TITLE: First homology group of a double torus (genus 2 surface) – intuition
QUESTION [6 upvotes]: First homology group of a double torus is $H_1(T^2\#T^2)=\mathbb Z^4,$ (where # stands for a connected sum) which – for me – intuitively means there are 4 different cycles up to homotopy, the black ones. But what about those two yellow? Are they some kind of combination of the four?

REPLY [6 votes]: Here's some pictures that show how to get the big yellow loop from the two horizontal black loops and the smaller yellow loop from the two vertical black loops.
Big yellow:

Small yellow: