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Suppose that I have two Hawaiian earrings, and i connect them by a line segment (through the wedge points). I would like to inquire whether the inclusion of the line segment is a cofibration.

My confusion is:

On one hand, it seems to me that the line segment is not a neighborhood deformation retract since every open set containing the segment contains infinitely many loops attached to the segment. On the other hand, it seems to me that morally speaking the quotient out by the line should be a homotopy equivalence.

My intuition tells me that the quotient is not a homotopy equivalence but is a weak homotopy equivalence, and the inclusion of the line segment is not a cofibration. Is this correct?

Thanks in advance!

902
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    I feel like the quotient doesn't act as nicely as you think it does. Suppose you have a loop in the quotient which goes around a ring on the left, then a smaller ring on the right, then a smaller ring on the left, then... This is a loop in the quotient, but that class has no representative in the preimage, because such a 'loop' would have to have 'infinite length' so to speak. That is, I don't think that the quotient induces a surjection on fundamental groups. – Dan Rust Sep 30 '16 at 08:06
  • Oh i see.. yes I think you are correct; your argument makes sense to me. Thank you! – 902 Sep 30 '16 at 14:39

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