By Uwe Kaiser

Any topological idea of knots and hyperlinks may be in keeping with basic rules of intersection and linking. during this booklet, a normal thought of hyperlink bordism in manifolds and common structures of linking numbers in orientated 3-manifolds are built. during this manner, classical strategies of hyperlink concept within the 3-spheres are generalized to a definite type of orientated 3-manifolds (submanifolds of rational homology 3-spheres). The strategies wanted are defined within the publication yet uncomplicated wisdom in topology and algebra is thought. The publication may be of interst to these operating in topology, particularly knot thought and low-dimensional topology.

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It is universal if p(π(E , e)) = {e}. Clearly a covering space is regular or universal if and only if its associated covering of fundamental groupoids is regular or universal. A covering of groupoids is universal if and only if π(B, b) acts freely on Fb , and then Fb is isomorphic to π(B, b) as a π(B, b)-set. Specializing to covering spaces, this sharpens our earlier claim that the elements of Fb and π1 (B, b) are in bijective correspondence. 5. The classification of coverings of groupoids Fix a small connected groupoid B throughout this section and the next.

Show that the space of G-maps G/H −→ X is naturally homeomorphic to X H . In particular, O(G/H, G/K) ∼ = (G/K)H . 8. THE CONSTRUCTION OF COVERINGS OF SPACES 33 6. Let X be a G-space. Show that passage to fixed point spaces, G/H −→ X H , is the object function of a contravariant functor X (−) : O(G) −→ U . CHAPTER 4 Graphs We define graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space. 1. The definition of graphs We give the definition in a form that will later make it clear that a graph is exactly a one-dimensional CW complex.

For (ii), there is a map g : e −→ e′ since E is connected. Conjugation by g gives a homomorphism π(E , e) −→ π(E , e′ ) that maps under p to conjugation of π(B, b) by its element p(g). For (iii), the surjectivity of p on St(e) gives that any f ∈ π(B, b) is of the form p(g) for some g ∈ St(e). If e′ is the target of g, then p(π(E , e′ )) is the conjugate of p(π(E , e)) by f . The fibers Fb of a covering of groupoids are related by translation functions. Definition. Let p : E −→ B be a covering of groupoids.