Let $G$ be a torus which acts on a topological space $X$. Then consider the Borel fibration $X\longrightarrow X_{G}\longrightarrow B_{G}$. Let $% \left( E_{r}^{\ast ,\ast },d_{r}\right) $ be the Leray-Serre spectral sequence of the Borel fibration. We know the edge homomorhism $% H_{G}^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow E_{\infty }^{0,n}\subset E_{2}^{0,n}\subset H^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} \right) $. Let $i^{\ast }:H_{G}^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ be the map determined by the fiber inclusion $i:X\longrightarrow X_{G}$.
Furthermore, consider the edge homomorphism $H^{n}\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \rightarrow E_{2}^{n,0}\twoheadrightarrow E_{3}^{n,0}\twoheadrightarrow \cdots \twoheadrightarrow E_{n+1}^{n,0}=E_{\infty }^{n,0}\subset H_{G}^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $. Note that $\pi ^{n}:H^{n}\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H_{G}^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is the combination of these maps.
Suppose that for some $a\in H_{G}^{n}\left( X\right) $, $i^{n}\left( a\right) =b\neq 0$ and $b\in E_{\infty }^{0,n}$. And assume for all $n$, $% \pi ^{n}:H^{n}\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H_{G}^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is injective.
Now, consider $H_{G}^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ as $H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $-module via $\pi ^{\ast }$: the product is defined by $rx=\pi ^{\ast }\left( r\right) \cup x$ for $r\in H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ and $x\in H_{G}^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $. The product $\cup $ denotes the cup product in $H_{G}^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $.
I want to show that $b$ is non-torsion in $E_{\infty }$ with respect to $% H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $, moreover I want to show that $a$ is non-torsion in $H_{G}^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ with respect to $H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $. Set $R=H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $.
I think that the map $i^{n}$ is not an $R$-module homomorphism.
A map $\psi $ is expressed (as $R$-module) in the following article https://www.jstor.org/stable/1971075
(In the proof of Lemma 2)
I don't quite understand how this works. How to determine the map $\psi $?
The following source states that each row of the spectral sequence has the structure of an $R$-module?
In the following source (page 7) (https://www.google.com.tr/books/edition/_/nfJdDwAAQBAJ?hl=tr&gbpv=1), it is stated that in order for each row of the spectral spectral to be an $R$-module, an increasing filtration should be used instead of the usual descending one, but there is no such explanation in other sources. I don't quite understand what the fine point is here.
I found the following source https://www.jstor.org/stable/1971074 if that helps.
(In the proof of Lemma 3.4)
Actually, my question is related to the following question that I could not find a solution before.
