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"decompasable" should be "decomposable", I believe
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gmvh
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A torus bundle whose vertical tangent bundle is indecompasableindecomposable

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not decompasabledecomposable? Which homotopy groups does one need to look at to construct such an example?

(This problem has been edited using inputs from the comments.)

A torus bundle whose vertical tangent bundle is indecompasable

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not decompasable? Which homotopy groups does one need to look at to construct such an example?

(This problem has been edited using inputs from the comments.)

A torus bundle whose vertical tangent bundle is indecomposable

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not decomposable? Which homotopy groups does one need to look at to construct such an example?

(This problem has been edited using inputs from the comments.)

added 56 characters in body; edited title
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Anon
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A torus bundles with non-trivialbundle whose vertical tangent bundle is indecompasable

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent spacebundle $\ker(d\pi) \to X$ is not a trivial vector bundledecompasable? Which homotopy groups does one need to look at to construct such an example?

(This problem has been edited using inputs from the comments.)

A torus bundles with non-trivial vertical tangent bundle

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent space $\ker(d\pi) \to X$ is not a trivial vector bundle? Which homotopy groups does one need to look at to construct such an example?

A torus bundle whose vertical tangent bundle is indecompasable

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not decompasable? Which homotopy groups does one need to look at to construct such an example?

(This problem has been edited using inputs from the comments.)

edited body
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Anon
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I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent space $\ker(d\pi) \to P$$\ker(d\pi) \to X$ is not a trivial vector bundle? Which homotopy groups does one need to look at to construct such an example?

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent space $\ker(d\pi) \to P$ is not a trivial vector bundle? Which homotopy groups does one need to look at to construct such an example?

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent space $\ker(d\pi) \to X$ is not a trivial vector bundle? Which homotopy groups does one need to look at to construct such an example?

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Anon
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