Search Results

For $k\neq 0,2n+1$ the homology groups with coefficients in a commutative ring $R$ are $$ H_k(L_n(p),R)=\left\{\begin{array}{cc} R/pR & \text{if $k$ is odd}\\ T_p(R) & \text{if $k$ is even } \end{array … Fix an extension $\Gamma$ of $R$ by $R^{\vee}$: $$ 1\rightarrow R^{\vee}\rightarrow \Gamma \rightarrow R\rightarrow 1 \ . $$ This induces a long exact sequence of homology and cohomology groups, with connecting …

{Z} $$ for the torsion part of the homology groups. … Maybe this is a very trivial generalization, but is there an analogous notion for homology groups with coefficients in an arbitrary abelian group $A$? …

\ \rangle : \ \Gamma _k(X)_{\text{free}} \times H^k(X,\mathbb{Z})_{\text{free}} \rightarrow \mathbb{Z} \end{equation} obtained by restricting the the Kronecker pairing between the free subgroups of homology …

In a similar way one can define the Bockstein map in homology $b : H_k(X,A_3)\rightarrow H_{k-1}(X,A_1)$. On a chain $c\in Z_k(X,A_3)$ this is defined such that $\iota(b(c))=\partial s(c)$. … On the other hand, if $X$ is orientable, by using Poincarè duality $\Phi_A :H_p(X,A)\rightarrow H^{d-p}(X,A)$ for homology/cohomology we induce a map in homology $b': H_p(X,A_3)\rightarrow H_{p-1}(X,A_ …

Consider the homology with $\mathbb{Z}_n$ coefficients, and the Bockstein of the sequence $$ \mathbb{Z}_n\rightarrow \mathbb{Z}_{n^2}\rightarrow \mathbb{Z}_n $$ Given $\Sigma \in H_2(M_3,\mathbb{Z}_n)$ …