7 дек. 2016 г. · It is a purely natural algebraic construction. It so happens that the cohomology of a group G is the singular cohomology of a homotopy type ...

28 мая 2018 г. · Is there a confusion between 1-cycle in homology and 1-cocyle in cohomology ? Also, you might want to learn the difference between inhomogeneous ...

4 мая 2013 г. · One of my favorite application of cohomology is to classify wallpaper groups. Most wallpaper patterns are symmetric under a subgroup of all ...

26 нояб. 2016 г. · For any simplicial set X its simplicial homology coincides with the cellular (and hence singular) homology of its geometric realization: HSimpl.

7 сент. 2017 г. · The map ˉ∂ is the zero map, so the homology of this chain complex is Z in degrees 0,1, and is trivial in other degrees. Hence Hn(Z,Z)≅{Z,n=0,10, ...

6 мая 2019 г. · For a simple counterexample, consider G=Z and A=Z/2Z. Then H1(G,A)≅Gab⊗A≅Z/2Z but Hom(A,Gab)=0.

1 апр. 2022 г. · In general, homology and cohomology are not quite the same. The torsion ends up being in different degrees. This is a consequence of one version of the UCT.

10 июн. 2020 г. · We want to define the group homology of a group G with values in a Z[G]-module M as the derived functor. Hi(G,M)=TorZ[G]i(Z,M). This is a ...

12 февр. 2016 г. · I am interested in computing group homology with rational coefficients. I would like to know an example of a perfect group which has non-trivial rational group ...

13 окт. 2020 г. · The Hurewicz theorem computes some homology groups from the knowledge of homotopy groups; it indicates that homotopy groups are harder to compute than homology ...