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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Cohomology group of pairs (X,A) where X is thickening figure 8 and A is its 3 boundary components

Calculate Cohomology group of pairs (X,A) where X is thickening figure 8 and A is its 3 boundary components using simplicial structure.(without using UCT for relative pairs) I have no idea how to ...
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Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
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1answer
45 views

Trying to make sense of this proof in Hatcher

So I'm trying to understand this proof in Hatcher's Algebraic Topology. Lemma: The composition $\Delta_n(X)\xrightarrow{\partial_n} \Delta_{n-1}(X)\xrightarrow{\partial_{n-1}}\Delta_{n-2}(X)$ is zero ...
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Cohomology groups as sub vector spaces over $ \mathbb{Q} $. [on hold]

Let $X$ be a non singular algebraic complex projective variety. How to construct a sub-vector space of the vector space $ H^k ( X , \mathbb{Q} ) $ ?. Is it true that $ H^k (Z , \mathbb{Q} ) $ is a ...
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1answer
62 views

What is a homology of chain complex?

First of all I don't know much about homological algebra, and algebraic topology I just took a class using Kinsey book (Topology of Surfaces) which is undergraduate math class and I learned what is ...
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42 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
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How to recover the cohomology of a torus from its description of a quotient

Note: here, "cohomology" means "De Rham cohomology". I know how to compute the De Rham Cohomology of a torus $T=\left(S^1\right)^n$ using Kunneth formula. But a torus can also be obtain as a ...
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Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
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37 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
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37 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
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1answer
28 views

With field coefficients homology and cohomology coincide

With field coefficients the universal coefficients theorem takes the form: $$H^n(X;F)=Hom_{F-modules}(H_n(X;F),F)$$. Now in all computations I have seen with field coefficients we have ...
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1answer
23 views

Homology and Reduced homology coincide on non trivial pair.

In Hatcher page 118, he says that There is a completely analogous long exact sequence of reduced homology groups for a pair $(X;A)$ with $A\not = \emptyset$ ; This comes from applying ...
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Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
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1answer
52 views

Computing Klein bottle's cohomology ring in $\mathbb{Z}$

Well I've been struggling with this one. This is the picture of the Klein Bottle. It has two triangles (U upper, V lower), three edges (the middle one is "c") and only one vertex repeated 4x. So my ...
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1answer
93 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
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115 views

On defining homology groups

I have been trying to understand what homology groups are "talking about," and now I am wondering if the following works as a definition of homology. But first, some illustration of what it is ...
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1answer
81 views

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$. I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces ...
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38 views

Why “singular” in “singular homology/cohomology”?

As the title suggests, I'm curious to know whether there is any reason why the word "singular" appears in "singular homology/cohomology".
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Homology of connected sum of CW-complexes

Let $X$ and $Y$ be finite and connected CW-complexes of dimension $n$ with exactly one $n$-cell. Then we can define their connected sum $X\#Y$ just like in the manifold case: extract an ...
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32 views

Existence of Moore spaces for modules over commutative rings.

Let $R$ be a commutative ring, $A$ a $R$-module and $n$ a natural number. Does there exist a CW complex $M(A,n)$ with $\tilde{H}_i(M(A,n),R)=0$ if $i\neq n$ and $\tilde{H}_n(M(A,n),R)\cong A$ as ...
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Double complex with exact rows

Let $(K^{p,q},\delta,d)$ be a double complex of modules. We assume that $\delta$ of degree $(1,0)$, $d$ has degree $(0,1)$ and $d$ and $\delta$ commute. Since $d$ and $\delta$ commute, then $\delta$ ...
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Is it the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?

So as the question statement asks, is it necessarily the case that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism? I suspect the answer is yes, but I don't know ...
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Homology and fundamental group of closed ball minus a “T”

I wish to solve the following problem: Let $D = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1\}$. Let $Y = \{(x, 0, 1/2) \in \mathbb{R}^3 \mid x \in \mathbb{R}\} \cup \{(0, 0, z) \in ...
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Confused with notations about Leray's theorem for singular cohomology

The following theorem is copied from Bott's book Differential Forms in Algebraic Topology in Page 192: Theorem 15,11 {Leray's theorem for singular cohomology with coefficients in a commutative ...
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1answer
47 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
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1answer
36 views

Equivalence between derived categories preserve distinguished triangles

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!
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1answer
19 views

Homology of a pair of simplicial complexes

Let $K=\{a, b, c, ab, bc\}$ (in graph-theoretic terms, a path $a,ab,b,bc,c$, where $a, b,$ and $c$ are vertices and $ab$ and $bc$ are edges) be a simplicial complex and let $L=\{a, b, c\}$ be a ...
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31 views

definition of homology via spectra

Let $K(\mathbb{Z}, n)$ denote a Eilenberg-Mac Lane space, characterized by $H^n(X, \mathbb{Z})=[X, K(\mathbb{Z}, n)]$ for all spaces $X$. In stable homotopy theory, the corresponding homology theory, ...
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68 views

Euler characteristic: dependence on coefficients

Let $X$ be a finite CW complex and $\chi(X)$ its Euler characteristic (defined using integer coefficients). When is it true that $\chi(X)=\sum (-1)^i \dim H_i(X;F)$, where $F$ is a field? I thought ...
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3answers
92 views

Homeomorphic manifolds have the same dimension

So I want to prove: If two manifolds $M$ and $N$ are homeomorphic then $dim(M) = m = n = dim(N)$. My idea was to use the property of the manifolds that they are locally homeomorphic to the ...
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1answer
116 views

Non-orientable one dimensional manifold.

I was trying to solve a question from Hatcher's book in section 3.3. Question is: Show that there exist a non-orientable 1-dimensional manifold if Hausdroff condition is droped from the definition ...
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Simplicial homology and cohomology of Möbius band

I'm trying to calculate simplicial homology and cohomology groups, with coefficients in $\mathbb{Z}$, of Möbius band $M$. I use triangulation given on the following picture: We have in this case ...
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Basis of (first de Rham) cohomology: $y^n=f(x)$

Let $K$ be a field, $f(x) \in K[x]$ be a monic polynomial with distinct roots, $\deg(f)=d$. Let $R=K[x,y]/(y^n-f(x))$ and $C=Spec(R)$. $\:\:\;\:\:\:\:\quad$ ($n>2$ integer) What is the basis ...
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Calculate the homology group of $S^3/G$, an Harvard qualifying exam problem with “unclear” solution

Problem Suppose that $G$ is a finite group whose abelianization is trivial. Suppose also that $G$ acts freely on $S^3$. Compute the homology groups (with integer coeffcients) of the orbit space ...
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1answer
53 views

Show that $f:X\to \mathbb{RP}^n$ factors through $S^n\to\mathbb{RP}^n$ iff $f^*:H^1(\mathbb{RP}^n,Z_2)\to H^1(X,Z_2)$ is zero

Problem Show that (1) $f:X\to \mathbb{RP}^n$ factors through $S^n\to\mathbb{RP}^n$ iff $f^*:H^1(\mathbb{RP}^n,Z_2)\to H^1(X,Z_2)$ is zero (2) $f:X\to \mathbb{CP}^n$ factors through ...
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1answer
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group cohomology of 3-manifolds

If $M$ is a closed aspherical 3-manifold with first fundamental group $G$, then cohomology groups of $G$ and $M$ are isomorphic because $M$ is a Eilenberg-MacLane space $K(G,1)$. In particular there ...
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1answer
43 views

Degree of a map over a different ring in homology

The degree of a map $f: S^n \to S^n$ is definied as the unique integer $H_n(f;\mathbb{Z} ): H_n(S^n;\mathbb{Z}) \to H_n(S^n;\mathbb{Z})$ since $H_n(S^n;\mathbb{Z}) \cong \mathbb{Z}$. Now my question ...
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50 views

Counting chain maps

Let $\mathbb{K}$ be a field and let $C_{\cdot}$ and $K_{\cdot}$ be bounded chain complexes with coefficients in $\mathbb{K}$. Then the set of chain maps $f_{\cdot}:C_{\cdot}\to K_{\cdot}$ is a ...
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Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
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2answers
61 views

Book recommendation: Homology and Cohomology

I need to learn some concepts about Homology and Cohomology theory to apply in riemannian geometry basically, but really I have not time to read about that. I know just two books of W. S. Massey, ...
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Relations between $R^fG$ and either $\mathbb{C}^fG$ or $\mathbb{Z}^fG$.

Denote by $RG$ the group ring of the group $G$ over the commutative ring $R$. A result by Passman saying that if $R$ is a commutative ring then $$RG=R\otimes_{\mathbb{Z}}\mathbb{Z}G.$$ As a result, ...
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A Ham Sandwich type problem

If $A_1,...,A_n$ are measurable subsets of $S^n$, then there is a great $S^{n-1}$ cutting each $A_i$ exactly in half. The tools I have at my disposal are the Borsuk Ulam theorem and the Ham ...
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Is this a correct understanding of what de Rham's Theorem is saying?

De Rham's Theorem states that $H_{dR}^k(M) \simeq H^k(M;\mathbf R)$ for all $k$. This is what it states. But I've been struggling to understanding what it's telling me. Here's my understanding so far: ...
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1answer
69 views

Compute the fundamental and homology groups of $S^3 \setminus K$, where $K$ is two linked copies of $S^1$ in $\mathbb R^3$

Compute the homology groups of $S^3 \setminus K$, where $K$ is two linked copies of circles in $\mathbb R^3$. How about the homology group of $S^3 \setminus K'$ where $K'$ is just one copies ...
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23 views

recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
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natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
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when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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1answer
24 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
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64 views

Singular homology: Change of coefficients

Let $f: X \to Y$ be a map of topological spaces which induces isomorphisms $H_*(f;\mathbb{Z})$ on singular homology with $\mathbb{Z}$-coefficients. Show that $f$ induces isomorphisms ...
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1answer
58 views

How can I get a cohomology of hypersurfaces by using their equation?

While studying about complex projective hypersurfaces, I attempts to find a cohomology of this hypersurface : $$X_n=\{(x_0:x_1:x_2:x_3) \in \mathbb{C}\mathbb{P}^3~|~x_0^n+x_1^n+x_2^n+x_3^n=0\}$$ I ...