Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to understand the geometric interpretation of two simplicial cycles being

homologous to each other.

Let C_k(X) be the k-th chain group in the simplicial complex X, and let c_k be

a chain in C_k(X)

The algebraic definition is clear: two k-cycles x=c_k and y=c_k' are homologous,

i.e., x~y , iff (def.) x-y is a boundary, i.e., if there is a cycle c_(k+1) in C_(k+1)(X)

with del(c_(k+1))= c_k-c_k' .

Still: how about geometrically: is there a nice geometric way of telling that two

cycles are homologous.?. I am having trouble translating the subtraction of cycles

into a geometric situation; it would seem like we could translate the expression

of c_k-c_k' is a boundary into saying that the curves c_k and -c_k' (i.e., c_k with

reversed orientation) are cobordant, in that there is a surface embedded in X--

the ambient complex--that is bounded by c_k and -c_k' .

Is this correct.?

Thanks.

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Geometric description of (simplicial)homologous cycles

Loading...

Similar Threads - Geometric description simplicial | Date |
---|---|

B Trying to represent a written geometrical description | Sep 28, 2016 |

A A question about coordinate distance & geometrical distance | Jul 19, 2016 |

A Geometrical interpretation of Ricci and Riemann tensors? | Jul 15, 2016 |

Geometric Sets and Tangent Subspaces - McInnerney, Example 3 | Feb 18, 2016 |

**Physics Forums - The Fusion of Science and Community**