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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16 views

Show that $(f+g)_{\ast}=f_{\ast}+g_{\ast}$

I have the following problem: Let $X$ be some (path-connected) topological space. I have to show that for two $f,g\in\pi_{n}(X)$ we have that $(f+g)_{\ast}=f_{\ast}+g_{\ast}$, where $\ast$ denotes the ...
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14 views

Representable cohomology theories in motivic homotopy theory

I am reading Mazza, Voevodskys and Weibels book on Lecture Notes on Motivic Cohomology and have grown curious about the following question: Which cohomology theories on $Sm/k$ is representable, i.e. ...
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51 views

About the functoriality of the long exact sequence in cohomology

I'm writing some notes on homological algebra and there I proved the long exact sequence in cohomology using the snake lemma. (I can give more details if anyone wants.) The next natural step is to ...
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30 views

Why does the exactness of a Koszul complex require commutativity?

Most references on Koszul complexes seem to assume that the elements $x_1,\ldots, x_n$ live in a commutative ring or are central. It appears to me that the proof also works provided the weaker ...
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1answer
35 views

Same winding number implies homotopic

In the lecture we defined the winding number $\omega$ to be the unique integer $d$ such that for a continuous map $f:S^{n}\to S^{n}$ it holds that $f_{\ast}:H_{n}(S^{n})\to H_{n}(S^{n}), [a]\mapsto d[...
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45 views

Computing cohomology of tautological bundles on Fano threefold

Consider the Fano threefold $V=V_5$ of index $2$ and degree $5$. Note that $V = \mathrm{Gr}(2,5) \cap \mathbb{P}^6 \subset \mathbb{P}^9$. Let $U$ and $Q$ be the tautological and tautological quotient ...
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(Massey) map of pairs induces commutative diagram between exact sequences of relative homologies

I am reading Massey "A basic course in Algebraic Homology". In chapter 7 Massey asserts the commuativity of all squares in the diagram (7.6.1). Unfortunately the details of the proof are ...
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21 views

Boundary Map of Simplicial Chain Complexes over $\mathbb{Z}/4$

There are several computer programs (e.g. Sage) that are able to compute the boundary maps of simplicial chain complexes over finite fields. However, I have not been able to find a computer program ...
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15 views

Constructing a singular simplex from other two singular simplexes.

Suppose $\sigma_1:\Delta^k \rightarrow X$ is a singular $k$-simplex and $\sigma_2:\Delta^l \rightarrow X$ is a singular $l$-simplex. Is there a singular $(k+l)$-simplex, $\sigma: \Delta^{k+l} \...
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42 views

Simple proof of Kunneth formula for coefficients in a field

The usually proven Kunneth formula, when taking coefficients in a DIP, results in a split exact sequence relating the (co)homology of $X\times Y$ to the (co)homology of $X$, $Y$ and a Tor functor ...
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1answer
42 views

How is the sum of simplices defined?

I have recently started to learn about simplices, simplicial complexes and simplicial homology. I understand that for some set of points $\{ p_0, \ldots, p_k \} \subset \mathbb{R}^n$ a simplex $\sigma$...
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36 views

When is a map inducing isomorphisms on homology with rational and mod $p$ coefficients a weak equivalence?

Suppose $f \colon X \to Y$ is a map of simply connected topological spaces that induces isomorphisms on homology with rational coefficients as well as homology with $\mathbb{Z}/p$ coefficients for all ...
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55 views

Computing the Cohomology of a genus-2 surface

I know that it has already been discussed which is the Cohomology of a genus-2 surface but I have a very concrete question about the procedure. I want to compute the Cohomology of the following genus-...
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38 views

General linear group of tensor product.

Given two noetherian algebras $A$ and $B$ over a field, is there any description of $H_n(GL(A\otimes B))$ in terms of $H_i(GL(A))$ and $H_i(GL(B))$? Let's assume homologies for $A$ and $B$ are trivial,...
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47 views

Künneth formula for cohomology

I see that most authors prove only the Künneth formula for Homology, and get the formula for cohomology as a consequence. My question here is how to get from one to the other. I assume you need to use ...
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1answer
40 views

Long exact sequence for mapping torus from Mayer-Vietoris

Note: I have actually done the thing I am trying to do. My question is whether there is an easier way to do it. Question: Let $f:X\to X$ be a map of topological spaces, and define the mapping torus of ...
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53 views

Manifolds with Euler characteristic equal to $\pm 1$

A compact connected smooth surface has Euler characteristic equal to $\pm 1$ if and only it is homeomorphic to the real projective plane or the connected sum of $3$ real projective planes. What are ...
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54 views

Homotopy with rational coefficients of simply connected manifolds that are not-rational homology spheres

I have been reading a book on differential geometry/dynamical systems and this question comes from a fact used in a proof. The author says the following : Note first that $W$ is simply connected and ...
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78 views

Homology groups of spherical $3$-manifolds

Let $G$ be a finite subgroup of $SO(4)$, acting freely on $\mathbb{S}^3$. How can we compute $H_2(\mathbb{S}^3/G; \mathbb{Z})$? Is it zero?
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23 views

Induced homomorphism in homology is surjective for finite regular covering maps

Let $\pi : M^n \to N^n$ be a regular and finite sheeted smooth covering map between smooth $n$-manifolds. I know that the induced map $\pi^\ast : H^p_{dR}(N) \to H^p_{dR}(M)$ in de Rham cohomology is ...
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Cohomology group $H^{i}(G;\mathbb{Z})$ where $G= (\mathbb{Z}_{p_1})^{n_1} \times \cdots \times (\mathbb{Z}_{p_k})^{n_k} $. [closed]

How to calculate the cohomology group $H^{i}(G;\mathbb{Z})$ where $G= (\mathbb{Z}_{p_1})^{n_1} \times \cdots \times (\mathbb{Z}_{p_k})^{n_k} $, and $p_1, \cdots , p_k$ are prime numbers and $n_1, \...
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67 views

$E^*(\mathbb{C}P^{\infty})=\bigoplus_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$ or $\prod_{n\in\mathbb{Z}}E^n(\mathbb{C}P^{\infty})$?

PRELIMINARY DEFINITIONS: Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have: $$ \tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt) $$ So there is a special ...
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Homology group of weird space using Mayer-Vietoris

For the space $X$ depicted above compute the homology groups. Attempt: I tried to use Mayer Vietoris with $A=X/D_1$ and $B=D_2$ ($D_1$ a disc at the center and $D_2\supset D_1$) getting an exact ...
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73 views

Cohomology ring of complex grassmanian as a quotient

This is about the complex version of this question, I am interested in understanding the integral cohomology ring of $\mathrm{Gr}_k(\mathbb{C}^n)$ as the following quotient of the integral cohomology ...
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1answer
83 views

Homology of Alexander Horned Sphere

I am taking a course in homology this semester, and so far we have only examined spaces/surfaces that the simplicial structures are rather easy to find. I was curious about the Alexander Horned Sphere,...
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52 views

Proof of universal coefficient theorem in cohomology

In the notes of a course of algebraic topoplogy, the universal coefficient theorem (for homology) is proven as it follows. Given the chain complex $\mathcal C$: $$\dots\to C_2\xrightarrow{\partial_2}...
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35 views

Explicitly Calculating the Cohomology of $n$-simplex

My algebraic topology lecturer has set me what I think is a tough exercise in persistence. I am meant to calculate the cohomologies of an $n$-simplex, $\Delta[n]$ which I know is $\mathbb{Z}$ for $H^0(...
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1answer
93 views

Basic computation in Steenrod algebra

In Homology operations for $H_\infty$ and $H_n$ spectra (pdf), Steinberger makes the computation of the Dyer-Lashof operations in $H\mathbb F_p$, and at some point uses the following "basic fact&...
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67 views

Cohomology of a smoothly embedded space curve

Let $\mathbb{P}^3 = P(\mathbb{C}^4)$ and $\gamma:C\rightarrow \mathbb{P}^3$ be a smoothly embedded algebraic space curve. Then its total Chern class is $c(C) = c(TC) = 1+a\in H^*(C)$, where $a^2=0$. ...
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54 views

Question about surjective map $S^n \to S^n$ of degree zero

There exists surjective map $f:S^n \to S^n$ of degree zero. By Hopf theorem, $f$ is homotopic to some constant map $c_{y}:S^n \to \{y\}$. Let the homotopy be $H(x,t)$, $H(x,0) = f(x)$, $H(x,1) = c_y(x)...
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1answer
52 views

How to find generators in simplicial cohomology? For example, for $S^1$ and $S^2\dots$

Is there a way to find generators in cohomology groups? Any algorithm or even philosophical remark would be highly appreciated. For example, I have $S^1\cong K$ represented by $1$-dimensional ...
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22 views

Finding a natural homomorphism between two relative cohomology groups

I'm trying to better understand how relative homology/cohomology works in order to make use of them for a specific example in mind. Here I'll be working with cohomology over the integers. Let $A \...
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1answer
85 views

$\mathbb{R}^n$ minus a simple closed curve is simply connected for $n\geq 4$?

I would like to prove the statement in my question, at least for the case $n=4$. Here is some context for it; in the study of the general Jordan curve theorem, one confronts with the following result (...
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1answer
28 views

Homology Groups of $S^n$

Is it possible to consider $S^n$ as a $0$-simplex and a singular map of an $n$-simplex so that the $n$-simplex forms the surface of $S^n$ minus a point, and the point is the singular map of the $0$-...
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40 views

Singular homology on $S^n$ [duplicate]

Is it possible to consider $S^n$ as a $0$-simplex and a singular map of an $n$-simplex so that the $n$-simplex forms the surface of $S^n$ minus a point, and the point is the singular map of the $0$-...
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1answer
21 views

Dimensions of cycles and boundaries in a full simplex

$\newcommand\rk{\operatorname{rk}}$Let $\Delta_n$ denote the full $n$-simplex $\{n,\dotsc,0\}$. It is clear that there are $\binom{n+1}{d+1}$ many $d$-simplices, since a $d$-simplex corresponds to a ...
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1answer
26 views

Homotopical factorization of maps that are zero in homology

Suppose $f: M \rightarrow N$ is a cellular map between CW-complexes of dimension $m$ and $n$ respectively such that the induced maps in homology are zero after a while (i.e. $H_i(f) = 0$ for some $k$, ...
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50 views

Product structure of $H^*(\mathbb RP^{2n+1};\mathbb Z_2)$ using spectral sequence

I am trying to compute the product structure of $H^*\left(\mathbb RP^{2n+1};\mathbb Z_2\right)$ using spectral sequence from $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$, assuming that we already know $H^...
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41 views

Central extension of the Heisenberg Lie algebra

I'm trying to compute $H^2(\frak h_3, \mathbb R)$ where $\frak h_3$ is the Heisenberg algebra and $\mathbb R$ is a trivial $\frak h_3$-module. From my attempt the cohomology seems to be two ...
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13 views

Filtered complex and cohomology (Differential Forms in Algebraic Topology by Bott and Tu)

In Section 14, for a filtered complex the following is defined: \begin{equation} A = \bigoplus_{p\in\mathbb Z}K_p. \end{equation} Define $i:A\rightarrow A$ is the inclusion as \begin{equation} K_{p+1}\...
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1answer
108 views

When does $H^{n-k}(M,\Bbb R)\simeq \Bbb R$ imply $H_{k}(M,\Bbb Z)\simeq \Bbb Z$?

I am not sure that my question is a trivial fact or not or even make sense or not. Anyway I want to know When does cohomology group $H^{n-k}(M,\color{blue}{\Bbb R})\simeq \Bbb R$ imply homology group ...
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22 views

Applying the Leray-Hirsch theorem on certain manifolds

Let $M$ be a connected closed orientable manifold with Euler characteristic $0$. Let $TM$ be the tangent bundle over $M$ and $SM$ be the induced sphere bundle with fiber $S^{n-1}$. Looking at page 432 ...
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40 views

Homology of $X \times I$

I'm trying to compute $H_n(\Sigma_g \times S^1,\mathbb{R})$ (where $\Sigma_g$ is the a closed orientable surface of genus g). As part of the answer I wanted to use Mayer–Vietoris sequences since I ...
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1answer
54 views

Homology of the unit ball $D^3$ with identification of boundary points by $180^\circ$ degree rotation around the vertical axis

This problem comes from Topology and Geometry of Bredon : Let $X$ result from $D^3$ by identifying points on its boundary $S^2$ taken into one another by the $180^\circ$ rotation about the vertical ...
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37 views

Understanding the Eilenberg-Zilber map on singular chains

I am having trouble to understand how the image of an element $a\otimes b\in C_p(X)\otimes C_q(Y)$ under the Eilenberg-Zilber map on singular chains is an element of $C_{p+q}(X\times Y)$. I am using ...
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1answer
52 views

Intuitive Question about Klein Bottles and Torsion in Surface Groups

This is just a geometric sort of question about how to picture/intuit on homology, hope it's considered attractive and not just "vague" haha So if you take a typical generator for the ...
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1answer
55 views

$\mathbb Z_2$-equivariant cohomology of tori

I consider the tori $X=(\mathbb R/ 2 \pi \mathbb Z)^d$ on which the group $\mathbb Z_2$ acts by $x \mapsto -x$ for the nontrivial element of $\mathbb Z_2$. This action has $2^d$ fixed points, which ...
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1answer
39 views

Different Representations of Dunce Cap

I am having a bit of confusion concerning the Dunce Cap while studying simplicial homology, hope someone can help! Given a solid triangle with vertices $a, b, c$ I usually see the Dunce Cap defined as ...
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62 views

Can one set and use some axioms for calculating intersection forms of manifolds?

I want to calculate the intersection form of some (four?) manifolds, and I wonder is there any axioms that one can compute the intersection form of (four?) manifolds just by them? like axioms of ...
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188 views

Examples of Finite-Dimensional Space with Non-Vanishing Homology in Higher Dimensions?

The Barratt-Milnor Sphere $X_n$ is an $n$-dimensional space which has non-vanishing singular homology in arbitrarily high dimensions. The space $X_n$ is a generalized Hawaiian Earring, i.e. the $n$-...

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