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Let $C(I,X)$ be the set of continuous functions from $I=[0,1]$ to some topological space $X$ (could be path-connected). Consider $\xi:C(I,X)\to S_1(X)$ given by $c\mapsto \xi(c)=c\circ\pi$ with $\pi$ the projection $\pi:\Delta_1=\{(t,1-t)\}_{t\in\mathbb R}\to \mathbb R$, $(t,1-t)\mapsto 1-t$.

My question is:

Given $c,d\in C(I,X)$ s.t. $c(1)=d(0)$, is it true that $\xi(c*d)=\xi(c)+\xi(d)$ ?

Note: If we sustitute $C(I,X)$ by $\Omega(X,x_0)=\{c\in C(I,C):c(1)=c(0)=x_0\}$, it is well know that $\xi(c*d)-\xi(x)-\xi(d)$ is in $B_1(X)$ for any $c,d\in \Omega(X,x_0)$. From here, can we solve my question?

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    What is $S_1$...? – 2017-01-10
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    $S_1(X)$ is the set of the "formal sum": $T=\sum_{\mu=1}^n n_\mu T_\mu$ with $n_\mu$ is an integer and $T_\mu$ is a continuous map $\Delta_1\to X$. – 2017-01-10

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