(b) Show that if f: RP2n!Xis a covering map of a CW complex X, then f is a homeomorphism. 5. For each of the following statements, either give speci c examples of (connected) topological spaces Xand Ywhich satisfy the statement or explain (in one or two sentences) why no such examples exist. Homology and cohomology groups are considered with Z-

COVERING SPACES DAVID GLICKENSTEIN 1. Introduction and Examples We have already seen a prime example of a covering space when we looked at the exponential map t ! exp(2ˇit); which is a map R ! S1: The key property is tied up in this de–nition. De–nition 1. A covering space of a space X is a space X~ together with a map p : X~ ! to an 8-sheeted covering space for S1 ∨ S1, with the vertices of the octagon all mapping to the basepoint. For the rest of this example, we’ll denote this covering space by Xe. Now that we’ve constructed a covering space for S 1∨ S, the rest of the problem amounts to proving that this covering is indeed the correct one.

Finally, Section 7 covers critical groups of signed graphs. In particular, given an induced surjection K(G′) →K(G) of signed graphs where G′ is a 2-sheeted cover of G, the kernel can be described in terms of another signed graph. As an application, we provide an interpretation of Bai’s proof [2] of the Sylow Covering Spaces Section 1.3 59 of disjoint circles, as is the union of the bedges.Choosing orientations for all these circles gives a 2 orientation. It is a theorem in graph theory that inﬁnite graphs with four edges incident at

Theorem 1. The complex manifold R2 is a homogeneous space and hence Oka. Our main theorem is the following. Theorem 2. There exists a 6-sheeted branched covering space of R3 that is an Oka manifold. It must be noted that it is not known whether the Oka property can be passed down a branched covering map. Hence we are unable to conclude that R3 ... (b) Show that if f: RP2n!Xis a covering map of a CW complex X, then f is a homeomorphism. 5. For each of the following statements, either give speci c examples of (connected) topological spaces Xand Ywhich satisfy the statement or explain (in one or two sentences) why no such examples exist. Homology and cohomology groups are considered with Z-