Articles of homology cohomology

Bockstein homomorphism and Steenrod square

question: What is the relation between Bockstein homomorphism and Steenrod square? For example, can one explain why the following relation works in the case of cohomology group with $\mathbb{Z}_2$ coefficient? For $x \in H^m(M^d,\mathbb{Z}_2)$, $$ \beta_2 x=\frac{1}{2} d x \text{ mod } 2 \in H^{m+1}(M^d,\mathbb{Z}_2) $$ which is the Bockstein homomorphism. It turns out that […]

The Abelianization of $\langle x, a \mid a^2x=xa\rangle$

I wish to verify the following statement (which comes from Fox, “A Quick Trip Through Knot Theory”, although that is probably not important). “$\Gamma=\pi_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so the homology of $M$ is infinite cyclic.” So, I need to find the Abelianization of the fundamental group. Using the relations I get $$y_1:=[x,a]=x^{-1}ax,\qquad y_2:=[a,x]=x^{-1}a^{-1}x$$ […]

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of nodes/vertices/points X, let $C_{l}(X)$ denote the subsets of $X$ with cardinality $|C_{l}(X)|=l+1$. $\partial_{l}$ and $\delta_{l}$ are bounded linear maps with […]

integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher’s book and Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to $\mathbb{Z}_{2}[x]/(x)^{n+1}$. For the graded module structure, it is obtained from the homology struture.

Real projective plane: $f_*$ isomorphism $\implies f$ surjective

Suppose we have a continuous map $f: \mathbb{R}P^2\rightarrow \mathbb{R}P^2$ that induces an isomorphism in homology $f_*: H_p(\mathbb{R}P^2) \rightarrow H_p(\mathbb{R}P^2)\ \ \ \forall p$. How do I show that $f$ is surjective?

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems to me like the natural way of calculating something of this sort although I know that it is also possible to […]

If two chain maps over a PID induce the same homomorphism, then they are homotopic

If two chain maps $f,g:\mathcal{X} \rightarrow \mathcal{Y}$, where $\mathcal{X},\mathcal{Y}$ are chain complexes with free modules $X_p$ and $Y_p$ over a PID, $R$, induce the same homomorphism in the homology, then how to prove that they are homotopic?

Skew symmetry of indices in cocycles of Cech cohomology

In Cech cohomology, the coboundary operator $$\delta:C^p(\underline U, \mathcal F)\to C^{p+1}(\underline U, \mathcal F) $$ is defined by the formula $$ (\delta \sigma)_{i_0,\dots, i_{p+1}} = \sum_{j=0}^{p+1}(-1)^j \sigma_{i_0,\dots, \hat {i_j},\dots i_{p+1}}{\Huge_|}_{U_{i_0}\cap\dots\cap U_{i_{p+1}}}. $$ In the book by Griffiths and Harris, the authors claim that $\delta\sigma=0$ implies the skew-symmetry condition $$ \sigma_{i_0,\dots,i_p} = -\sigma_{i_0,\dots, i_{q-1},i_{q+1},i_q,i_{q+2},\dots,i_p}. $$ How […]

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 – \{(0,0)\})$ but then realised that for better understanding I had to look at different examples, too. For the purpose of this question I am only interested in finding just one […]

Removing the star without changing homology

I know that if the link of a simplex $\sigma$ in a finite simplicial complex $K$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type. Basically, if the link of $\sigma$ is contractible then it can be reduced to a point by a sequence of collapses and anti-collapses and, […]