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Determine the covering space of S 1 ∨ S 1 corresponding to the subgroup of π 1 (S 1 ∨ S 1 ) generated by the cubes of all elements. The covering space is 27 sheeted and can be drawn on a torus so that the complementary regions are nine triangles with edges labeled aaa , nine triangles with edges labeled bbb , and nine hexagons with edges ... $\begingroup$ @Fredrik It is arguably "obvious" from the fact that a cover must be a local homeomorphism. Also, by my count there are a total of three 2-sheeted covers and seven 3-sheeted covers. Two of the 2-sheeted covers are isomorphic as graphs but have different labelings. $\endgroup$ – Jim Belk Apr 22 '12 at 22:37

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are (q + 1)-many total spaces ofq2-sheeted covering maps overX which are mu-tually non-homeomorphic and(q +1)-many toroidal groups which admitq2-sheeted covering maps from X and which are mutually non-homeomorphic. (4) Let p be a prime and t =1 and let αn =1 if n=3k for some k and αn =0 otherwise. Title ON BOUNDED ANALYTIC FUNCTIONS ON TWO-SHEETED COVERING SURFACES(Analytic Function Spaces and Their Operators) Author(s) Kobayashi, Yasuyuki a universal covering space C.Then every subgroup G⊆ π(X,x 0) comes from a covering space X˜ of X. Corollary 7 Suppose Xis connected and locally pathwise connected, and suppose Xhas a universal covering space. Then there is a one-to-one correspondence between conjugacy classes of subgroups of π(X) and covering spaces of X.

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X and (X,pe ) or Xe is called a covering space of X. We deﬁne the dual concept of covering space of a topological space as follows. Deﬁnition 2.1. If X is a topological space, then a co-covering space of X is a pair (p,Xe) such that p: X → Xe is a covering map. The category of all co-covering spaces of X is denoted by CoCov(X). Then a torus-like continuum Y = T(α 0 ) generated by α 0 admits three 4-sheeted covering maps whose total spaces are pairwise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map whose total space is not homeomorphic to Y . In particular, Y is not a self-covering space. $\begingroup$ If you are asking about characterization of covering spaces of the given degree rather than covering maps, then the answer to your question is trivially positive since a closed oriented surface is uniquely determined by its genus and genus of the covering surface is computed by the formula that you wrote. So, what exactly is your ...

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X1 and X2 that have a common ﬁnite-sheeted covering space p1: X → X1, p2: X → X2, but do not commonly cover another space, i.e., they are not both covering spaces of a single space Y. 19. Show that if a group G acts freely (x = gx ⇒ g = 1) and properly discontinuously (for all x ∈ X there is a nbhd U of x such that {g : g(U) ∩ U 6 ... Math 6520 Final, Spring 2014 1. Let X = RP2 ∨ RP2. Give an example of an irregular covering space X˜ → X. You should draw the picture of the 1-skeleton of X˜ (make sure 2-cells lift). 2. Compute the fundamental group of the space obtained from the disjoint union of two 2-tori by identifying them along a pair of points in each torus. 3.

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2. Rotating the circle S1 by an integer multiple of 2π/n is a deck transformation of the covering space z 7→zn.The group of deck transformations is cyclic of order n. 3. Each translation of R2 by a vector (m,n), where m and n are integers, is a deck

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(a) Find the fundamental group of M, perhaps by relating it to another space. State carefully any theorems that you use. (b) Describe all of the connected, nite-sheeted covering spaces of M, up to equivalence. Explain why you have found all of them. (c) How many homeomorphism types are there among the covering spaces you found in part (b)? 10. $\begingroup$ If you are asking about characterization of covering spaces of the given degree rather than covering maps, then the answer to your question is trivially positive since a closed oriented surface is uniquely determined by its genus and genus of the covering surface is computed by the formula that you wrote. So, what exactly is your ...

2 (1968) 115-159 HOMOGENEOUS SPACES DEFINED BY LIE GROUP AUTOMORPHISMS. II JOSEPH A. WOLF & ALFRED GRAY 7. Noncompact coset spaces defined by automorphisms of order 3 We will drop the compactness hypothesis on G in the results of §6, doing this in such a way that problems can be reduced to the compact case. This Math 6510 Homework 11 Tarun Chitra May 19, 2011 §2.2 Problems 40 Problem. From the long exact sequence of homology groups associted to the short exact sequence of chain complexes 6. Let X be a topological space. We say that two covering spaces f : Y → X and g : Z → X are isomorphic if there exists a homeomorphism h : Y → Z such that g h = f. If X is a compact oriented surface of genus g (that is, a g-holed torus), how many connected 2-sheeted covering spaces does X have, up to isomorphism? 2 (1968) 115-159 HOMOGENEOUS SPACES DEFINED BY LIE GROUP AUTOMORPHISMS. II JOSEPH A. WOLF & ALFRED GRAY 7. Noncompact coset spaces defined by automorphisms of order 3 We will drop the compactness hypothesis on G in the results of §6, doing this in such a way that problems can be reduced to the compact case. This This paper is a continuation of the study of those 3-manifolds which are obtained by Dehn filling on a surface bundle, N, over S 1 and directed toward the question: which 3-manifolds, M, have fundamental group, π 1 (M), virtually Z--representable (have a finite sheeted covering space M̃→M with β 1 (M ̃)=rank H 1 (M ̃)>0)?

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6. Let X be a topological space. We say that two covering spaces f : Y → X and g : Z → X are isomorphic if there exists a homeomorphism h : Y → Z such that g h = f. If X is a compact oriented surface of genus g (that is, a g-holed torus), how many connected 2-sheeted covering spaces does X have, up to isomorphism? Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, the interior of O is the quotient of hyperbolic 3-space by a lattice in PSL(2, C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {Oi} with linear growth of p-homology, for some prime p.

and X is a 2-sheeted covering of X with 2g + 2 branch points. The proof given here is considerably simpler than the version in [1], and at the same time it holds in a much more general situation. The major tool that made this possible was the device of lifting maps to the universal covering space. The analogous problem in higher-dimensional ... A is the covering space corresponding to the kernel of the homomorphism ˇ1(A) ! ˇ1(X). We have seen in the previous course that pjp 1(A): p 1(A) ! A is a covering space. Also, it is easy to check that the restriction of a covering space to a path component is also a covering space. Now let x0 2 A and y0 2 Ae be basepoints with p(y0) = x0. Let ...

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X and (X,pe ) or Xe is called a covering space of X. We deﬁne the dual concept of covering space of a topological space as follows. Deﬁnition 2.1. If X is a topological space, then a co-covering space of X is a pair (p,Xe) such that p: X → Xe is a covering map. The category of all co-covering spaces of X is denoted by CoCov(X).

2-Dimensional torus Kleinbottle Compactgroup Solenoid 2-Dimensional compact abeliangroup Weaksolenoidal space We consider ﬁnite-sheeted covering maps from 2-dimensional compact connected abelian groups to Klein bottle weak solenoidal spaces, metric continua which are not groups. We m 2;m is shown in grey. Figure 2. The covering space S~ for m= 2. The red curves are some con-nected components of pre-images of the generator a 1 of ˇ 1(S). The green curve is a connected component of the lift of b 1b 2, and the black curves are connected components of the lift of [a 1;b 1]. This subsection is now concluded with a useful ... 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 and X is a 2-sheeted covering of X with 2g + 2 branch points. The proof given here is considerably simpler than the version in [1], and at the same time it holds in a much more general situation. The major tool that made this possible was the device of lifting maps to the universal covering space. The analogous problem in higher-dimensional ...