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Fg(i)c Fi(=)f(0)p Fg(;)14 b(:::;)g(n)27 b Fi(as)g(c)n(harts\).)g(Let) 1121 1727 y Fg(\031)g Fi(:)c Fg(S)1297 1693 y Fa(n)1365 1727 y Ff(!)g Fb(R)p Fg(P)1590 1693 y Fa(n)1641 1727 y Fg(;)14 b Fi(\()p Fg(x)1757 1739 y Fc(0)1795 1727 y Fg(;)g(:::;)g(x)1985 1739 y Fa(n)2030 1727 y Fi(\))24 b Ff(7!)f Fi([)p Fg(x)2262 1739 y Fc(0)2323 1727 y Fi(:)g Fg(:::)g Fi(:)g Fg(x)2554 1739 y Fa(n)2600 1727 y Fi(])440 1910 y(b)r(e)h(the)g(restriction)e(of)h(the)h(canonical)e(pro)5 b(jection.)23 b(Sho)n(w)f(that)i Fg(\031)j Fi(is)c(surjectiv)n(e)g(and) g(that)440 2009 y(the)29 b(preimage)e(of)h(ev)n(ery)f Fg(p)c Ff(2)i Fb(R)p Fg(P)1519 1979 y Fa(n)1598 2009 y Fi(consists)i(of)h(t)n(w)n(o)g(p)r(oin)n(ts)g(\(i.e.,)h Fg(S)2653 1979 y Fa(n)2726 2009 y Fi(is)f(a)f(t)n(w)n(o-sheeted)440 2109 y(co)n(v)n(ering)e(of)j Fb(R)p Fg(P)979 2079 y Fa(n)1030 2109 y Fi(\).)440 2457 y(4.)f(Similarly)g(one)h(de\014nes)f(complex)g (pro)5 b(jectiv)n(e)27 b(n-space)f(as)739 2640 y Fb(C)15 b Fg(P)859 2606 y Fa(n)933 2640 y Fi(:=)22 b(\()p Fb(C)1129 2606 y Fa(n)q Fc(+1)1283 2640 y Ff(n)c Fi(0\))p Fg(=)1459 2652 y Fe(\030)1514 2640 y Fg(;)97 b Fi(where)83 b Fg(z)26 b Ff(\030)d Fg(w)j Ff(,)d(9)p Fg(\025)g Ff(2)h Fb(C)39 b Ff(n)18 b Fi(0)23 b(:)g Fg(z)j Fi(=)d Fg(\025w)r(:)440 2823 y Fi(As)f(ab)r(o)n(v)n(e,)g(one)g(can)g(sho)n(w)f(that)i(this)f (is)h(a)f(compact)g(2n-dimensional)f(di\013eren)n(tiable)h(ma-)440 2922 y(nifold.)k(Consider)f(the)h(restriction)f(of)h(the)g(canonical)f (pro)5 b(jection)25 b(to)h Fg(S)2722 2892 y Fc(2)p Fa(n)p Fc(+1)2907 2922 y Ff(\032)c Fb(R)3048 2892 y Fc(2)p Fa(n)p Fc(+2)3240 2900 y Ff(\030)3240 2927 y Fi(=)440 3022 y Fb(C)494 2992 y Fa(n)p Fc(+1)629 3022 y Fi(.)j(Sho)n(w)f(that)g(this)h (is)f(a)g(submersion.)g(What)g(is)h(the)f(preimage)g(of)g(a)g(p)r(oin)n (t)g(in)h Fb(C)15 b Fg(P)3214 2992 y Fa(n)3265 3022 y Fi(?)440 3321 y(Please)23 b(hand)h(in)h(y)n(our)e(solutions)g(on)h(Th)n (ursda)n(y)-7 b(,)23 b(the)i(6th)f(of)g(No)n(v)n(em)n(b)r(er,)f(in)i (the)g(lecture.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF