Geometry/Topology Qualifying Exam

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1 Geometry/Topology Qulifying Exm Spring 202 Solve ll SEVEN prolems. Prtil credit will e given to prtil solutions.. (0 pts) Prove tht compct smooth mnifold of dimensionncnnot e immersed inr n. 2. (0 pts) Let Σ, e the compct oriented surfce with oundry, otined from T 2 = R 2 /Z 2 with coordintes(x,y) y removing smll disk{(x 2 )2 +(y 2 )2 = }. 00 () Compute the homology ofσ,. () Let Σ 2 denote closed oriented surfce of genus 2. Use your nswer from () to compute the homology ofσ (0 pts) LetS e n oriented emedded surfce inr 3 ndω e n re form ons which stisfies ω(p)(e,e 2 ) = for ll p S nd ny orthonorml sis (e,e 2 ) of T p S with respect to the stndrd Eucliden metric on R 3. If(n,n 2,n 3 ) is the unit norml vector field ofs, then prove tht ω = n dy dz n 2 dx dz +n 3 dx dy, where(x,y,z) re the stndrd Eucliden coordintes on R (0 pts) Consider the spcex = M M 2, wherem ndm 2 re Möius nds ndm M 2 = M = M 2. Here Möius nd is the quotient spce([,] [,])/((,y) (, y)). () Determine the fundmentl group ofx. () Is X homotopy equivlent to compct orientle surfce of genusg for someg? 5. (0 pts) Determine ll the connected covering spces ofrp 4 RP (0 pts) Let f : M N e smooth mp etween smooth mnifolds, X nd Y e smooth vector fields on M nd N, respectively, nd suppose tht f X = Y (i.e., f (X(x)) = Y(f(x)) for ll x M). Then prove tht f (L Y ω) = L X (f ω), where ω is -form on N. Here L denotes the Lie derivtive. 7. (0 pts) Consider the linerly independent vector fields onr 4 {0} given y: X(x,x 2,x 3,x 4 ) = x +x 2 +x 3 +x 4 x x 2 x 3 x 4 Y(x,x 2,x 3,x 4 ) = x 2 +x x 4 +x 3. x x 2 x 3 x 4 Is the rnk 2 distriution orthogonl to these two vector fields integrle? Here orthogonlity is mesured with respect to the stndrd Eucliden metric on R 4.

2 Geometry-Topology Qulitfying exm Fll 202 Solve ll of the prolems. Prtil credit will e given for prtil nswers.. Denote y S R 2 the unit circle nd consider the torus T 2 = S S. Now, define A T 2 = S S y A = {(x, y, z, w) T 2 (x, y) = (0, ) or (z, w) = (0, ) }. Compute H (T 2, A). Here we regrd S s suset of the plne, hence we indicte points on S s ordered pirs. 2. Denote y S nd S 2 the circle nd sphere respectively. Recll tht the definition of the smsh product X Y of two pointed spces is the quotient of X Y y (x, y 0 ) (x 0, y). Show tht S S nd S S S 2 hve isomorphic homology groups in ll dimensions, ut their universl covering spces do not. 3. Let X e CW-complex with one vertex, two one cells nd 3 two cells whose ttching mps re indicted elow. -skeleton () Compute the homology of X. 2-skeleton () Present the fundmentl group of X nd prove its nonelin. (Justify your work.) 4. Does there exist smooth emedding of the projective plne RP 2 into R 2? Justify your nswer. 5. Let M e mnifold, nd let C (M) e the lger of C functions M R. Explin the reltionship etween vector fields on M nd C (M). If we consider the vector fields X nd Y s mps C (M) C (M) is the composition mp XY lso vector field? Wht out [X, Y ] = XY Y X? Explin. 6. Let S e the unit sphere defined y x 2 +y 2 +z 2 +w 2 = in R 4. Compute S ω where ω = (w + w2 )dx dy dz. 7. Does the eqution x 2 = y 3 define smooth sumnifold in R 3? Prove your clim.

3 GEOMETRY TOPOLOGY QUALITFYING EXAM SPRING 203 Solve ll of the prolems tht you cn. Prtil credit will e given for prtil solutions. () Consider the form ω =(x 2 + x+y)dy dz on Ê 3. Let S 2 ={x 2 + y 2 + z 2 = } Ê 3 e the unit sphere, nd i: S 2 Ê 3 the inclusion. () Clculte S 2 ω. () Construct closed form α on Ê 3 such tht i α = i ω, or show tht such form α does not exist. (2) Find ll points in Ê 3 in neighorhood in which the functions x,x 2 + y 2 + z 2,z cn serve s locl coordinte system. (3) Prove tht the rel projective spce ÊP n is smooth mnifold of dimension n. (4) () Show tht every closed -form on S n, n> is exct. () Use this to show tht every closed -form on ÊP n, n> is exct. (5) Let X e the spce otined from Ê 3 y removing the three coordinte xes. Clculte π (X) nd H (X). (6) Let X = T 2 {p,q}, p qe the twice punctured 2-dimensionl torus. () Compute the homology groups H (X, ). () Compute the fundmentl group of X. (7) () Find ll of the 2-sheeted covering spces of S S. () Show tht if pth-connected, loclly pth connected spce X hs π (X) finite, then every mp X S is nullhomotopic. (8) () Show tht if f : S n S n hs no fixed points then deg( f)=( ) n+. () Show tht if X hs S 2n s universl covering spce then π (X)={} or 2. Dte: Ferury, 203.

4 Geometry/Topology Qulifying Exm Fll 203 Solve ll SEVEN prolems. Prtil credit will e given to prtil solutions.. (5 pts) Let X denotes 2 with the north nd south poles identified. () (5 pts) Descrie cell decomposition ofx nd use it to computeh i (X) for lli 0. () (5 pts) Computeπ (X). (c) (5 pts) Descrie (i.e., drw picture of) the universl cover of X nd ll other connected covering spces ofx. 2. (0 pts) Show tht ifm is compct nd N is connected, then every sumersionf : M N is surjective. 3. (0 pts) Show tht the orthogonl group O(n) = {A M n (R) AA T = id} is smooth mnifold. HereM n (R) is the set ofn nrel mtrices. 4. (0 pts) Compute the de Rhm cohomology ofs = R/Z from the definition. 5. (0 pts) Let X, Y e topologicl spces nd f,g : X Y two continuous mps. Consider the spce Z otined from the disjoint union (X [0,]) Y y identifying (x,0) f(x) nd (x,) g(x) for ll x X. Show tht there is long exct sequence of the form: H n (X) H n (Y) H n (Z) H n (X) (0 pts) A lens spce L(p,q) is the quotient of S 3 C 2 y the Z/pZ-ction generted y (z,z 2 ) (e 2πi/p z,e 2πiq/p z 2 ) for coprimep, q. () (5 pts) Computeπ (L(p,q)). () (5 pts) Show tht ny continuous mpl(p,q) T 2 is null-homotopic. 7. (0 pts) Consider the spce of ll stright lines inr 2 (not necessrily those pssing through the origin). Explin how to give it the structure of smooth mnifold. Is it orientle?

5 Geometry nd Topology Grdute Exm Spring 204 Solve ll SEVEN prolems. Prtil credit will e given to prtil solutions. Prolem. Let X n denotes the complement of n distinct points in the plne R 2. Does there exist covering mp X 2 X? Explin. Prolem 2. Let D = {z C; z } denote the unit disk, nd choose se point z 0 in the oundry S = D = {z C; z = }. Let X e the spce otined from the union of D nd S S y gluing ech z S D to the point (z,z 0 ) S S. Compute ll homology groups H k (X;Z). Prolem 3. Let B n = {x R n ; x } denote the n dimensionl closed unit ll, with oundry S n = {x R n ; x = }. Let f: B n R n e continuous mp such tht f(x) = x for every x S n. Show tht the origin 0 is contined in the imge f(b n ). (Hint: otherwise, consider S n B n f R n {0}.) Prolem 4. Consider the following vector fields defined in R 2 : X = 2 x +x y, nd Y = y. Determine whether or not there exists (loclly defined) coordinte system (s, t) in neighorhood of (x,y) = (0,) such tht X = s, nd Y = t. Prolem 5. Let M e differentile (not necessrily orientle) mnifold. Show tht its cotngent undle T M = {(x,u);x M nd u: T x M R liner} is mnifold, nd is orientle. Prolem 6. Clculte the integrl ω where S 2 is the stndrd unit sphere in S 2 R 3 nd where ω is the restriction of the differentil 2 form (x 2 +y 2 +z 2 )(xdy dz +ydz dx+zdx dy) Prolem 7. Let M e compct m dimensionl sumnifold of R m R n. Show tht the spce of points x R m such tht M R n is infinite hs mesure 0 in R m.

6 Geometry/Topology Qulifying Exm - Fll 204. Show tht if (X, x) is pointed topologicl spce whose universl cover exists nd is compct, then the fundmentl group π (X, x) is finite group. 2. Recll tht if (X, x) nd (Y, y) re pointed topologicl spces, then the wedge sum (or -point union) X Y is the spce otined from the disjoint union of X nd Y y identifying x nd y. Show tht T 2 (the 2-torus S S ) nd S S S 2 hve isomorphic homology groups, ut re not homeomorphic. 3. Suppose S n is the stndrd unit sphere in Eucliden spce nd tht f : S n S n is continuous mp. i) Show tht if f hs no fixed points, then f is homotopic to the ntipodl mp. ii) Show tht if n = 2m, then there exists point x S 2m such tht either f(x) = x or f(x) = x. 4. If M is smooth mnifold of dimension d, using sic properties of de Rhm cohomology, descrie the de Rhm cohomology groups HdR (S M) in terms of the groups HdR (M) (long the wy, plese explin, quickly nd riefly, how to compute HdR (S )). 5. Show tht if X R 3 is closed (i.e., compct nd without oundry) sumnifold tht is homeomorphic to sphere with g > hndles ttched, then there is non-empty open suset on which the Gussin curvture K is negtive. 6. Suppose M is (non-empty) closed oriented mnifold of dimension d. Show tht if ω is differentil d-form, nd X is (smooth) vector field on X, then the differentil form L X ω necessrily vnishes t some point of M. 7. Let V e 2-dimensionl complex vector spce, nd write CP for the set of complex -dimensionl suspces of V. By explicit construction of n tls, show tht CP cn e equipped with the structure of n oriented mnifold.

8 Geometry nd Topology Grdute Exm Spring 206 Prolem. Let Y e the spce otined y removing n open tringle from the interior of compct squre in R 2. Let X e the quotient spce of Y y the equivlence reltion which identifies ll four edges of the squre nd which identifies ll three edges of the tringle ccording to the digrm elow. Compute the fundmentl group of X. Prolem 2. Let X e pth connected spce with π (X; x 0 ) = Z/5, nd consider covering spce π : X X such tht p (x 0 ) consists of 6 points. Show tht X hs either 2 or 6 connected components. Prolem 3. Compute the homology groups H k (S S n ; Z) of the product of the circle S nd the sphere S n, with n. Prolem 4. Let M e compct oriented mnifold of dimension n, nd consider differentile mp f : M R n whose imge f(m) hs non-empty interior in R n. () Show there there is t lest one point x M where f is locl diffeomorphism, nmely such tht there exists n open neighorhood U M of x such tht restriction f U : U f(u) is diffeomorphism. () Show tht there exists t lest two points x, y M such tht f is locl diffeomorphism t x nd y, f is orienttion-preserving t x, nd f is orienttion-reversing t y. Possile hint: Wht is the degree of f? Prolem 5. Consider the rel projective spce RP n, quotient of the sphere S n y the equivlence reltion tht identifies ech x S n to x. Is there degree n differentil form such ω Ω n (RP n ) such tht ω(y) 0 t every y RP n? (The nswer my depend on n.) Prolem 6. Let S n denote the n dimensionl sphere, nd rememer tht for n its de Rhm cohomology groups re { H k (S n ) 0 if k 0, n = R if k = 0, n. Consider differentile mp f : S 2n S n, with n 2. If α Ω n (S n ) is differentil form of degree n on S n such tht S n α =, let f (α) Ω n (S 2n ) e its pull-ck under the mp f. () Show tht there exists β Ω n (S 2n ) such tht f (α) = dβ. () Show tht the integrl I(f) = S 2n β dβ is independent of the choice of β nd α.

9 Geometry nd Topology Grdute Exm Spring 207 Prolem. Let ω Ω 2 (M) e differentil form of degree 2 on 2n dimensionl mnifold M. Suppose tht ω is exct, nmely tht ω = dα for some α Ω (M). Show tht ω n = ω ω ω Ω 2n (M) is exct. Prolem 2. Consider the unit disk B 2 = {x R 2 ; x } nd the circle S = {x R 2 ; x = }. The two mnifolds U = S B 2 nd V = B 2 S hve the sme oundry U = V = S S. Let X e the spce otined y gluing U nd V long this common oundry; nmely, X is the quotient of the disjoint union U V under the equivlence reltion tht identifies ech point of U to the point of V tht corresponds to the sme point of S S. Compute the fundmentl group π (X; x 0 ). Prolem 3. Compute the homology groups H n (X; Z) of the topologicl spce X of Prolem 2. Prolem 4. For unit vector v S n R n, let π v : R n v e the orthogonl projection to its orthogonl hyperplne v R n. Let M e n m dimensionl sumnifold of R n with m n 2. Show tht, for lmost every v Sn, the restriction of π v to M is injective. Possile hint: Use suitle mp f : M M S n, where = {(x, x); x M} is the digonl of M M. Prolem 5. Let G e topologicl group. Nmely, G is simultneously group nd topologicl spce, the multipliction mp G G G defined y (g, h) gh is continuous, nd the inverse mp G G defined y g g is continuous s well. Show tht, if e G is the identity element of G, the fundmentl group π (G; e) is elin. Prolem 6. Let f : M N e differentile mp etween two compct connected oriented mnifolds M nd N of the sme dimension m. Show tht, if the induced homomorphism H m (f): H m (M; Z) H m (N; Z) is nonzero, the sugroup f ( π (M; x 0 ) ) hs finite index in π (N; f(x 0 )). Hint: consider suitle covering of N.

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam

440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP