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