I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients.

Let me first give the definition of a local coefficient system that I found in [2, p. 257] and [3, p. 35]:

Let $X$ be a topological space. A local coefficient system is a functor from the category $\Pi_1(X)$ (= the fundamental groupoid) to the category AbGrp of abelian groups. Such a functor assigns to each $x \in X$ an abelian group $G(x)$ and to each homotopy class $\xi \in \pi_1(X; x_1, x_2)$ (the set of all endpoint-preserving homotopy classes of paths from $x_1$ to $x_2$) a homomorphism $G(\xi): G(x_2) \to G(x_1)$; these are required to satisfy

(i) if $\xi \in \pi_1(X, x) = \pi_1(X; x, x)$ is the identity, then $G(\xi): G(x) \to G(x)$ is the identity;

(ii) if $\xi \in \pi_1(X; x_1, x_2)$, $\eta \in \pi_1(X; x_2, x_3)$, then $G(\xi \eta) = G(\xi) \circ G(\eta): G(x_3) \to G(x_1)$

So my question is:

It is easy to see that every bundle of groups (defined in [1, p. 330]) is a local coefficient system, but I think the converse is not true (as stated without proof in [3, p. 35]), so I am looking for a topological space that has a (non-trivial) local coefficient system which is not a bundle of groups.

Thank you in advance!


[1] Hatcher, Allen: Algebraic Topology, 2002, https://www.math.cornell.edu/~hatcher/AT/AT.pdf, p. 330

[2] Whitehead, George: Elements of Homotopy Theory, New York: Springer, 1978 (Graduate Texts in Mathematics Vol. 61), p. 257

[3] Hutchings, Michael: Introduction to higher homotopy groups and obstruction theory, 2011, https://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf, p. 35


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