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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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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17 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 ...
3
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20 views

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
55 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, ...
1
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1answer
27 views

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, ...
6
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1answer
51 views

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 ...
2
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0answers
19 views

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: ...
2
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0answers
53 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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21 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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0answers
28 views

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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51 views

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
20 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 ...
2
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0answers
60 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 ...
2
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1answer
53 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 ...
2
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0answers
30 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
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0answers
31 views

Does anybody know a good introduction to homology?

Essentially what the title says. I need something that will give me a decent introduction into homology theory. I don't need too deep of an understanding, just enough to get through a paper I'm ...
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24 views

A “non-degenerate pairing” between $\operatorname{Gal}(K/k)$ and $K/k$

In this post, I'd like to compare Galois theory and homology theory. Due to the limit of my knowledge, I'm not sure if my consideration is right. I hope you can show me the right way. In topology, ...
2
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1answer
37 views

homomorphism of $H$-spaces between a monoid and loop space of its classifying space

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. The geometric realization ...
5
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1answer
75 views

Proof of Kunneth's formula in Bott & Tu

Let $M, F$ be smooth manifolds and let's assume all (henceforth, de Rham) cohomologies of $F$ are finite-dimensional. Let $\pi : M \times F \rightarrow M$ and $\rho : M \times F \rightarrow F$ be the ...
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0answers
54 views

the geometric realization of a simplicial set is contractible

Let $M$ be a monoid up to homotopy. The simplicial set $WM$ is defined by setting $$ WM_n=M^{n+1}=\{(g_0, g_1,\cdots,g_n)\mid g_i\in M\} $$ with faces and degeneracies given by \begin{eqnarray*} ...
0
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1answer
42 views

Calculate the first homology group of $P^2\#T$, that is $H_1(P^2\#T)$

I've already found that the fundamental group of the connected sum $P^2\#T$, by the labelling scheme $aabcb^{-1}c^{-1}$, to be $F_3/<aabcb^{-1}c^{-1}>$. How would I find the first homology ...
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51 views

Relationship between relative homology and reduced homology.

Prove for any homology theory ($H_*, \partial_*$) with values in R-mod that satisfies the dimension axiom there is an isomorphism $H_n(X,A)$ $\cong$ $\tilde H_n(X/A)$ where $A$ $\subset$ $X$ is a ...
3
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27 views

Transfer homomorphism in transformation groups

I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups. But suppose ...
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28 views

Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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54 views

Show $L_{3,1}\sharp L_{3,1}$ and $L_{3,1}\sharp \overline{L_{3,1}}$ are not homotopy equivalent [closed]

Let $L_{p,q}$ with $(p,q)=1$ the usual Lens space, I must show that $L_{3,1}\sharp L_{3,1}$ and $L_{3,1}\sharp \overline{L_{3,1}}$ are not homotopy equivalent using homology/cohomology tools. Here, ...
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0answers
14 views

Elementary Cube vs. Elementary Chain

I am reading Computational Homology by Kaczynski, Mischaikow, and Mrozek. On page 47, for every elementary cube, $Q \in \mathcal{K}_k^d$ they associate an object $\widehat{Q}$ that they call an ...
2
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1answer
39 views

Can we see directly from the cocycle condition that 2-cocycles are symmetric?

Let $A$ be an abelian group and let $C$ be a cyclic group. All central extensions of $C$ by $A$ are abelian because in any such extension $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
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136 views

Generalization of De Rham cohomology for spinor fields

Is there a generalization of De Rham cohomology for spinors fields? I can see that one can construct p form fields out of spinor field by contraction of the type $\bar{\psi} \gamma^{a_1} ...
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41 views

Are there any open topological spaces other than R3 with overall zero curvature (and asymptotic to R3 towards infinity)?

What I mean by this is as follows: Take an infinite flat manifold $\mathbb{R}^3$ with zero curvature. Then subtract out a knotted torus or linked tori. And sew them back in using Dehn surgery. (In ...
3
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1answer
74 views

Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$

I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$. A non-orientable ...
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0answers
27 views

Compute the singular homology group of a “rational optical grating”

Let $X$ be the subspace of the square $I \times I$ consisting of the four boundary edges plus all points in the interior whose first coordinate is rational. Calculate the singular homology ...
2
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1answer
94 views

The augmentation ideal of $\mathbb{Z}G$

Let $G$ be a cyclic group of order $p$ and let $IG$ denote the augmentation ideal of the group ring $\mathbb{Z}G$. I need to find $H^1(G,IG)$. Since $$0 \rightarrow IG \rightarrow \mathbb{Z}G ...
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0answers
53 views

homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...
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0answers
53 views

Computing the cohomology of the pair $(S^n\times S^n,D)$

Let $D=(x,x)\subset S^n\times S^n$ be the diagonal, and assume $n$ is even. I need to prove that the following sequence (taken from l.e.s of the pair) is exact $$0 \rightarrow H^n(S^n\times ...
4
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1answer
44 views

Prove that $w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$

Let $\gamma^n$ be the canonical $n$-plane bundle over the infinite Grassmann manifold $G_n(\mathbb{R}^{\infty})$. I'm asked to prove that $$w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$$ (exercise 9-A ...
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0answers
14 views

Local homology of a fibred product

Let $A,B$ be topological spaces and suppose that for $a\in A$ and $b\in B$ the local singular homology groups $H_k(A,A\setminus\{a\};\mathbb{Q})$, $H_k(B,B\setminus\{b\};\mathbb{Q})$ are known for all ...
1
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1answer
85 views

Homotopy equivalence and chain complexes

This is from the book: Hilton and Stammbach, A Course in Homological Algebra, Chapter IV, Derived Functors, exercise 4.2. Let $\varphi:C \to D$ be a chain map of the projective complex $C$ into ...
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0answers
17 views

Čech cohomology and fundamental class

I have a notational question. Simplified, I have a Cech cohomology on a simplical complex $\Sigma$ generated from the nerve of a covering of a set $X$. I also have a map $f: \Delta^n \to \Sigma$. In ...
4
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1answer
116 views

Homology of a co-h-space manifold

Let $M$ be a compact connected topological manifold of dimension $n>1$. Suppose the corepresented functor $[M,-]\colon Top_{\ast}\rightarrow Set$ lifts to monoids or equivalently that $M$ is a ...
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37 views

Cohomology of permutation representation

Consider the action of $S_n$ over $\{1,...,n\}$ consider the associated representation with integral coefficients $X_n$. What are $H^r(S_n,X_n)$? More in general is there a nice way to predict the ...
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2answers
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Objects that are quotient of two projective objects and cohomology in degree>1

1) What is an example of an abelian group which is not the quotient of two free abelian groups? For the abelian group $X$ for which this is true then for all Right exact functors F, i would have ...
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1answer
64 views

Computing homology w/ Mayer Vietoris

Let $(H_*, \partial_*)$ be a homology theory satisfying the dimension axiom. Let $n\ge1$ and $X= S^n \cup_f D^{n+1}$ where $f:S^n\to S^n$ is a degree $k$ map. Compute each homology group of $X$. I ...
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0answers
31 views

Viewing Koszul complex as an algebra

I keep coming across notes which says that the Koszul complex can be viewed as an algebra. Is it true that complexes can be viewed as an algebra. If the complex is not exact, can the homologies also ...
3
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2answers
61 views

Whether or not such a simple CW complex can be made a $C^{\infty}$ manifold?

Problem Let $X$ be the space obtained by attaching two disks to $S^1$, the first disc being attached by the 7 times around,i.e. $z \to z^7$, and the second by the 5 times around. Can $X$ be made ...
2
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1answer
65 views

Mayer-Vietoris in reduced homology for a torus.

By using the Mayer-Vietoris sequence in reduced homology : I have to calculate the homology groups of : The torus $\mathbb{T}^2 :=[0;1]^2 /\mathcal{T}$ by using the following decomposition $X_1 := ...
3
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1answer
50 views

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ ...
6
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2answers
206 views

Is there an analogue of Eilenberg-Maclane spaces for homology?

Let $G$ be a group and $n$ a positive integer. A connected topological space $Y$ is called an Eilenberg–MacLane space of type $K(G, n)$, if $\pi_n(Y) \cong G$ and all other homotopy groups of $Y$ are ...
5
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1answer
78 views

Try to generalize a problem in Hatcher: finite vs. infinite CW-complexes

While solving a problem in Hatcher I got this doubt in my mind, In the 2nd chapter (Homology) Hatcher asked us to prove the following question... If $X$ is a finite dimensional CW-complex then, ...
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0answers
130 views

Representation of sum of homology classes

Let $X$ be a path-connected topological space, let $x, x' \in H_{k}(X)$ for $k>0$ be represented by two connected manifold i.e. there exist two compact oriented connected manifolds $M$, $N$ and two ...
5
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1answer
71 views

Not null homotopic map from $S^3$ to $S^2 \vee S^2$

I'm asked to present a continuous function $\alpha \colon S^3 \rightarrow S^2 \vee S^2$ s.t. it is not null homotopic but taken both projections $pr \colon S^2 \vee S^2 \rightarrow S^2$ the ...