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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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Shub Conjecture and polynomial entropy

The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the ...
Ali Taghavi's user avatar
5 votes
0 answers
147 views

Representing some odd multiples of integral homology classes by embedded submanifolds

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
Zhenhua Liu's user avatar
13 votes
1 answer
489 views

Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces

Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
Zhenhua Liu's user avatar
6 votes
0 answers
241 views

Reference to a definition of a graph homology

Let $G$ be a graph, and define $C_k$ to be the free abelian group on the homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge ...
Sean Longbrake's user avatar
3 votes
1 answer
111 views

When are homologous embedded surfaces in 3-manifolds related by embedded cobordisms?

Let $M$ be an orientable closed 3-manifold and suppose $A$ and $B$ are embedded incompressible closed orientable surfaces in $M$ with $[A] = [B]$ in $H_2(M,\mathbb{Z})$. In general, there are a ...
Edgar A. Bering IV's user avatar
4 votes
2 answers
297 views

On a generalized homotopy transfer theorem

In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
groupoid's user avatar
  • 215
5 votes
1 answer
528 views

Betti numbers of non-orientable $3$-manifolds

Let $M^3$ be a compact $3$-manifold with boundary $\partial M$. If $M$ is orientable, then it is known (see Lemma 3.5 here) that $2\dim(\ker(H_1(\partial M,\mathbb{Q})\rightarrow H_1(M,\mathbb{Q})))=\...
Alessio Di Prisa's user avatar
4 votes
1 answer
436 views

Cohomology of finite symmetric products of manifolds

Let $M$ be a closed, orientable manifold of dimension $k$. I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational ...
KFJ2611's user avatar
  • 506
13 votes
1 answer
353 views

Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds

Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
Zhenhua Liu's user avatar
1 vote
0 answers
46 views

Homology groups of moduli of parabolic bundles with fixed determinant

I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant. In particular, what is the second homology group of such ...
yors's user avatar
  • 195
7 votes
4 answers
981 views

Amending flawed "proof" that homology groups are zero

I am trying to prove a certain statement that seems true based on computational data, and there is a nice argument that proves it, assuming all cycles are the simplest ones (e.g., when the only 1-...
Marcel K. Goh's user avatar
1 vote
0 answers
82 views

Projective dimension and certain subideals

My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it. Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{...
LeviathanTheEsper's user avatar
20 votes
2 answers
801 views

Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds

Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
Zhenhua Liu's user avatar
4 votes
0 answers
122 views

Weaker condition for the excision axiom

This comes from a question I asked on mathstackexchange (link: here) The excision axiom in homology states that if $\overline Z\subseteq\operatorname{Int}A$, then $h_n(X\setminus Z,A\setminus Z)$ is ...
Sardines's user avatar
  • 141
8 votes
1 answer
208 views

Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle

Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $...
Zhenhua Liu's user avatar
2 votes
0 answers
115 views

Triple insersection number of a surface in three-manifolds

I heard something about the triple intersection number $\text{mod}(2)$ (but maybe also $\text{mod}(n)$) of a surface in an orientable three-manifold but I couldn't find a precise definition. My guess ...
Andrea Antinucci's user avatar
3 votes
3 answers
381 views

Pairing between cohomology and the image of the Hurewicz homomorphism

Let $X$ be a compact manifold of dimension $\geq k$. Denote by \begin{equation} h: \pi _k(X) \rightarrow H_k(X,\mathbb{Z}) \end{equation} be Hurewicz homomorphism and by $\Gamma _k(X)\subset H_k(X,\...
Andrea Antinucci's user avatar
3 votes
1 answer
140 views

Linking form for homology with general coefficients

For integral homology groups there is the notion of linking form (http://www.map.mpim-bonn.mpg.de/Linking_form) $$ Tor(H_{l}(X,\mathbb{Z}))\times Tor(H_{n-l-1}(X,\mathbb{Z}))\rightarrow \mathbb{Q}/\...
Andrea Antinucci's user avatar
3 votes
2 answers
638 views

Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?

Given a manifold $X$ and short exact sequence of abelian groups $$ 1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1 $$ we get the Bockstein map in cohomology ...
Andrea Antinucci's user avatar
5 votes
1 answer
372 views

Computation of the linking invariant on Lens spaces

Let $L_n(p)$ be the $2n+1$ dimensional Lens space $$ S^{2n+1}/\mathbb{Z}_p $$ where the action is given as $z_i\rightarrow e^{\frac{2\pi}{p}}z_i$, $i=1,...,n+1$, with $z_i$ the coordinates of $\mathbb{...
Andrea Antinucci's user avatar
4 votes
1 answer
227 views

Euler class of vertical tangent bundle of the surface bundle over circle

Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...
Faniel's user avatar
  • 653
5 votes
1 answer
270 views

Reference for Künneth Theorem in (co)homology with local coefficients

Is there a discussion in the literature of Künneth-type theorems for (co)homology with local coefficients? The sources I know of that discuss local coefficients (Whitehead's Elements of Homotopy ...
Dan Ramras's user avatar
  • 8,528
1 vote
0 answers
138 views

Trivial homology groups for p-torsion groups

Let $G$ be a group where each element has a $p$-power order. Let $M$ be a $G$-module without $p$-torsion. Here $G$ is a discrete infinite subgroup of a complete group. Then, it cannot be assumed pro-...
Claudio Bravo's user avatar
2 votes
0 answers
84 views

Arf invariant for characteristic surfaces in closed 4-manifolds depends on homeomorphism type

Let $X$ be a closed smooth oriented 4-manifold with $H_1(X;\Bbb Z)=0$. Then for a smoothly embedded orientable surface $F\subset X$ which is characteristic, there is a well-defined invariant $\text{...
blancket's user avatar
  • 189
1 vote
2 answers
162 views

Lattices formed by unions of elements in an antichain

Let $A_1, \dots, A_k$ be incomparable subsets (of $\{1, \dots, n\}$) and consider the poset $P$ consisting of all possible unions of these under inclusion. Its not hard to see that this is a lattice, ...
Moty Katzman's user avatar
1 vote
0 answers
99 views

bott element in periodic cyclic homology

I am reading a paper by Thomason "Algebraic K-theory and étale cohomology", which deals with algebraic K theory localized by inverting the bott element $\beta \in K_2(\overline{k})^{\wedge}...
K.M.'s user avatar
  • 91
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0 answers
201 views

pullback square in abelian category and derived categories

Let $\mathcal{A}$ be an Abelian category. Take objects $A,B,C$ and $D$ in $\mathcal{A}$, and morphisms $b:B\to A$, $b':B\to A$, $c:C\to A$, $e:D\to C$, $e':D\to C$ and $f:D\to B$ such that diagrams $\...
user145752's user avatar
5 votes
1 answer
345 views

"Singular homology = simplicial homology" relative to a fibration

Let $p:E\to B$ be a fibration. Suppose $B$ has a simplicial decomposition. For each $n\in\mathbb{Z}_{\ge0}$, let $C_n$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\...
Yeah's user avatar
  • 347
6 votes
1 answer
381 views

Two surfaces in a 4-manifold whose algebraic intersection number is zero

Suppose $X$ is a smooth closed oriented 4-manifold, and $\Sigma_1,\Sigma_2$ are smoothly embedded compact oriented surfaces in $X$. Suppose they intersect transversally at two points with different ...
user302934's user avatar
4 votes
1 answer
250 views

Homology classes in connected sum of $\Bbb CP^2$'s that can be represented by smoothly embedded spheres

Let $h=[\Bbb CP^1]\in H_2(\Bbb CP^2;\Bbb Z)$. By a theorem of Kronheimer and Mrowka (Theorem 1 of this paper: https://people.math.harvard.edu/~kronheim/thomconj.pdf), a class $nh \in H_2(\Bbb CP^2;\...
user302934's user avatar
5 votes
1 answer
211 views

Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?

In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3_{\Bbb Z}\to \Bbb Z$, where $\Theta^3_{\Bbb Z}$ is ...
user302934's user avatar
5 votes
0 answers
177 views

Bar constructions and pushouts

Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout. Is there any hope of ...
Pedro's user avatar
  • 1,554
2 votes
1 answer
1k views

Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology

I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet ...
6 votes
0 answers
359 views

Singular homology using singular cubes

When singular homology is defined using cubes instead of simplices it is important to factor out the degenerate cubes in the course of building the singular chain complex. If you omit this step it is ...
Peter Kropholler's user avatar
3 votes
0 answers
130 views

Homology groups of a certain simplicial complex

I've run across a simplicial complex which, according to Sage, seems to have a very easily-described homology. However, proving this fact has been rather difficult. Fix $s\ge 2$ (though I would be ...
Marcel K. Goh's user avatar
9 votes
0 answers
216 views

Are multiples of representable homology classes still representable by smooth submanifolds?

Recently the following question comes up in my research. Suppose we have a closed compact connected smooth manifold $M,$ of dimension $d+c$ and an integral homology class $[N]$ induced by a compact ...
Zhenhua Liu's user avatar
3 votes
0 answers
154 views

Chekanov-Eliashberg Legendrian DGA with positive grading?

I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...
Nikhil Sahoo's user avatar
  • 1,205
2 votes
2 answers
367 views

Spaces homotopy dominated by $S^2 \times S^2\times S^2$

We say that a topological space $A$ is homotopy dominated by a topological space $X$ if there exist continuous maps $f:A\to X$ and $g:X\to A$ such that $g\circ f\simeq 1_A$. Let $X$ be $S^2 \times S^2 ...
M.Ramana's user avatar
  • 1,182
2 votes
0 answers
118 views

Different definitions of p-fusion and Mislin's theorem

Currently, I am trying to understand and compute homology of finite groups with coefficients in a field of positive characteristic. So, I was searching for some results that could reduce this problem (...
Guillerme C. Cruz's user avatar
4 votes
1 answer
232 views

Mayer–Vietoris sequence for coproduct of Hopf algebras

Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in Agore - Categorical Constructions for Hopf ...
Grisha Taroyan's user avatar
1 vote
0 answers
105 views

Modular cycles?

It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive ...
DaveWasHere's user avatar
21 votes
1 answer
816 views

What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?

Consider the following partial order. The objects are unordered tuples $\{V_1,\ldots,V_m\}$, where each $V_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V_1 \oplus \cdots \oplus V_m =...
Inna's user avatar
  • 1,025
9 votes
2 answers
594 views

What is the rank of the period lattice of modular forms?

Let $f$ be a weight $2$ cusp form for the group $\Gamma_0(N)$. I was experimenting with integrals of the form $$ \int_r^s f(z) \, dz$$ where $r, s \in \mathbf{P}^1(\mathbf{Q})$ and the integral above ...
Adithya Chakravarthy's user avatar
5 votes
1 answer
348 views

Homology of singular chain complex modulo subdivision

Let $S_p(X)$ be the $p$-th singular chain group and $\mathcal S(X)$ be the singular chain complex of a topological space $X$. There is a barycentric subdivision operator (which is also a chain map) $\...
Zhang Yuhan's user avatar
1 vote
0 answers
181 views

Homological stability and relative homology [closed]

Given a sequence of topological spaces $\{Y_{k}\}_{k\geq1},$ the homological stability is a property that, there is a function $g:\,\mathbb{N}_{0}\rightarrow \mathbb{N}_{0}$ such that for any $i\geq0$ ...
Michael jordan's user avatar
2 votes
1 answer
149 views

Is a certain map a quasi-isomorphism?

$\DeclareMathOperator\Hom{Hom}$Assume $F$ and $M$ are respectively right and left modules over a ring $R$ and let $I^\bullet$ be a left-bounded exact complex of $R$-$R$-bimodules. We know there is a ...
H. Ali's user avatar
  • 161
-1 votes
1 answer
176 views

If $H_i(U_j)=0$ for infinitely many $j$ then $H_i(X)=0$ [closed]

Let $X$ be a topological space and $U_i$ open subsets. If $U_i\subset U_{i+1}$ and $\bigcup^{\infty}_{i=1}U_i=X$. How can I prove that if for infinitely many $j$, the $i$-th homology vanishes $H_i(U_j)...
Mary Susy's user avatar
6 votes
0 answers
615 views

Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary?

Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary? I am also interested in several variations of this question. ...
Arshak Aivazian's user avatar
4 votes
0 answers
234 views

Being a product - from homology to topology

The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras. Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes ...
Andrea Marino's user avatar
1 vote
0 answers
77 views

Optimality condition of the harmonic form representatives of a homology class

In "Hodge theory on metric spaces, Smale et al." the $d$-th harmonic forms of the Hodge Laplacian $\Delta_d=\delta^* \delta+\delta \delta^*$ satisfying $\Delta_d(f)=0$ are claimed to be ...
Alex Keros's user avatar

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