Fiber transport and monodromy action

Let $p:E\to B$ be a Hurewicz fibration, in ordinary spaces or in the explicitly chosen CGWH convention. Put $D_p=\{(e,\gamma ):p(e)=\gamma (0)\}\subseteq E\times B^I$, with the corresponding path and pullback topology. Evaluation and the interval exponential law of Interval exponential law and quotient homotopies make $((e,\gamma ),t)\mapsto \gamma (t)$ a continuous homotopy with initial lift $(e,\gamma )\mapsto e$. One application of Hurewicz and serre fibrations gives a universal lifting function $\mathrm{\Lambda }:D_p\times I\to E$ with $\mathrm{\Lambda }(e,\gamma ,0)=e,p\mathrm{\Lambda }(e,\gamma ,t)=\gamma (t)\mathrm{.}$ It is not assumed regular: $\mathrm{\Lambda }(e,c_{p(e)},t)$ may move in its fiber. Selecting this one map is a single existential choice, not an application of AC.

For a path $\gamma :b\to c$, define its fiber transport by $T_{\gamma }:F_b\to F_c$, $T_{\gamma }(e)=\mathrm{\Lambda }(e,\gamma ,1)$; fibers have the meaning of Fiber and fiber homotopy equivalence. The following proposition proves that its homotopy class is independent of the lifting function and of endpoint-fixed path homotopy, that $T_{\gamma \ast \eta }\simeq T_{\eta }{T}_{\gamma }$, and that it is a homotopy equivalence. Those claims, used in the next definitions, are licensed by that declared justifier.

For every abelian coefficient group $G$ and $q\ge 0$, the maps $H_q(T_{\gamma };G)$ consequently form a path-groupoid local system: to each point assign $H_q(F_b;G)$ and to each endpoint-fixed path class assign its induced isomorphism. Here the path groupoid has points as objects and endpoint-fixed path classes as arrows, composed in traversal order. Homology functoriality and invariance are those used in Homotopy equivalences induce isomorphisms on singular homology. Restriction to loops at $b$ is monodromy, written as a right action $z\cdot [\gamma ]=H_q(T_{\gamma };G)(z)$ under the first-loop-first convention.

For $n\ge 1$, the induced based map instead has type $(T_{\gamma }{)}_{\ast }:{\pi }_n(F_b,e)⟶{\pi }_n(F_c,T_{\gamma }(e))\mathrm{.}$ It is an isomorphism by homotopy equivalence with moving-basepoint correction. The correction is Higher homotopy basepoint transport and moving homotopies: a path $a:e^{\mathrm{\prime }}\to T_{\gamma }(e)$ in $F_c$ gives ${\beta }_a(T_{\gamma }{)}_{\ast }$ with target ${\pi }_n(F_c,e^{\mathrm{\prime }})$. Existence of such a path is additional data; its choice can change the resulting map. To obtain a fixed based group action one must supply endpoint paths with composition compatibility, or prove hypotheses making their effect independent of choices. A homology monodromy action alone supplies neither. Precisely, if $F_b$ is path connected and ${\beta }_l=\mathrm{i}\mathrm{d}$ on ${\pi }_n(F_b,e)$ for every loop $l$ at $e$, define $x\cdot [\gamma ]={\beta }_a(T_{\gamma }{)}_{\ast }x$ using any $a:e\to T_{\gamma }(e)$. The declared justifier proves independence of $a$ and of the lifting function, and the right-action law. This hypothesis applies separately to each $n\ge 1$; for $n=1$ it is equivalent to abelianness, and a simply connected fiber satisfies it for every $n$. No unqualified fixed-basepoint action is part of the definition.

Empty fibers are allowed; the following proposition shows emptiness is constant along path components of the base. Point fibers give identity maps on their invariants. No canonical pointwise transport homeomorphism is asserted.