Let $G$ be a torus which acts on a topopological 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 $?

[![enter image description here][1]][1]

The following source states that each row of the spectral sequence has the
structure of an $R$-module?

[![enter image description here][2]][2]

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.

[![enter image description here][3]][3]

I found the following source https://www.jstor.org/stable/1971074 if that helps.

(In the proof of Lemma 3.4)

[![enter image description here][4]][4]

Actually, my question is related to the following question that I could not
find a solution before.

https://mathoverflow.net/questions/471438/a-question-about-spectral-sequences


  [1]: https://i.sstatic.net/YFaWlEtx.png
  [2]: https://i.sstatic.net/8q0oH8TK.png
  [3]: https://i.sstatic.net/EDp0bv7Z.png
  [4]: https://i.sstatic.net/M66twfhp.png