Let $G$ be a topological group and let $H$ be a closed subgroup, assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG $$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ it is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence ; but I want to consider cases where $G$ is not connected.

EDIT 28/02

Looking around  , I realize that maybe the Leray-Hirsch theorem might play a role here in some specific situations: If the spectral sequence collapses, and $H^*(G/H)$ is a free $R$-module, then $H^*(BH)$ is a free $H^*(BG)$-module.

Conversely, if I assume that $H^*(G/H)$ is a free $R$-module, and $H^*(BH)$ is a free $H^*(BG)$-module, does it follows that the spectral sequence collapses and $$H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$$

Let $G$ be a topological group and let $H$ be a closed subgroup. Assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG.$$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected Lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence; but I want to consider cases where $G$ is not connected.

EDIT 28/02

Looking around, I realize that maybe the Leray-Hirsch theorem might play a role here in some specific situations: if the spectral sequence collapses, and $H^*(G/H)$ is a free $R$-module, then $H^*(BH)$ is a free $H^*(BG)$-module.

Conversely, if I assume that $H^*(G/H)$ is a free $R$-module, and $H^*(BH)$ is a free $H^*(BG)$-module, does it follow that the spectral sequence collapses and that $$H^*(BH) \cong H^*(BG) \otimes H^*(G/H)?$$

Let $G$ be a topological group and let $H$ be a closed subgroup, assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG $$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ it is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence ; but I want to consider cases where $G$ is not connected.

EDIT 28/02

Looking around , I realize that maybe the Leray-Hirsch theorem might play a role here in some specific situations: If the spectral sequence collapses, and $H^*(G/H)$ is a free $R$-module, then $H^*(BH)$ is a free $H^*(BG)$-module.

Conversely, if I assume that $H^*(G/H)$ is a free $R$-module, and $H^*(BH)$ is a free $H^*(BG)$-module, does it follows that the spectral sequence collapses and $H^(BH) \cong H^(BG) \otimes H^*(G/H)$$

Let $G$ be a topological group and let $H$ be a closed subgroup, assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG $$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ it is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence ; but I want to consider cases where $G$ is not connected.

EDIT 28/02

Looking around , I realize that maybe the Leray-Hirsch theorem might play a role here in some specific situations: If the spectral sequence collapses, and $H^*(G/H)$ is a free $R$-module, then $H^*(BH)$ is a free $H^*(BG)$-module.

Conversely, if I assume that $H^*(G/H)$ is a free $R$-module, and $H^*(BH)$ is a free $H^*(BG)$-module, does it follows that the spectral sequence collapses and $$H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$$

Let $G$ be a topological group and let $H$ be a closed subgroup, assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG $$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ it is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence ; but I want to consider cases where $G$ is not connected.

Let $G$ be a topological group and let $H$ be a closed subgroup, assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces

$$G/H \rightarrow BH \rightarrow BG $$

I am interested in the case where the associated spectral sequence degenerates and leads to an isomorphism $H^*(BH) \cong H^*(BG) \otimes H^*(G/H)$.

This holds, for instance, when $G/H$ is contractible or in the case of $G$ a compact connected lie group, and $H$ the maximal torus on $G$.

Specifically, I am wondering if just assuming that $H^*(BH)$ is a free $H^*(BG)$-module with the structure induced by the inclusion $BH \rightarrow BG$ it is enough to talk about the degeneracy of the spectral sequence. If I assume that $G$ is connected , then $BG$ is simply connected and my statement will hold under the Eilenberg-Moore spectral sequence ; but I want to consider cases where $G$ is not connected.

EDIT 28/02

Looking around , I realize that maybe the Leray-Hirsch theorem might play a role here in some specific situations: If the spectral sequence collapses, and $H^*(G/H)$ is a free $R$-module, then $H^*(BH)$ is a free $H^*(BG)$-module.

Conversely, if I assume that $H^*(G/H)$ is a free $R$-module, and $H^*(BH)$ is a free $H^*(BG)$-module, does it follows that the spectral sequence collapses and $H^(BH) \cong H^(BG) \otimes H^*(G/H)$$