(an annulus and a double annulus) (two circles Also, determine the Euler characteristics for Tg = T#T#T# ··· #T (g copies) and RP2#RP2# ··· #RP2 (g copies)
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(an annulus and a double annulus) (two circles Also, determine the Euler characteristics for Tg = T#T#T# ··· #T (g copies) and RP2#RP2# ··· #RP2 (g copies)
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Exercises for Lecture #1 (Dugger)
1. Classify all of the capital letters of the Latin alphabet, up to homeomorphism.
2. For each of the following spaces, nd a way to regard it as a nite cell complex and compute its Euler
characteristic:(an annulus and a double annulus) (two circles glued together at a point, and two spheres glued together at a point) (a cone without its base, and a pair of pants) and the following quotient space: Also, determine the Euler characteristics forTg=T#T#T##T(gcopies) andRP2#RP2##RP2 (gcopies). The spaceTgis called the \genusgtorus".3. (a) Using cut-and-paste techniques, show that the following quotient space is homeomorphic toRP2#T.
(Hint: One way is to start by cutting along the dotted line). (b) Use cut-and-paste techniques to show thatRP2#K=RP2#T. (Note: This might take some experimentation before you get it.)4. IfXis a topological space, thecone onXis the spaceCXobtained fromXIby collapsingXf1g
to a point (recall thatI= [0;1]). ThesuspensionofXis the space Xobtained fromXIby collapsing all ofXf0gto a single point, and also collapsing all ofXf1gto a dierent point. So X consists of two conesCXglued together along their base. IfXis a cell complex, determine how to put related cell complex structures onXI,CX, and X.Prove the formulas
(a)(CX) = 1 (b)(XI) =(X) (c)(X) = 2(X). (Hint: To get started, maybe do all of this in the special case whereXis the torus).5. Calculate the Betti numbers ofRP2,K,T,Tg, and (RP2)#g. Also calculate them for the following
spaces: (a)S2_S2 (b)S2_T (c)T_K. Guess a formula for the Betti numbers ofX_Y, whereXandYare connected spaces. As a challenge, compute the Betti numbers of T.Page 2
Extra problems
6. (a) Convince yourself that ifXis a nite cell complex andAis a subcomplex, then(X) =(A) +
(X=A)1. Also convince yourself that ifAandBare two subcomplexes such thatX=A[B andA\Bis a subcomplex, then(X) =(A) +(B)(A\B). (b) SupposeXis a nite cell complex andAis a nite set of points inX. What is(X=A)? (c) Suppose thatXandYare spaces and that we have certain pointsx2X, andy2Y. Thewedge ofXandYis the spaceX_Yobtained by gluingXandYtogether by collapsingxandyinto a single point. Give a formula for(X_Y) in terms of(X) and(Y). Also give a formula for (X1_X2_ _Xn). (d) Use all the above facts to recalculate the Euler characteristics of the genusgtorusTgand of the spaceRP2#RP2##RP2(gfactors). In other words, calculate the Euler characteristics without giving cell structures on the space.(e) Also, recalculate the Euler characteristic of the surface shown below (again, without giving a cell
structure):7. Show that for everyn2Zthere exists a connected spaceXsuch that(X) =n.8. LetWbe the quotient space dened by the following diagram:α
δ(a) Compute(W).
(b)Wis a compact 2-manifold, and it is a fact that all of these are given by the listfS2;Tg;(RP2)#ggg1.
Which of these \standard models" is homeomorphic toW? (c) (Challenge) Find a cut-and-paste proof thatWis homeomorphic to your answer in (b).