The fundamental group of preimage of covering map

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Discussion Overview

The discussion revolves around the properties of the preimage of a covering map, particularly focusing on the path-connectedness of the preimage set and the implications of fundamental groups. Participants explore conditions under which the preimage of a subset under a covering map remains path-connected, as well as the fundamental group of that preimage.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant poses two questions regarding the path-connectedness of the preimage set D under certain conditions related to the fundamental groups of B, Y, and X.
  • Another participant argues that it is possible to find a path-connected subset of Y whose inverse image under the covering map is not path-connected, particularly in the case of non-trivial coverings.
  • A different participant suggests that if the inclusion map is an isomorphism on the level of fundamental groups, then the preimage of B under the covering map will likely be path-connected, although they do not provide a detailed proof.
  • One participant acknowledges the previous point about isomorphism but indicates that it does not capture the entire situation, suggesting there is more complexity to consider.
  • Some participants reiterate the argument about the existence of a path-connected subset whose inverse image is not path-connected, emphasizing the implications of non-trivial coverings.
  • Another participant repeats their intuition regarding the isomorphism of fundamental groups and its implications for path-connectedness, indicating confidence in their reasoning.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the preimage is path-connected, with some supporting the idea of isomorphism of fundamental groups as a sufficient condition, while others highlight counterexamples that challenge this notion. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants have not fully established the assumptions under which their claims hold, and there are unresolved aspects regarding the definitions of path-connectedness and the nature of the covering maps involved.

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i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the fundamental group of D at some point?
 
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One can always find a path connected subset whose inverse image is not path connected if the covering is non-trivial.

Take any open set whose inverse image is a collection of disjoint open sets. Such a set always exists around any point in Y. Choose a path in this set. Its inverse image will not be path connected.
 
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My intuition is telling me that, under your assumptions, if the inclusion

i: B → Y​

is an isomorphism on the level of fundamental groups:

i#: π1(B,b) →≈ π1(Y,b),​

then the inverse image of B under the covering map

p: X → Y,​

namely p-1(B), will be path-connected. I don't think the proof is difficult.
 
The above (#3) may be true, but it is definitely not the whole story. Stay tuned.
 
lavinia said:
One can always find a path connected subset whose inverse image is not path connected if the covering is non-trivial.

Take any open set whose inverse image is a collection of disjoint open sets. Such a set always exists around any point in Y. Choose a path in this set. Its inverse image will not be path connected.
Thanks
 
zinq said:
My intuition is telling me that, under your assumptions, if the inclusion

i: B → Y​

is an isomorphism on the level of fundamental groups:

i#: π1(B,b) →≈ π1(Y,b),​

then the inverse image of B under the covering map

p: X → Y,​

namely p-1(B), will be path-connected. I don't think the proof is difficult.
Thanks
 

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