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Homework Assignment # 3, due Sept. 18 1. Show that the connected sum RP2 #RP2 of two copies of the real projective plane is homeomorphic to the Klein bottle K. 2. a) Show that the surface of genus g Σg = T # . . . #T | {z } g and the connected sum RP2 # . . . #RP2 {z } | k are both homeomorphic to the topological space Σ(W ) associated to a word W . What is the word W in these two cases? b) Calculate the Euler characteristic of these manifolds. 3. Important examples of quotient spaces are orbit spaces of a group G acting on a topological space X. We recall that a (left) of a group G on a set X is given by a map G × X −→ X typically written as (g, x) 7→ gx, such that g1 (g2 x) = (g1 g2 )x for g1 , g2 ∈ G, x ∈ X (associativity) and ex = x for e the unit element of G, x ∈ X (unit property). Given a G-action on a topological space X the orbit space denoted X/G is the quotient space X/ ∼ where two elements x, y ∈ X are declared equivalent if and only if there is some g ∈ G with gx = y. In particular, the equivalence class of x is the subset {gx | g ∈ G}, which is called the orbit of x, and hence X/G is the space of orbits. Although we haven’t used this terminology, we’ve already encountered orbit spaces, namely RPn = S n /{±1} and CPn = S 2n+1 /S 1 . Here the actions are given by {±1} × S n → S n (t, (v0 , . . . , vn )) 7→ (tv0 , . . . , tvn ) S 1 × S 2n+1 → S 2n+1 , (z, (z0 , . . . , zn )) 7→ (zz0 , . . . , zzn ) and where z ∈ S 1 ⊂ C and (z0 , . . . , zn ) ∈ S 2n+1 ⊂ Cn+1 . (a) Consider the action Z2 × R2 → R2 , (m, n), (x, y) 7→ (x + m, y + n). Show that the quotient space R2 /Z2 is homeomorphic to the torus, described as the quotient space b a a b Use without proof the fact that this orbit space is Hausdorff (this will come up later this semester). 1 (b) Consider the action G × R2 → R2 where G is the subgroup of the group of isometries of the metric space R2 generated by the isometries g, h : R2 → R2 defined by g(x, y) = (x + 1, y) and h(x, y) = (−x, y + 1) Show that the quotient space R2 /G is homeomorphic to the Klein bottle, described as the quotient space of the square [0, 1] × [0, 1] with edge identifications b a a b Again, use without proof the fact that this orbit space is Hausdorff. Hint: Show that every orbit can be represented by a point (x, y) ∈ [0, 1] × [0, 1]. To do this, it might be helpful to argue that the composition ghgh−1 is the identity and to use this to show that every element of G can be uniquely written in the form g m hn with m, n ∈ Z. 4. a) Show that a subspace of a Hausdorff space is Hausdorff and that a product of Hausdorff spaces is Hausdorff. b) Show that a subspace of a second countable space is second countable and that a product of second countable spaces is second countable. 2