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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1answer
125 views

Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
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158 views

Where can I find W. Browder's thesis

I've been looking for W. Browder's thesis Homology of loop spaces for a while now, and I really found nothing except for articles and book having it in their bibliography. Does someone know if it can ...
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Euler class and the real homological class of the fiber in an orientable sphere bundle

In the paper Foliations transverse to the fiber of a bundle, Plante considers the following example. Let $p:E\longrightarrow B$ a orientable fiber bundle with fiber $\mathbb{S}^k$. We have the Gysin ...
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89 views

A stronger generalized Jordan curve theorem

The generalized Jordan curve theorem is usually stated as such: Given $X\subseteq S^n$ such that $X$ is homeomorphic to $S^k$, $$\tilde{H}_i(S^n\setminus X)\cong\begin{cases}\mathbb{Z},\quad i=n-k-...
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If all 2-faces of a polytope are $2n$-gons, is the edge-graph bipartite?

This question on MSE has not received a satisfying answer. It can be summarized as follows: Question: Is is true that the edge-graph of a (convex) polytope is bipartite if and only if all 2-faces ...
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Homology of configuration space of punctured projective spaces?

Let $M=\mathbb{C}P^n$ or $\mathbb{R}P^n$ with $m$ punctures, is it known what the homology of the configuration space, $H_*(C_k(M))$ is? How are cases $\mathbb{C}P^n$ and $\mathbb{R}P^n$ different?
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Fibrant objects in $\mathbb{S}$-local model structure on $Top_*$

Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$. Equipping $Top_*$ with the Quillen model structure (...
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Explicit action of the Dehn twist in the homology of punctured sphere with local coefficients

Let $X=\mathbb{P}^1\setminus S$, where $S=\{a_1,\dots, a_k\}$ is a finite subset of $\mathbb{P}^1$ and we may assume that $|S|\geq 4$. Let $\mathbb{L}$ be a local system on $X$ given by a monodromy ...
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1answer
247 views

How to identify cup product with intersection

What's the standard generalization and reference for the following statement: If two oriented submanifolds $L$, $L'$ of an oriented compact manifold $M$ intersect transversally, then the Poincare ...
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mod $p$ homology of Thom spectra MSU

Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big(...
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60 views

Embeddedness and homology of a limit of minimal surfaces

Consider the following theorem, proved in this paper: Theorem (Theorem 6.1). Suppose we have a sequence $(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$ of immersed free boundary minimal $...
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1answer
570 views

When homology isomorphism implies homotopy isomorphism

Let's suppose that $f:X\rightarrow X$ is a continuous map such that $H_{\ast}(f): H_{\ast}(X)\rightarrow H_{\ast}(X)$ is a homology isomorphism (with integral coefficients) $X$ is a finite ...
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1answer
293 views

Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $

Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph: For any scheme of finite type over a ground field ...
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2answers
318 views

Alexander duality and homology equivalence

While reading the paper of Kauffman and Taylor "Signature of links" I found the following situation. In the proof of Theorem 2.6 they suppose that two links $L_1, L_2\in \mathbb{S}^3$ are ...
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Computation of mod p homology of $MSU$

I am trying to proof Novikov theorem \begin{equation} MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i. \end{equation} This can be proved by using ...
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1answer
548 views

Geometric intuition behind this chain homotopy

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion $$C_\bullet^\mathcal{...
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1answer
240 views

Tor functor and invertible elements

Let $A$ be a commutative ring, $a \subset A$ be an ideal. For $A$-module $M$ let $S \subset A$ be the set of elements, which are invertible in $M$, so $M$ is actually a $S^{-1}A$-module. It is not ...
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1answer
211 views

Easier ways to compute homology/cohomology by adding extra structure

Suppose $X$ is a topological space and I want to talk about its “homology”. There is this notion of singular homology obtained from singular chain complex. This is not very easy to compute. Suppose ...
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136 views

Homology of homotopy fiber of inclusion

We consider an inclusion $j: A \hookrightarrow X$. Let $A_j = A \times_X PX$ be the homotopy fiber (which is the fiber of the fibration associated to $j$). The space $PX$ there is the Moore path space ...
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180 views

Simply put Floer homology

I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would ...
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1answer
167 views

Cup product in Tate Cohomology Ring

Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^*(G, \mathbb{F}_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}_pG$-module. There is a ...
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229 views

Funtoriality of twisted K-theory

I posted this question on math.stackexchange, but received no answer there. In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed ...
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Vietoris-Rips complex and coarse geometry

Let $K$ be an infinite countable subset of Euclidean space $E$ such any point of $E$ is within distance 1 of some point of $K$. In the language of John Roe's "coarse geometry", this implies that $K$ ...
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Finite generation of group homology

I am reading 'Subgroups of direct products of limit groups' of Bridson, Howie, Miller and Short (http://annals.math.princeton.edu/wp-content/uploads/annals-v170-n3-p11-p.pdf) and I am finding similar ...
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97 views

Discrete Morse theory, choice of Morse function, and removing noise

If I have a simplicial complex, and a discrete Morse function defined on the simplices, I can use persistent homology to produce a barcode which helps me distinguish "persistent" shape from noise. To ...
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111 views

Naturality of Poincaré–Lefschetz

Let $X$ be compact and Hausdorff, $A\subseteq B\subseteq X$ both closed such that $X\setminus A$ is an open orientable $d$-manifold. Then also $X\setminus B$ is an open orientable $d$-manifold. We ...
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Finite generation of the image of the induced homomorphism on homotopy groups of infinite loops spaces

Let $f:X\rightarrow Y$ be a map of infinite loop spaces such that image of homology groups $H_i(X,\mathbb{Z})$ for $i\geq 1$ under $f_*$ are finitely generated. Does this imply that the image of ...
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4answers
385 views

Homology of solvable (nilpotent) Lie algebras

Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...
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89 views

Do closed hypersurfaces separate the euclidean space?

The following extension of the Jordan Curve Theorem is well known: every closed connected hypersurface of the sphere $\mathbb S^N$ separates $S^N$ into exactly two connected components. As a ...
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112 views

Fibre transfer of $\mathbb{S}^1$-bundles

Let $p:E\to X$ and $p':E'\to X'$ be two orientable $\mathbb{S}^1$-bundles. Denote their homological transfers by $p_!:H_*(X)\to H_{*+1}(E)$ resp. $p'_!$. Now let $(u,f)$ be a bundle morphism ($u:E\to ...
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Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$

I read that, using the fact that $\text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $\text{Ext}^* (A,B)$ (which I understand), we can give $\bigoplus_{i} \text{Ext}^i (A,...
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1answer
197 views

Homology of universal abelian cover of a manifold

If one define the universal abelian covering $M_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, ...
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Singular homology: Lifting simplices gives map in homology

Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$. Then the ...
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65 views

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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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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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
530 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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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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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
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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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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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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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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
872 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
330 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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1answer
95 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
304 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
102 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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185 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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143 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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