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I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first describe the situation:

There are two approaches in defining Homology with local coefficients of a topological space $X$ (see [1, p. 328 – 331]). The first one is via modules (see [1, p. 328]) while the second is via bundles of groups (see [1, p. 330]). However, the first approach only works if $X$ admits a universal cover. So my question is:

Are there examples of topological spaces that admit a non-trivial bundle of groups but not a universal cover?

I have tried to use a covering of the Hawaiian earrings as described in [1, Section 1.3 Exercise 6 p. 79]. However, if I understand correctly, this is no group bundle since the lifts of closed loops are no group homomorphisms of the fiber.

I‘d be very thankful if you helped me!

Kathy

References:

[1] Hatcher, Allen: Algebraic Topology, 2002, https://www.math.cornell.edu/~hatcher/AT/AT.pdf, p. 79 and p. 327 - 331

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  • $\begingroup$ You might like to give a link/reference to local systems/coefficients in the book at groupoids.org.uk/nonab-a-t.html (pdf available), though this won't help with your question since it requires a filtered space, which can be obtained from a CW-structure or just by taking the singular complex. On the other hand, groupoids, a key feature of our book, are surely relevant. $\endgroup$ Jul 24, 2016 at 9:43
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    $\begingroup$ In my humble opinion, the cleanest way to define a local coefficient system is as functor from the fundamental groupoid to abelian groups. This works for all spaces without any assumptions and has good functoriality. The most comprehensive account of homology with local coefficients that Im aware of is in Whitehead's book "Elements of homotopy theory". $\endgroup$ Jul 24, 2016 at 14:05
  • $\begingroup$ Have you tried solenoids? $\endgroup$ Jul 24, 2016 at 19:06

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You are correct that the covering space of the Hawaiian earrings in the picture on p.79 of Hatcher's book cannot be made into a group bundle. There are other coverings of the Hawaiian earrings which can, however. So the Hawaiian earrings are an example of the spaces you are looking for.

The total space of a group bundle always contains one copy of the base space, corresponding to all trivial elements in all fibers.

So try to construct a non-trivial group bundle as follows:

Let $X$ be the Hawaiian earrings, then construct first a non-trivial double cover $f:Y\to X$. Now take the covering $id \cup f: X\cup Y\to X$. Finally you have to equip this with a group bundle structure.

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