I know very well about the cup product in ordinary cohomology. My question is that how we compute the different cup product with twisted or compactly supported cohomology i.e, $$\cup:H^{*}(M,\mathbb{Q}^{w})\times H^{*}(M,\mathbb{Q}^{w})\longrightarrow H^{*}(M,\mathbb{Q})$$ $$\cup:H^{*}(M,\mathbb{Q}^{w})\times H^{*}(M,\mathbb{Q}^{w})\longrightarrow H_{c}^{*}(M,\mathbb{Q})$$ $$\cup:H_{c}^{*}(M,\mathbb{Q}^{w})\times H_{c}^{*}(M,\mathbb{Q}^{w})\longrightarrow H_{c}^{*}(M,\mathbb{Q})$$ $$\cup:H_{c}^{*}(M,\mathbb{Q})\times H_{c}^{*}(M,\mathbb{Q})\longrightarrow H_{c}^{*}(M,\mathbb{Q})$$ where $H^{*}(M,\mathbb{Q}),\,H_{c}^{*}(M,\mathbb{Q})$ and $H^{*}(M,\mathbb{Q}^{w})$ are ordinary, compactly supported and twisted cohomology ring of manifold respectively.

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