# 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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### 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 ...

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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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### 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 ...

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### Tate cohomology and cup product: functoriality in $G$

Let $G$ be a finite group, and let $A, B$ be $G$-modules.
See Atiyah and Wall [AW] for the definition of the Tate cohomology groups $H^q(G,A)$ for all $q\in\mathbb Z$
and of the cup product pairings
\...

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### 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 ...

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### 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 =...

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### 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 ...

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### 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) $\...

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### 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 $i\geq0$ ...

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### 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 ...

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### 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)...

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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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### 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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### 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 ...

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### Regular mapping space vs continuous mapping space for affine schemes

Let $A$ and $B$ be Zariski open affine sub-schemes of $\mathbb{A}_{\mathbb{C}}^{n}$ and $\mathbb{A}_{\mathbb{C}}^m$ respectively. We denote the infinite symmetric product of $B$ by $Sym^{\infty}(B)$. ...

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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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### Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem.
Link to the statement of the theorem
...

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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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### Conclusion of Hurewicz for $H_3$ without vanishing fundamental group?

Fix a space $X$, which I want to assume is a manifold. Under the assumption of simple-connectivity, Hurewicz's theorem tells us that
$$
\pi_3(X)\to H_3(X,\mathbb{Z})\qquad (*)
$$
is surjective, hence ...

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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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### Rational systole of a manifold

I also posted this question on MSE, but since it may be a delicate question, I decided to post it here.
Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ ...

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### Cellular homology of the universal cover

Let $X$ be a connected pointed CW complex. Let $\tilde{X}$ be its universal covering space and $G=\pi_{1}(X)$.
Lets denote $(C^{Cell}_{\ast}(\tilde{X}),d)$ the cellular chain complex associated to $\...

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### Explicit $BP_*BP$-comodule structure on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$

So as it says in the title, how can one explicitly calculate the comodule structures on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$ for a prime $p$?
For example, $\mathbb{C}P^2$ sits in a ...

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### Conditions under which the preimage of a submanifold in nontrivial in homology

Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...

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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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### 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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### $E_\infty$-space structure of $B\mathrm{GL}(\mathbb S_{(p)})$

In Geometric Topology - Localization, Periodicity, and Galois Symmetry by Dennis Sullivan, we can read that there is a decomposition
$$B\mathrm{SL}(\mathbb S_{(p)})\times K((\mathbf Z_{(p)})^\times)\...

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### A covariant functor on a given abelian category and comparison of homology in target and source

The definition of cohomology of a complex is based on the following:
We have a complex (of appropriate objects) $$0\leftarrow C_0\leftarrow C_1\leftarrow C_2\ldots \leftarrow C_n\ldots$$
Then for an ...

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### Relative homology of free loop space with respect to constant loops

Let $Q$ be a closed manifold with $\dim Q\geq2$ and let $\Lambda_0Q$ be the connected component of the free loop space of $Q$ whose elements are contractible loops. I am looking for conditions on the ...

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### Every disk in $(S^2 \times S^1) \setminus B$ whose boundary lies in $\partial B$ separates

Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. Is it true to assert that every embedded $2$-disk $D \subset M$ such that $D \cap \partial M = \partial D$ ...

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### What topological spaces can be realized as cell complexes?

What are the topological spaces can be realized as cell complexes, up to homeomorphism? It seems for instance that all manifolds can be built from cell complexes. It is clear however that one can ...

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### regular CW complex and incidence matrices

Suppose that we have a regular CW-complex $X$. I want to define the incidence matrix of $k$-skeleton of $X$ with respect to the $k-1$ skeleton and I wonder what might go wrong in this case.
If it's ...

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### Representing relative homology classes orientable surfaces with boundary

Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies ...

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### loop space of a finite CW-complex

Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups ...

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### Is an (n-1)-sphere quotient by an (n-1)-sphere contractible? [closed]

I am thinking about the homotopy type of the following quotient space:
Let $X$ be a topological space and $A$ be a subspace of $X$. If both $X$ and $A$ have homotopy type of a sphere $S^{n-1}$ (of the ...

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### Nontrivial integer homology class implies orientability

I posted this question on MSE and I would like to see if my reasoning is correct.
Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and ...

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### Homotopy equivalent Postnikov sections but not homotopy equivalent

Two pointed, connected CW complexes with the same homotopy groups need not be homotopy equivalent (Are there two non-homotopy equivalent spaces with equal homotopy groups?). Moreover, having the same ...

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### Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of
$G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf
...

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### Analog of Cartan model for equivariant homology

Let $X$ be a manifold, acted on by a Lie group $G$.
(For example $X$ real-even-dimensional acted on by $G=U(1)$ with only finitely many isolated fixed points.) The Cartan model for $G$-equivariant ...

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### Questions about a structure related to simplicial complexes

While researching some superficially unrelated theory, a structure similar to the one described below presented itself to me. I'm having trouble with identifying what's the structure name. It seems to ...

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### 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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### 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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### 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 (...