I have the following problem understanding the notion of coefficients in Singular Homology. Let $X$ be a topological space (for example closed differential manifold). Let $x:[0,1]\rightarrow X$ be a continous path such that $x(0)=x(1)$. Then $x$ defines a homology class $[x]\in H_{1}(X;\mathbb{Z})$. Define a different path $\tilde{x}:[0,1]\rightarrow X$ by $\tilde{x}(t)=x(2t)$, for $t\in [0,1/2]$ and $\tilde{x}(t)=x(2t-1)$, for $t\in [1/2,1]$ (essentially the loop $x$ run twice).
Question: Do we have that $2[x]=[\tilde{x}]$ in $H_{1}(X;\mathbb{Z})$?
If yes or no, then can you provide an explanation?
Greetings, Milan
Yes. Consider an oriented equilateral triangle $\Delta=e_0e_1e_2$, and let $f(P)=P’$ for every $P \in \Delta$, where $P’ \in [e_0,e_2]$ and the lines $PP’$ and $e_0e_2$ are perpendicular.
Then the boundary of $\tilde{x} \circ f$ is exactly $x+x-\tilde{x}$.