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



Group bundles for topological spaces without universal cover
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.


