$-algebra (e.g. a polynomial ring in some variables).
Let $M$ be a finitely generated graded vector space so the graded module $\widetilde{M}...
14
votes
1answer
414 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 ...
2
votes
0answers
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 ...
1
vote
0answers
114 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.
...
1
vote
0answers
64 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 $...
5
votes
2answers
515 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 $\pi_1(M)$, ...
5
votes
2answers
290 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
votes
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
votes
2answers
265 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_{\...
3
votes
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 ...
9
votes
0answers
141 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$...
3
votes
0answers
130 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 ...
5
votes
0answers
160 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 ...
13
votes
2answers
724 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 ...
0
votes
1answer
108 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 (...
4
votes
1answer
248 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 ...
5
votes
1answer
263 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 ...
2
votes
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
votes
1answer
219 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
votes
1answer
298 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 ...
0
votes
1answer
239 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
votes
1answer
412 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 ...
6
votes
0answers
121 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 "...
2
votes
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 \...
6
votes
2answers
330 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?
4
votes
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
vote
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 ...
0
votes
0answers
55 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 ...
9
votes
4answers
535 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 \...
4
votes
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
votes
1answer
198 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
votes
1answer
141 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 $X$, ...
7
votes
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 ...
5
votes
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
$$
H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \...
3
votes
1answer
193 views
Homology groups of compact subset of $\mathbb{R}^2$
I am working over the paper: Target Enumeration via Euler Characteristic Integrals and in order to follow a proof I need to prove:
If $A$ is compact nonempty subset of $\mathbb{R}^2$, then the ...
11
votes
1answer
357 views
Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$
Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$
$$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
6
votes
1answer
228 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
...
13
votes
0answers
548 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
votes
1answer
570 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})\...
1
vote
0answers
209 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 $\...
3
votes
0answers
85 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 \...
1
vote
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
votes
0answers
192 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.
22
votes
3answers
738 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-...
1
vote
1answer
252 views
Addition of two homology classes is zero in construction of Poincare Sphere
I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
4
votes
1answer
190 views
“Small” simplicial complex with torsion trees
I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the ...
