If $H_n(X)$ is the homology group, and $\sum_{\lambda}H_n(X_{\lambda})$ is a direct sum such that $X$ is partitioned by path components with index $\lambda$, what does an element of $\sum_{\lambda}H_n(X_{\lambda})$ look like in terms of symbols?
I'm trying to show that, if $\gamma = \sum_{\lambda} \gamma$, $\theta_n:H_n(X) \rightarrow \sum_{\lambda}H_n(X_{\lambda})$ by cls $\gamma \mapsto$ cls $\gamma_{\lambda}$ is well defined but I'm not sure how I'd even define an element in $\sum_{\lambda}H_n(X_{\lambda})$.
$x \in \sum_{\lambda}H_n(X_{\lambda}) \Rightarrow x = ?$
The direct sum, which I'll write as $\bigoplus_i A_i$, of Abelian groups comprises finite formal sums of elements of the $A_i$. A typical element is then $a_{i_1}+\cdots+a_{i_k}$ with $a_{i_j}\in A_{i_j}$.
In your application, you can define your map provided that each of your $\text{cls}\,\gamma$ has the property that $\text{cls}\,\gamma_\lambda$ is only nonzero for finitely many $\lambda$. This may or may not be true...