So, I am trying to compute some products on chains and their duals, but I have difficulties in understanding some operations.
The cup product of cochains is quite easy to understand, especially when computing them. However, I cannot grasp why starting from chains I hit a non-associative wall. Let me explain, if I have a duality $\phi : C_p \rightarrow C^p$ (on a complex $K$ with values in an abelian group $G$), I can in theory calculate a product of two chains on the same complex as $$ c_k \cdot c_\ell = \sum_i (c^k \smile c^\ell)(\sigma_{i}) \sigma_{i} = \sum_i \phi(c_k)(\sigma_i|_{v_0 \ldots v_k}) \cdot \phi(c_\ell)(\sigma_i|_{v_k \ldots v_{k+\ell}}) \sigma_i \in C_{k+\ell}, $$
This seems to be non associative.
Am I doing something non kosher here? I expected to be able to maintain all the properties of cup products in a very simple chain product like that one, but I was wrong...
Another question is probably easy. Can you enlighten me on homology cross products? Cup products are intuitive, at least to me, since their application is stunningly easy with the front/back faces of a simplex. I cannot grasp how cross products work on a chain complex.
Reading Hatcher (Ch. 3.B page 268) and Munkres (Ch. 7, page 346) didn't help. I still cannot figure out how to actually compute a cross product. In particular, I cannot understand what happens if I restrict myself to a single complex, so having $\times : C_k(K, G) \times C_\ell(K, G) \rightarrow C_{k+\ell}(K, G)$, obviously, with $K$ being a $k+\ell$ (simplicial) complex.
