Iterated monodromy group

From Topospaces
Jump to: navigation, search

Definition

Motivation

The notion of iterated monodromy group is defined in the context of a covering map from a subspace of a space to the whole space (the covering map obviously differs from the inclusion of the subspace). The key word here is iterated -- the fact that the cover itself can also be identified as a subset of the space allows us to consider iterating the function used for the covering map.

Full definition

Suppose X is a path-connected locally path-connected topological space, t \in X, X_1 \subseteq X, and f:X_1 \to X is a covering map. We define the iterated monodromy group of f, denoted \operatorname{IMG} f, as follows:

\operatorname{IMG} f := \frac{\pi_1(X,t)}{\displaystyle \bigcap_{n \in \mathbb{N}} \operatorname{Ker} F^n}

where:

  • \pi_1(X,t) is the fundamental group of X at basepoint t.
  • F^n represents the homomorphism from \pi_1(X,t) to \operatorname{Sym}(f^{-n}(t)) viewed in the context of the covering map f^{-n}(X) \to X. Note that in the case n = 1, we get the usual monodromy action for F and the quotient by the kernel is the usual monodromy group.

In other words, the iterated monodromy group stores the fundamental group modulo those loops that act trivially on all f^{-n}(t) sets.