Suppose that $T$ is a torus acting on a topological space $X $. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the associated universal bundle. This is called Borel fibration.

Localization Theorem: Let $T$ be a torus and $X$ be a compact $T$-space. Then the fixed point $X^{T}\neq \emptyset $ if and only if $\pi ^{\ast }:H^{\ast }\left( B_{T};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H_{T}^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is injective.

Recall that the compositions $ H^{q}\left( B_{T};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) =E_{2}^{q,0}\twoheadrightarrow E_{3}^{q,0}\twoheadrightarrow \cdots \twoheadrightarrow E_{q}^{q,0}\twoheadrightarrow E_{q+1}^{q,0}=E_{\infty }^{q,0}\subset H_{T}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right)$ and $ H_{T}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \twoheadrightarrow E_{\infty }^{0,q}=E_{q+1}^{0,q}\subset E_{q}^{0,q}\subset \cdots \subset E_{2}^{0,q}=H^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ are the homomorphisms $\pi ^{\ast }:H^{q}\left( B_{T};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \rightarrow H_{T}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ and $i^{\ast }:H_{T}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \rightarrow H^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $, where $i:X\longrightarrow X_{T}$ and $\pi :X_{T}\longrightarrow B_{T}$ and $E^{p,q}$ is the spectral sequence of Borel fibration $% X\longrightarrow X_{T}\longrightarrow B_{T}$.

Now, let a torus $T$ act on a compact Poincaré duality $n$-space $X$. Suppose that $i^{\ast }:H_{T}^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H^{n}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is non-trivial. It is claimed that from the spectral sequence argument of the fiber bundle $X\longrightarrow X_{T}\longrightarrow B_{T}$ and the Borel localization theorem, the fixed point set $X^{T}$ is not empty. I didn't see this.