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.

learn more… | top users | synonyms (2)

1
vote
0answers
17 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 ...
2
votes
0answers
48 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 ...
0
votes
1answer
19 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
votes
0answers
55 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 ...
1
vote
1answer
44 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
votes
0answers
29 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 ...
0
votes
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 ...
1
vote
0answers
23 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
votes
1answer
35 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
votes
1answer
72 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 ...
1
vote
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
votes
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 ...
1
vote
0answers
49 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
votes
0answers
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 ...
1
vote
0answers
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 ...
-1
votes
0answers
54 views

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

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, ...
2
votes
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
votes
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 $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
4
votes
0answers
135 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} ...
0
votes
0answers
40 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
votes
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 ...
0
votes
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
votes
1answer
92 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 ...
1
vote
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{ ...
1
vote
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
votes
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 ...
0
votes
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
vote
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 ...
0
votes
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
votes
1answer
115 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 ...
1
vote
0answers
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 ...
1
vote
2answers
45 views

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 ...
1
vote
1answer
62 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 ...
0
votes
0answers
30 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
votes
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
votes
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
votes
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
votes
2answers
205 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
votes
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, ...
1
vote
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
votes
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 ...
2
votes
1answer
37 views

Does every n-chain have a homology class?

I was under the impression that not every (singular) $n$-chain has a homology class, since $H_n(X) = Z_n(X)/B_n(X)$, and not every $n$-chain is an $n$-cycle. But I came across the following in ...
8
votes
1answer
72 views

Can (singular) homology classes always be represented by images of closed manifolds?

My intuition tells me that if $A \in H_2(M;\mathbf Z)$, then $A$ can be represented by a map $\Sigma \to M$, where $\Sigma$ is a closed (= compact boundaryless) surface, i.e., the connected sum of ...
1
vote
1answer
52 views

Mayer-Vietoris sequence in reduced homology.

By using the Mayer-Vietoris sequence in reduced homology : $...\overset{\Delta_{n+1}}{\longrightarrow} \tilde{H_n}(A)\overset{E_{n}}{\longrightarrow} \tilde{H_n}(X_1)\times ...
3
votes
1answer
83 views

Homology of mapping cone

Let $f:X\to Y$ be a map, and $\text{cone}(f) = CX \sqcup_f Y$ its mapping cone. Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be a homology theory with values in the category of $R$-modules. ...
7
votes
0answers
148 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
3
votes
1answer
58 views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
1
vote
1answer
70 views

Homology functors defined on $\mathsf{Top} \times \mathsf{Top}$ in Eilenberg-Steenrod axioms?

The Eilenberg-Steenrod axioms state that a homology theory is a sequence of functors $H_n : \mathsf{Top} \times \mathsf{Top} \to \mathsf{Ab}$ satisfying some additional properties. What I don't ...
7
votes
4answers
310 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
2
votes
1answer
65 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is ...