# Questions tagged [homology-cohomology]

Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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### Cech cohomology cap product

This is a rephrasing of a recent post I made, then swiftly deleted. I am reading this paper, where on page 95 it is mentioned that if $X$ is a closed oriented manifold and $K \subseteq X$ is a closed ...
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### The Motivation of Axioms for Homology [closed]

I am reading Algebraic Topology by Allen Hatcher and I'm curious about the motivation of axioms for singular homology. Singular homology is easy to be defined and understood,so I think singular ...
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### The First Singular Homology Group $H_1(X)$ and the Fundamental Group

THEOREM. Let $X$ be a topological space and let $x_0\in X$ a point. The map $\varphi\colon\pi_1(X,x_0)\to H_1(X)$ defined by $[\sigma]_{\simeq}\mapsto[\sigma]_{\sim}$ is well defined and it is a ...
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### How can both the Čech complex and the alpha complex have the same homotopy type as the union of balls if they are constructed differently?

I understand that, according to the nerve theorem, both Čech and alpha complexes have the same homotopy type as the union of balls. However, consider the following four points: A = $(1,0)$ B=$(-1,0)$,...
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### Cohomology group for trivial group

Let $G$ be a group, $A, B$ be a $G$ - modulo. We can define the n-th Cohomology group of $G$ with coefficient in $A$. $$H^n(G,A) =\text{Ext }_G^n(\mathbb{Z},A)$$ And the n-th Homology group of $G$ ...
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### Homeomorphism of $[0,1]^m$ with subspaces of $S^n$

I was reading the Borsuk-Ulam theorem, which states that there is no continuous map $f$ from $S^2$ to $S^1$ which satisfies $f(-x)=-f(x)$. One question came to my mind: Is there any subspace of $S^n$ ...
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### Calculating homology of cobordism of 3-manifolds from Kirby diagram

I've been reading Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci and Stipsicz, and have gotten stuck on the following exercise on p. 44. Below $Y_1, Y_2$ are closed 3-manifolds, and $Q$ ...
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### The definition and properties of the restriction of integral （rational） cohomology to a subspace within the ambient space.

This seems to be a fairly standard question, but I am somewhat confused when considering it using Čech cohomology theory. Let $X$ be a paracompact topological space admitting good cover and $Y$ a ...
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### Cup products in the rational cohomology of products of spaces

I'm starting to learn cup products and I have been stuck at a problem. I need to find the cup product structure in the rational cohomology ring of $X = S^3 \times T^3$, where $S^3$ is the $3$-sphere ...
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### Rational homology sphere has the same cohomology ring as sphere

Let $M$ be a closed, $n$-dimensional manifold such that $H_0(M;\mathbb{Q}) = \mathbb{Q} = H_n(M;\mathbb{Q})$ and $H_i(M;\mathbb{Q}) =0$ for all $1 \le i \le n-1$. Then $M$ is a rational homology ...
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### How to get the algebraic expression of the prism operator in any dimension?

I am reading Hatcher's Algebraic Topology and am curious about how to design the algebraic expression of the prism operator even if I know the intuition is to create a cylinder such that the boundary ...
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### Reference for wiki claim about degree of spheres

I have seen the following claim on the wikipedia page for 'vector field' about degree theory on vector fields. The index is not defined at any non-singular point (i.e., a point where the vector is ...
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### Homology group of an attaching space

Consider a cylinder without bottom nor top: $S^1 \times [0,1]$ and another circle $S^1$ on the complex plain. We may define a map of $f: z \rightarrow z^2$ on $\mathbb{S}^1$, which is a map of degree ...
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### Did the Descartes-Euler conjecture influence Poincaré's theory of homology?

What is the Descartes-Euler Conjecture? all simply-connected polyhedra with simply-connected faces are Eulerian $V-E+F=2$. Additional explanation: The lemma which was falsified by the ring-shaped ...
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### How to set up this problem geometrically? (Hatcher AT Page 131 Problem 3)

I'm attempting to go through all of Hatcher's problems on homology. I was able to do 1, 2, 4, and 5 so far, but I don't know what he's asking for geometrically in 3. I see this thread (Hatcher ...
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### Spectral sequence of $\text { the fibration } E X \xrightarrow{\Omega X} X$

I'm studying Fomenko-Fuchs Lecture 22.3 and i'm stuck on the proof of the following theorem at page 333: Theorem: Let X be a topological space (with a base point), and let the space $X$ be $(n-1)$-...
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### A quick question on equivalence between cellular and singular homology

We know that for a CW complex $X$, the cellular and singular homology groups are isomorphic: $H_n (X) \cong H_n ^{CW} (X)$ for each $n \in \mathbb{N}$. My question is the following: suppose $Y$ is ...
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### Relation between two theories of degree of maps

For a holomorphic map between 2 Riemann surfaces, we can define its degree to be the sum of ramification index of all preimages of a point, or simply the cardinality of the preimage of a regular(not ...
Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
### Definition of $E^{\infty}_{p,q}$ in spectral sequence $\{E^r,d^r\}_{r\in\mathbb{Z}}$
Let $\{E^r,d^r\}_{r\in\mathbb{Z}}$ a spectral sequence, that is $E^r$ are differential bigraded $R$-modules $E^r=\bigoplus_{p,q} E^r_{p,q}$ with $d^r$ differential of degree $(-r,r-1)$ (that is \$d^r=\...