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Michael
Michael
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Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary).

Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

  1. Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

Suppose now that $M$ admits a lorentzian metric. Is it then possible for $M$ to be non-parallelizable ? (This[EDIT wikipedia page seems: The answer to say that this is not possible, but I don't understandyes : see the argument).answer of Danny Ruberman]

  1. Suppose now that $M$ admits a lorentzian metric and a spin structure [EDIT]. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).

Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary).

Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

Suppose now that $M$ admits a lorentzian metric. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).

Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary).

  1. Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

[EDIT : The answer to this is yes : see the answer of Danny Ruberman]

  1. Suppose now that $M$ admits a lorentzian metric and a spin structure [EDIT]. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).
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Michael
Michael
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Let $M$ be an oriented simply-connecteda connected orientable open 4-manifold (noncompact, without boundary).

Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

Suppose now that $M$ admits a lorentzian metric. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).

Let $M$ be an oriented simply-connected open 4-manifold (noncompact, without boundary).

Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

Suppose now that $M$ admits a lorentzian metric. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).

Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary).

Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

Suppose now that $M$ admits a lorentzian metric. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).

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Michael
Michael
  • 361
  • 2
  • 8

Open non-parallelizable 4-manifolds

Let $M$ be an oriented simply-connected open 4-manifold (noncompact, without boundary).

Is it possible for $M$ to be non-parallelizable ? If yes, what example of such $M$ is there ?

Suppose now that $M$ admits a lorentzian metric. Is it then possible for $M$ to be non-parallelizable ? (This wikipedia page seems to say that this is not possible, but I don't understand the argument).