I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to \mathbb{K}$ defined for all $i$ on chain complexes $\cdots \to C_{i+1} \to C_i \to C_{i-1} \to \cdots$ of $\mathbb{K}$-modules over a principal ideal domain $\mathbb{K}$. I am particularly interested in how these descend to homology groups and what invariants/classifications can be constructed from them (think of intersection theory in singular homology). Thanks.
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$\begingroup$ Is there a mistake in the indexing in the question? If I think of topological intersection theory, I would expect the intersection pairing to go $H_i\times H_{n-i}\to \mathbb{K}$ whenever the object has some sort of duality making it look like it is $n$-dimensional. I guess I would have an answer for this kind of indexing. $\endgroup$– Matthias WendtCommented Jan 16, 2016 at 15:38
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$\begingroup$ No, I think the question is valid as stated. Perhaps intersection theory is not the best example (although it was the only one I could think of) because the pairing $H_i \times H_i \to \mathbb{K}$ defined by the cup product is only possible when $2i=n$. I am interested in arbitrary bilinear forms defined on the chain groups of chain complexes, and under which conditions they induce bilinear forms on homology groups. Similarly to chain maps descending to maps on homology, when does the induced bilinear form satisfy a certain property (e.g. unimodularity)? $\endgroup$– AliCommented Jan 18, 2016 at 9:56
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$\begingroup$ I think linearity of the differential would imply that the form descends whenever it vanishes on the image of the differential. $\endgroup$– Matthias WendtCommented Jan 18, 2016 at 13:32
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$\begingroup$ The better theory exists for the geometrically relevant case where a Poincaré duality is involved. One can define forms as quasi-isomorphisms from a complex to its dual and view this as objects in a Waldhausen category with duality. Then forms have classes in the K-group, and one can use all sorts of machinery to get invariants. Unfortunately, this does not apply to the question with the indices as stated... $\endgroup$– Matthias WendtCommented Jan 18, 2016 at 13:33
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$\begingroup$ That still sounds interesting because in my particular case I will have some chain complexes of the form $0 \to C_0 \to C_1 \to C_2 \to 0$. Do you have a recommendation for where the things you mention are described? Elementary yet at the same time including as much as possible... Thanks! $\endgroup$– AliCommented Jan 18, 2016 at 15:02
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