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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Homology classes of subvarieties of toric varieties
Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety.
Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero?
Background
If $X$ is a Kaehler variety, this is of ...
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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 ...
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Singular chains generated by manifolds with corners --- does it really work?
Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...
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About maps inducing bijections on homotopy classes
Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
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Do homology classes have "special" representatives?
Recall that, according to Hodge, de-Rham cohomology classes of "nice enough" manifolds have "special" representatives - namely, harmonic forms.
Now, how does one choose a "special" one among ...
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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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Homology of Lie groups
Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...
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Singular chains as an HZ-module spectrum
For $R$ any ring and $H R$ its Eilenberg-MacLane spectrum -- a ring spectrum -- there is an equivalence between the $\infty$-categories of $H R$-module spectra and that of unbounded chain complexes of ...
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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 ...
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Low-Dimensional Spaces with High-Dimensional Homology
Barratt-Milnor Spheres $X_n$ are spaces with finite topological dimension $n$ but which have non-vanishing singular homology in arbitrarily high dimensions. Here, they prove that if $n > 1$ then ...
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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 until ...
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kernel of the mod $2$ Bockstein on the first cohomology group
Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
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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 ...
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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 ...
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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. ...
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Can a spherical simplicial complex have more than one "central" inversion?
Let $\Delta$ be a finite connected simplicial complex. Call a simplicial map $\phi:\Delta\to\Delta$ an inversion if
$\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and
$\phi$ is not ...
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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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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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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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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 ...
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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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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 ...
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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 ...
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Have mod $p^k$ Dyer Lashof operations been studied?
Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it ...
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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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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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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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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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Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher
It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a $2$...
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Uniform distribution of special homology classes mod-p
Let $X_0(N)$ be the usual modular curve. For $\chi$ a quadratic Dirichlet character of conductor $D$, define the homology class $c(\chi)=\sum_{i=0}^{|D|-1}\chi(i)\{\frac{i}{|D|}{\infty} \}$; here $\{...
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Does a Dehn twist in the mapping class group of an cobordism give a BV-operator in string topology?
In her article Higher string topology operations, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes d})...
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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 ...
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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 ...
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Homology of a fiber as a cotorsion product
Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let
$\...
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Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_\infty$-coalgebras?
Let $X_\bullet := ... X_2 \to X_1$ be a tower of connected and simple spaces
with the following properties:
The induced tower $H_\ast(X_\bullet; \mathbb{F}_p)$ of graded $\mathbb{F}_p$-vector spaces
...
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Milnor exact sequence for homology of hopf algebras
Let $K$ be a field and $\mathrm{Hopf}^K_{E_\infty}$ the $\infty$-category of
homotopy-coherent hopf algebras over $K$ that are coherently commutative and cocommutative.
Precisely, $\mathrm{Hopf}^K_{E_\...
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Localizations of spaces with respect to homology and right properness
Let $E$ be a spectrum (with corresponding homology theory denoted $E_\ast$).
In "Localization of spaces with respect to homology", Bousfield constructed a model category structure on the ...
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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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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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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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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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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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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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What is the formula for the homology class represented by the diagonal?
Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...
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Homology of derivations of Differential Graded Lie algebra
Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule).
On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...
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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 ...
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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{...
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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 (...
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Characteristic disks in $S^2 \times S^2$ for knots
I'm studying the article Genera and degrees of Torus Knots in $\mathbb{CP}^2$ and I ended up with a question.
We know that every knot is slice (i.e. bounds properly embedded smooth disk) in $S^2 \...
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