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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 ...
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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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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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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 $...
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1answer
271 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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1answer
253 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 ...
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1answer
430 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
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. ...
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1answer
422 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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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 ...
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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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65 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
531 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-...
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4answers
554 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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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 ...
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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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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 $...
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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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3answers
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When are the homology and cohomology Hopf algebras of topological groups equal?

Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology ...
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5answers
10k views

What is a cohomology theory (seriously)?

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"? I know that there exist generalized cohomology theories, Weil ...
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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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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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8answers
2k views

Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
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1answer
111 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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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{...
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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 ...
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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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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
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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 ...
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4answers
2k views

Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain $G_0 > G_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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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 ...
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0answers
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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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 ...
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3answers
1k views

A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms $h_{*+1}(...
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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 $X$, ...
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1answer
359 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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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
194 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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15answers
9k views

Why torsion is important in (co)homology ?

I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
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16answers
6k views

Teaching homology via everyday examples

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory? To be more precise, I am teaching a short course on homology, from ...
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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 ...