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Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.

This question is a generalization of this question by zygund. In my answer I recalled an idea of Wu Wen-Tsün which can be found in the introduction to his remarkable book A theory of embedding, immersion, and isotopy of polytopes in a Euclidean space: in order to distinguish, up to homeomorphism, two topological spaces that are homotopy equivalent (like any two lines bundles over the projective line), it is fruitful to consider the homotopy type of their configuration spaces. Wu Wen-Tsün's book shows that this is particularly useful in considering embedding problems in Euclidean spaces.

So my real question is:

Given two complex line bundles over ${\mathbb CP}^1$ such that their two-point configuration spaces are homotopy equivalent, are their Chern numbers equal up to sign?

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.

This question is a generalization of this question by zygund. In my answer I recalled an idea of Wu Wen-Tsün which can be found in the introduction to his remarkable book A theory of embedding, immersion, and isotopy of polytopes in a Euclidean space: in order to distinguish, up to homeomorphism, two topological spaces that are homotopy equivalent (like any two lines bundles over the projective line), it is fruitful to consider the homotopy type of their configuration spaces. Wu Wen-Tsün's book shows that this is particularly useful in considering embedding problems in Euclidean spaces.

So my real question is:

Given two complex line bundles over ${\mathbb CP}^1$ such that their two-point configuration spaces are homotopy equivalent, are their Chern numbers equal up to sign?

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$ prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.

This question is a generalization of this question by zygund. In my answer I recalled an idea of Wu Wen-Tsün which can be found in the introduction to his remarkable book A theory of embedding, immersion, and isotopy of polytopes in a Euclidean space: in order to distinguish, up to homeomorphism, two topological spaces that are homotopic (like any two lines bundles over the projective line), it is fruitful to consider the homotopy type of their configuration spaces. Wu Wen-Tsün's book shows that this is particularly useful in considering embedding problems in Euclidean spaces.

So my real question is:

Given two complex line bundles over ${\mathbb CP}^1$ such that their two-point configuration spaces are homotopic, are their Chern numbers equal up to sign?

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.

This question is a generalization of this question by zygund. In my answer I recalled an idea of Wu Wen-Tsün which can be found in the introduction to his remarkable book A theory of embedding, immersion, and isotopy of polytopes in a Euclidean space: in order to distinguish, up to homeomorphism, two topological spaces that are homotopy equivalent (like any two lines bundles over the projective line), it is fruitful to consider the homotopy type of their configuration spaces. Wu Wen-Tsün's book shows that this is particularly useful in considering embedding problems in Euclidean spaces.

So my real question is:

Given two complex line bundles over ${\mathbb CP}^1$ such that their two-point configuration spaces are homotopy equivalent, are their Chern numbers equal up to sign?

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$ prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.

This question is a generalization of this question by zygund. In my answer I recalled an idea of Wu Wen-Tsün which can be found in the introduction to his remarkable book A theory of embedding, immersion, and isotopy of polytopes in a Euclidean space: in order to distinguish, up to homeomorphism, two topological spaces that are homotopic (like any two lines bundles over the projective line), it is fruitful to consider the homotopy type of their configuration spaces. Wu Wen-Tsün's book shows that this is particularly useful in considering embedding problems in Euclidean spaces.

So my real question is:

Given two complex line bundles over ${\mathbb CP}^1$ with different Chern numbers, is the homotopy type of their two-point configuration spaces also different?

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$ prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.

This question is a generalization of this question by zygund. In my answer I recalled an idea of Wu Wen-Tsün which can be found in the introduction to his remarkable book A theory of embedding, immersion, and isotopy of polytopes in a Euclidean space: in order to distinguish, up to homeomorphism, two topological spaces that are homotopic (like any two lines bundles over the projective line), it is fruitful to consider the homotopy type of their configuration spaces. Wu Wen-Tsün's book shows that this is particularly useful in considering embedding problems in Euclidean spaces.

So my real question is:

Given two complex line bundles over ${\mathbb CP}^1$ such that their two-point configuration spaces are homotopic, are their Chern numbers equal up to sign?