Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorMurat_Aygen
    • CommentTimeFeb 18th 2018
    • (edited Feb 19th 2018)


    Let KK be a complex with rr combinatorial components. Then H 0(K)H_{0}(K) is isomorphic to the direct sum of (not rr but) 2r2r copies of the group ZZ of integers.

    Proof for one component:

    As 0-cycles are synonymous with 0-chains, they have α 0\alpha_{0} many dimensions-of-freedom for sure. How many the 0-boundaries of our component have? Since the incidence matrix is a α 0\alpha_{0} by α 1\alpha_{1} matrix (see my previous posting with ‘homology’ tag), they have at least α 1α 0\alpha_{1} - \alpha_{0} many for sure. And 1 more for the rank deficiency! The proof follows from this arithmetic.