Give the Definition of a Topological Manifold. Which of the Following Topo

DIFFERENTIAL GEOMETRY. PROBLEM LIST.

1. Manifolds (1) Give the definition of a topological manifold. Which of the following topological spaces are manifolds: • Rn with standard topology, • Rn with discrete topology, • the cross {(x, y) ∈ R2 | xy = 0}, • the cusp {(x, y) ∈ R2 | x2 = y3}, and • the circle {(x, y) ∈ R2 | x2 + y2 = 1}? (2) Give an example of a topological space that is not a topological manifold and that is • Hausdorff and second countable, • second countable and locally Euclidean, and • locally Euclidean and Hausdorff. (3) Give the definition of a smooth manifold. Which of the following topological manifolds is/are smooth manifolds: • the cusp C = {(x, y) ∈ R2| x2 = y3} with the atlas given by one coordinate chart U = C → R1,(x, y) → x, • the sphere {(x, y, z) ∈ R3 | x2 + y2 + y3 = 1}, • the graph of a smooth function, 2 • the general linear group GL(n, R) ⊂ Rn . (4) Show that the product of two smooth manifolds is a smooth manifold. (5) Give the definition of a smooth map at a point. Give the definition of a smooth map of manifolds. • Show that if a continuous map F : M → N of smooth manifolds is C∞ at a point p ∈ M with respect to a coordinate neighborhood about p and a coordinate neighborhood about F (p), then F is C∞ at p with respect to any pair of coordinate neighborhoods about p and F (p). • Show that the composition of smooth maps is a smooth map. (6) Give the definition of a diffeomorphism of two manifolds. Let R be the real line with the standard smooth structure. Let R0 denote the real line with the differentiable structure given by the maximal atlas of the coordinate chart ψ : R → R, ψ(x) = x1/3. Show that • the differentiable structures on R and R0 are distinct. • R is diffeomorphic to R0. (7) Show that the projection M × N → M of the product of two manifolds is a smooth map. (8) Show that the inclusion M → M × N, given by x 7→ (x, y) where y is a fixed point in N, is a smooth map. More generally, show that given smooth manifolds M,K and L, a map (f, g): M → K × L is smooth if and only if both f and g are smooth. 1 2 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

2. Quotients and Tangent spaces (1) Let ∼ be an equivalence relation on a topological space S. Give the definition of the quotient topology on S/∼. • Prove that the quotient topology is indeed a topology. • Let π : S → S/∼ denote the quotient map. Show that a map f : S/∼ → Y to a topological space is continuous if and only if f ◦ π is continuous. (2) Give the definition of an open equivalence relation on a topological space S. Suppose that ∼ is an open relation on S. • Show that S/∼ is Hausdorff if and only if the graph

R = {(x, y) ∈ S × S | x ∼ y}

is closed in S × S. • Show that if S is second countable, the S/∼ is also second countable. (3) Suppose a right action S × G → S of a topological group G on a topological space S is continuous. Define two points x, y of S to be equivalent if there is an element g ∈ G such that y = xg. Let S/G be the quotient space; it is called the orbit space of the action. Prove that the projection map π : S → S/G is an open map. (4) Give the definition of RP n. Show that RP n is a topological manifold and introduce a smooth structure on RP n. (5) Give the definition of a derivation at a point in Rn. Show that the vector space of derivations at a point p is isomorphic to the vector space of vectors at p. (6) Give the definition of a derivation at a point in a manifold M. Give the definition of a tangent vector, and the tangent space TpM of M at p. Show that TpM is a vector space. Give the definition of the differential df : TpM → Tf(p)N of a map f : M → N. Show that the differential of a map is well- defined, i.e., it takes a derivation at a point in M to a derivation at a point in N. (7) Prove the chain rule for differentials of maps. Prove that coordinate vectors at a point in a manifold form a basis for the tangent space TpM. (8) Give the definition of a Lie group. Let G be a Lie group with multiplication map µ: G × G → G, inverse map i : G → G, and identity element e. • Show that de,eµ(X,Y ) = X + Y . • Show that dei(X) = −X. (9) Given a smooth map F : M → N and a point p ∈ M, let (U, x1, ..., xm) denote a coordinate neighborhood in M about p, and (V, y1, ..., yn) a coordinate neighborhood in N about F (p). Relative to the coordinate bases {∂/∂xi} for TpM and {∂/∂yj} for TF (x)N, find the matrix representing the differential 2 3 dpF . Compute the differential df of the map f : R → R given by

f(x, y) = (x2 + 3y, xy + 2x, y3 + x2)

at any point (x, y), and ﬁnd df(∂x). DIFFERENTIAL GEOMETRY. PROBLEM LIST. 3

3. Submanifolds (1) Give the definition of a regular submanifold. Show that a regular submanifold has a structure of a smooth manifold. Which of the following topological subspaces are regular submanifolds. • the circle S1 = {(x, y) ∈ R2 | x2 + y2 = 1 } in R2, • the closed interval [0, 1] ⊂ R2, • the open interval (0, 1) ⊂ R2, • the union of the graph of the topologists’ sine curve y = sin(1/x) and the interval {0} × (−1, 1). (2) Give the definition of a local diffeomorphism. State the inverse function theorem for Rn and deduce the inverse function theorem for manifolds. Prove the corollary: let N be a smooth manifold of dimension n. Smooth functions F1, .., Fn on a coordinate neighborhood (U, x1, ..., xn) of a point p in N define a coordinate neighborhood (U, F1, .., Fn) if and only if the Jacobian determinant det(∂Fi/∂xj) 6= 0. (3) Give the definition of a critical point and regular point of a smooth map. Give the definition of a critical value and regular value. Find critical/regular points and critical/regular values of the wrinkling map:

w : Ry × Rx × Rz −→ Rt1 × Rt2 ,

w(y, x, z) = (y, z3 + 3(y2 − 1)z + x2).

(4) Prove the regular level set theorem: Let c be a regular value of a smooth map f : M → N of manifolds. If f −1(c) is non-empty, then it is a regular submanifold of M of dimension dim M − dim N. (5) Let N ⊂ R3 be the set of solutions to the system of equations 4x2+9y2+z2 = 1 and 2x + y + 3z = 0. Determine whether N is a regular submanifold of R3. (6) Give the definition of a transverse map. Prove the transversality theorem: if a smooth map f : M → N is transverse to a regular submanifold S ⊂ N, then f −1 is a regular submanifold of M. (7) Give the definitions of an immersion and submersion. Which of the following maps are immersions/submersions: • R2 → R3 given by (x, y) 7→ (x, y, 0) • R3 → R2) given by (x, y, z) 7→ (x, y) • R → R2 given by x 7→ (x2, x3). • R → R2 given by x 7→ (x2 − 1, x3 − x) • R → R2 with image letter ‘b’. (8) Formulate the constant rank theorem and prove the immersion theorem: Suppose f : M → N is an immersion at p ∈ M. Then there are a coordinate chart (U, x1, ..., xm) about p and a coordinate chart about f(p) such that with respect to coordinate charts f|U is given by f(x1, ..., xm) = (x1, ..., xm, 0, ..., 0). Formulate and prove the submersion theorem. (9) Give the definition of an embedding. Show that the image of an embedding is a regular submanifold, and that the inclusion of a regular submanifold is an embedding. 4 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

4. Tangent bundle (1) Give the deﬁnition of the tangent bundle TM of a manifold as a set. In- troduce a topology on TM and show that TM is a topological manifold. Introduce a smooth structure on TM. (2) Give the deﬁnition of a vector bundle. Show that for any smooth manifold M, the projection M × Rn → M is a vector bundle (called trivial). Show that the projection TM → M taking a pair (x, v) to x is also a vector bundle (called the tangent vector bundle).

5. Bump functions (1) Give the deﬁnition of a bump function on a manifold M at a point p ∈ M. Deﬁne explicitly a bump function on a manifold at a point. (2) Prove that for any functions ρ1, ..., ρm on a manifold M, we have X supp( ρi) ⊂ ∪supp(ρi).

(3) Let {Kα} be a locally finite family of subsets of a topological space X. Show that every compact set K in X has a neighborhood that intersects only finitely many sets Kα. (4) Give the definition of a partition of unity. Prove that on a compact manifold with an open cover {Ui} there exists a partition of unity subordinate to {Ui}. 6. Vector fields (1) Give the definition of a smooth vector field. Show that a vector field X on a manifold M is smooth if and only if for every smooth function f on M, the function X(f) is also smooth. (2) Give the definition of the Lie bracket of smooth vector fields. Prove that the Lie bracket of smooth vector fields is a smooth vector field. (3) Give the definition of a Lie algebra. Show that the vector fields on a smooth manifold together with the Lie bracket form a Lie algebra. (4) Show that [fX, gY ] = fg[X,Y ] + f · X(g)Y − g · Y (f)X. Use this property 2 to calculate the Lie bracket [x∂y + y∂x, ∂x] of vector fields on R . P P (5) Given vector fields X = ai∂/∂xi and Y = bi∂/∂xi, calculate the coeffi- P cients ci in [X,Y ] = ci∂/∂xi as functions in ai, bj. DIFFERENTIAL GEOMETRY. PROBLEM LIST. 5

7. Lie groups (1) Give the definition of a Lie group. Show that the following groups are Lie groups: • the general linear group GL(n, R), • the special linear group SL(n, R). (2) Give the definition of a matrix exponential. Prove that the series for eX is convergent for any matrix X. (3) Prove the following properties of exponential. d tX tX tX • dt e = Xe . In particular, the velocity vector of the curve t 7→ e in GL(n, R) at e0X = I is X, where I is the identity matrix. • eAXA−1 = AeX A−1. (4) Give the definition of the trace of a matrix. Show that tr(XY ) = tr(YX) and tr(AXA−1) = tr(X). (5) Prove that det(eX ) = etrX . (6) Prove that dI (det): X 7→ trX. In other words, the rate of change of the function det: GL → R at the identity matrix I in the direction X is trX. (7) Let G be a Lie group with multiplication µ: G × G → G. Show that

d(a,b)µ(Xa,Yb) = drb(Xa) + dla(Yb),

for the left multiplication la, right multiplication rb and any vectors Xa ∈ TaG and Yb ∈ TbG. 8. Lie algebras (1) Identify the tangent space at the identity of GL(n, R) and SL(n, R). (2) Give the definition of a left invariant vector field on a Lie group. Show that every left invariant vector field is smooth. (3) Show that the vector space of left invariant vector fields on a Lie group G is isomorphic to the vector space TeG, i.e., to the tangent space to G at the identity. (4) Show that if X and Y are left invariant vector fields on a Lie group G, then [X,Y ] is also a left invariant vector field on G. Conclude that the vector space of left invariant vector fields on a Lie group (which is isomorphic to TeG) is a Lie algebra. (5) Problem 16.11.

HW: (1), (7), (12). 6 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

9. Differential 1-forms ∗ (1) Give the definition of the contangent space Tp M of a manifold M at a point ∗ p. Given a coordinate chart about p, find aa basis for Tp M. (2) Give the definition of a 1-form on a manifold, and a smooth 1-form on a manifold. Show that a 1-form w is smooth if and only if for every smooth vector field X, the function w(X) is smooth. (3) Give the definition of the differential df of a function f on a manifold. Show that df is a smooth 1-form. (4) Give the definition of a pullback of a 1-form. Let F : N → M be a smooth map. Prove the following properties for smooth 1-forms w1, w2, and a function f on M: ∗ ∗ ∗ ∗ • F (fw1 + w2) = F f · F w1 + F w2, • F ∗(df) = dF ∗f. (5) Express the pullback of a smooth 1-form in coordinates. Show that the pullback of a smooth 1-form with respect to a smooth map is a smooth 1-form. 2 ∂ ∂ (6) On R with standard coordinates (x, y), put X = −y ∂x + x ∂y and Y = ∂ ∂ 2 x ∂x +y ∂y . Find a 1-form w on R \{(0, 0)} such that w(X) = 1 and w(Y ) = 0.

10. Differential k-forms (1) Give the definition of a k-tensor on a vector space V . Give the definition of an alternating k-tensor on V . Give the definition of the wedge product of a k-vector and an l-vector. State the graded symmetry and associativity properties for alternating tensors. Prove that if α1, ..., αk are 1-forms on a vector space V , then for any k-tuple of vectors v1, ..., vk in V , we have

(α1 ∧ · · · ∧ αk)(v1, ..., vk) = det[αi(vj)].

(2) Let V be a vector space with basis e1, ..., en. Let α1, ..., αn be the dual basis ∗ for V . Let I = (i1, ..., ik) be a tuple of k numbers 1 ≤ i1 ≤ · · · ≤ ik ≤ n.

Let J be another such a tuple. Put αI = αi1 ∧ · · · ∧ αik and eJ = (vj1 , ..., vjk ). Show that αI (eJ ) = δI,J where δI,J = 1 if I = J and δI,J = 0 otherwise. (3) Show that the alternating tensors αI form a basis for the vector space Ak(V ) of alternating k-vectors. (4) Give the definition of a differential k-form on a smooth manifold M. Give a representation of a differential k-form in local coordinates. Show that a k-form ω is smooth if and only if for any k smooth vector fields X1, ..., Xk, the function ω(X1, ..., Xk) is smooth. Show that the wedge product of smooth forms is smooth. (5) Give the definition of a pullback of a k-form ω ∈ Ωk(M) with respect to a smooth map F : N → M of smooth manifolds. Prove that for any k-forms ω and τ in Ωk(M), and any number a ∈ R, we have • F ∗(ω + τ) = F ∗(ω) + F ∗(τ), • F ∗(aω) = aF ∗(ω). • F ∗(ω ∧ τ) = F ∗(ω) ∧ F ∗(τ). HW: 4, 6 and 18.1-18.3 (page 208) DIFFERENTIAL GEOMETRY. PROBLEM LIST. 7

11. Exterior Differentiation (1) Give the definition of an antiderivation, and its degree. Give the definition of an exterior derivative. Prove that if the manifold M is covered by a single coordinate chart, then an exterior derivative D on M exists and it is unique. (2) Prove that an exterior derivative exists on any smooth manifold M. (3) Prove that if two differential forms w and w0 agree on a neighborhood of a 0 point p, then Dw|p = Dw |p for any exterior derivative D on M. (4) Prove that an exterior derivative on any smooth manifold M is unique. (5) Show that dF ∗w = F ∗dw for any differential form w on a smooth manifold M, and any smooth map F : N → M. 2 2 3 2 (6) Let F : Rx,y → Ru,v be a map given by (x, y) 7→ (x + 2xy − y , xy). Find F ∗(udu − dv). (7) Let f1, ..., fn be smooth functions on a neighborhood U of a point p. Show that there is a neighborhood W ⊂ U of p on which f1, ..., fn are coordinate functions if and only if df1 ∧ · · · ∧ dfn|p 6= 0. (8) Problem 19.8 on page 219 (9) Problem 19.11(b) on page 219 (read Example 19.8).

12. The Lie derivative

(1) Give the definition of a smooth parametric family of vector fields Xt, the d limit limt→t0 Xt of a parametric family of vector fields, and its derivative dt Xt. Give the definition of a smooth parametric family of k-forms wt, the limit d limt→t0 wt and its derivative dt wt. Let Xt be a parametric family of vector fields, and wt a parametric family wt of 1-forms. Show that if limt→0 Xt = X0 and limt→0 wt = w0, then limt→0 wt(Xt) = w0(X0). d d d (2) Prove that dt (wt ∧ τt) = ( dt wt) ∧ τt + wt ∧ ( dt τt). d d (3) Prove that dt dwt = d( dt wt). (4) Suppose that Xt and Yt are smooth parametric families of vector fields on a smooth manifold M, and wt is a smooth parametric family of differential 2-forms. Prove that wt(Xt,Yt) is a smooth function on R × M. (5) Give the definition of a flow of a vector field on a closed manifold M. Give the definition of the Lie derivative LX Y of a vector field Y with respect to a vector field X. Show that if X and Y are smooth, then LX Y is a smooth vector field. (6) Prove that if X and Y are smooth vector fields on a closed manifold M, then LX Y = [X,Y ]. (7) Give the definition of the Lie derivative of a a differential form. Show that the Lie derivative LX f of a function with respect to a smooth vector field X is the function Xf.

HW: 11.6, 11.7, 11.8, 11.9, 12.1, 12.2, 12.4. 8 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

13. Lie derivative II

(1) Give the definition of interior multiplication. Show that for 1-forms α1, ..., αk and a vector field v on a smooth manifolds, k X j iv(α1 ∧ · · · ∧ αk) = (−1) αj(v)α1 ∧ · · · ∧ αˆj ∧ · · · ∧ αk. j=1 (2) Prove that for a k-differential form β and a differential form γ, we have k iv(β ∧ γ) = (ivβ) ∧ γ + (−1) β ∧ (ivγ). ∂ ∂ Calculate iX α for the vector field X = x ∂x + y ∂y , and the differential form α = dx ∧ dy on R2. (3) Prove that LX (w ∧ τ) = LX w ∧ τ + w ∧ LX τ. (4) Prove that LX ◦ d = d ◦ LX . (5) Prove that LX = diX + iX d. (6) Prove that X LX (w(Y1, ..., Yk)) = (LX w)(Y1, ..., Yk) + w(Y1, ..., LX Yj, ..., Yk). (7) Prove that X (LX w)(Y1, ..., Yk) = X(w(Y1, ..., Yk)) − w(Y1, ..., [X,Yj], ..., Yk). (8) Prove that for a differential 1-form w on a smooth manifold M, dw(X,Y ) = Xw(Y ) − Y w(X) − w([X,Y ]).

(9) Prove that dw(Y0, ..., Yk) equals X i ˆ X i+j ˆ ˆ (−1) Yiw(Y0, ..., Yi, ..., Yk) + (−1) w([Yi,Yj],Y0, ..., Yi, ...., Yj, ..., Yk). 0≤i

14. Orientation (1) Give the definition of an orientation on a vector space. Show that for any two bases {fi} and {ej} of a vector space, and any n-covector β, we have P β(f1, ..., fn) = |A| · β(e1, ..., en), where fj = i aijei. (2) Give the definition of an orientation on a manifold. Which of the following manifolds are orientable: • The sphere S2 = {(x, y, z) | x2 + y2 + z2 = 1}. • The Möbiusband. Show that a manifold M is orientable if and only if there exists a nowhere vanishing n-form on M. (3) Give the definition of an oriented atlas on a manifold. Show that a manifold is orientable if and only if it admits an oriented atlas. (4) Show that every Lie group is an orientable manifold. (5) Show that the tangent space of any (smooth) manifold is an orientable manifold.

Not exam problems: 12.6, 13.6, 13.7, 13.9. HW: 13.8, 14.4, 14.5. DIFFERENTIAL GEOMETRY. PROBLEM LIST. 9

15. Manifolds with boundary (1) Give the definition of a smooth function f : S → Rm, where S is an arbitrary subset of Rn. Prove the invariance of domain theorem. (2) Show that a diffeomorphism f : U → V of open subsets U, V ⊂ Hn maps interior points to interior points and boundary points to boundary points. (3) Give the definition of a smooth manifold with boundary. Show that if M is a smooth manifold-with-boundary of dimension n, then ∂M is a smooth manifold-with-boundary of dimension n − 1. (4) Give the definition of inward and outward pointing vectors. Prove that for every manifold-with-boundary M, there is an outward pointing vector field over ∂M. (5) Prove that if M is an orientable manifold with boundary, then ∂M is also an orientable manifold. (6) Let M be an oriented manifold with boundary, and p a point on ∂M. Show that an ordered basis (v1, ..., vn−1) represents the orientation of ∂M at p if and only if (Xp, v1, ..., vn−1) represents the orientation of M at p. (7) Let S1 × [0, 1] be an oriented cylinder with coordinates (ϕ, t), where ϕ is the angular coordinate on S1 and t is a coordinate on [0, 1]. Suppose that S1 ×[0, 1] is oriented by (∂/∂ϕ, ∂/∂t). Determine the orientations on S1 ×{0} and S1 × {1}. (8) The unit ball D = {x ∈ Rn+1 | |x| ≤ 1} inherits an orientation from Rn+1. P i−1 i ˆ Show that w = (−1) x dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn+1 is the orientation form on ∂D. (9) The antipodal map a: Sn → Sn is the map x 7→ −x. Show that the antipodal map is orientation preserving if and only if n is odd. Deduce that RP 2n+1 is orientable for each n ≥ 0.

16. Integration on manifolds (1) Let w be a differential form over an open set U ⊂ Rn. Give the definition of an integral of w over a subset A ⊂ U. Let T : V → U be a diffeomorphism of n R ∗ R open subsets of R . Show that V T w = U w if T preserves the orientation R ∗ R and V T w = − U w if T reverses the orientation. (2) Give the definition of an integral of a differential form over a smooth manifold. Show that the integral is well-defined. (3) Formulate and prove the Stokes’s theorem. (4) Calculate the integral of xdy ∧ dz − ydx ∧ dz + zdx ∧ dy over the unit 2-sphere in R3 twice: directly, and by applying the Stokes’s theorem.

HW: 15.7-15.9, 16.4. 10 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

17. De Rham cohomology (1) Give the definition of a closed form on a manifold. Give the definition of an exact form on a manifold. Prove that each exact form is closed. (2) Give the definition of the de Rham cohomology of a manifold M. Prove that H0M ≈ Rr, where r is the number of path components in M. Calculate H0(S1 t S1). (3) Show that ephif dim M = n, then HiM = 0 for i > n. Calculate HiR for all i. (4) Show that for a smooth map F : N → M, the homomorphism F ∗ sends closed forms to closed forms and exact forms to exact forms. Prove that F ∗ defines a so-called pullback homomorphism HiM → HiN for all i. (5) Prove that there exists no nowhere zero vector field v on R2 such that v(x) = x for all x with |x| = 1. (6) Define a ring structure on H∗M = ⊕HiM.

18. Long exact sequence (1) Give the definition of a cochain complex and a differential. Give the definition of the de Rham complex of a manifold M. Give the definition of the cohomology of a cochain complex. Give the definition of a cochain map. Prove that a cochain map induces a homomorphism in cohomology. Deduce that • a smooth map F : M → N of manifolds induces a linear map F ∗ : H∗N → H∗N, and ∗ ∗ • the Lie derivative LX :Ω M → Ω M induces a linear map in cohomology. f g (2) Let → A → B → C → be a part of a an exact sequence. Show that f is surjective if and only if g is a zero map. Show that g is injective if and only if f is a zero map. f g (3) Let → A → B → C → 0 → be a part of an exact sequence. Show that C = B/ Im(f). In particular, the homomorphism g is an isomorphism if and only if f is a zero map. (4) Give the definition of a short exact sequence of chain complexes. Give the definition of the connecting homomorphism of a shot exact sequence of complexes. Prove that the connecting homomorphism is well-defined. (5) Prove the Zig-Zag Lemma: a short exact sequence of chain complexes gives rise to a long exact sequence in cohomology.

HW: 17.5 (Hint: show that the existence of v implies that the identity map on S1 factors as S1 → R2 → S1. Apply H1() to the sequence), 18.1-18.5. DIFFERENTIAL GEOMETRY. PROBLEM LIST. 11

19. Mayer-Vietoris long exact sequence (1) Let U and V be open subsets of a manifold M such that M = U ∪ V . • Show that there is a short exact sequence of cochain complexes 0 → Ω∗(M) → Ω∗(U) ⊕ Ω∗(V ) → Ω∗(U ∩ V ) → 0. • Prove the Mayer-Vietoris theorem. (2) Describe the construction of the connecting homomorphism in the Mayer- Vietoris long exact sequence. (3) Use the Mayer-Vietoris long exact sequence to calculate the cohomology groups of S1. Describe explicitly generators of H∗(U), H∗(V ), H∗(S1), and all ho- momorphisms in the Mayer-Vietoris long exact sequence. (4) Give the definition of the Euler characteristic of a manifold M with finite dimensional cohomology vector spaces Hk(M). Suppose that M is a union of two open subsets U and V such that all cohomology groups of U, V and M are finite dimensional. Show that χ(M) = χ(U) + χ(V ) − χ(U ∩ V ). (5) Given an exact sequence of finite dimensional vector spaces,

0 → A0 → A1 → · · · → Am → 0 P k show that (−1) dim Ak = 0.

20. Homotopy of maps (1) Give the definition of a homotopy of maps. Show that any two continuous maps f, g : X → Rn are homotopic. (2) Give the definition of a homotopy equivalence of topological spaces. Show that homotopy equivalence is an equivalence relation. Prove that Rn \{0} is homotopy equivalent to Sn−1. (3) Give the definition of a contractible topological space. Show that Rn is contractible. (4) Give the definitions of a retraction and a deformation retraction. Show that a point is a deformation retract of Rn, and Sn−1 is a deformation retract of Rn \{0}. (5) Show that if S is a deformation retract of M, then S and M are homotopy equivalent. (6) State the Homotopy axiom for de Rham cohomology. Prove that if f : M → N is a smooth homotopy equivalence, then the induced map f ∗ in cohomology is an isomorphism. (7) Calculate the cohomology groups of the following spaces: • The Euclidean space Rn. • The punctured plane Rn \{0}. • The open Möbiusband. • The cylinder S1 × R.

HW: 19.4, 19.5, 20.4, 20.5, 20.7. 12 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

21. Computation of de Rham cohomology (1) Given a long exact sequence

αi−1 αi αi+1 · · · −→ Ai−1 −→ Ai −→ Ai+1 −→ · · · ,

Show that Ai is isomorphic to Im(αi−1) ⊕ Im(αi). (2) Consider a long exact sequence

αi−1 αi αi+1 · · · −→ Ai−1 −→ Ai −→ Ai+1 −→ · · · .

Let Ki denote the kernel of αi. Then there is a commutative diagram in which the diagonal sequences are exact: 0 0

αi−1 αi Ai−1 Ai Ai+1

Ki+1

0 0. (3) Calculate the cohomology groups Hk(M) of a torus M. (4) Let D be a small closed disc in a torus M. Calculate the cohomology groups of M \ D. (5) Calculate the cohomology groups of a surface with g handles. DIFFERENTIAL GEOMETRY. PROBLEM LIST. 13

Midterm I (1) Give the definition of a topological manifold. Which of the following topological spaces are manifolds: • Rn with standard topology, • Rn with discrete topology, • the cross {(x, y) ∈ R2 | xy = 0}. (2) Give the definition of a critical point and regular point of a smooth map. Give the definition of a critical value and regular value. Find critical/regular points and critical/regular values of the wrinkling map:

w : Ry × Rx × Rz −→ Rt1 × Rt2 , w(y, x, z) = (y, z3 + 3(y2 − 1)z + x2).

(3) Given a smooth map F : M → N and a point p ∈ M, let (U, x1, ..., xm) denote a coordinate neighborhood in M about p, and (V, y1, ..., yn) a coordinate neighborhood in N about F (p). • Suppose that near p the map F is given by

y1 = y1(x1, ..., xm), ···

yn = yn(x1, ..., xm).

Relative to the coordinate bases {∂/∂xi} for TpM and {∂/∂yj} for TF (x)N, find the matrix representing the differential dpF . • Compute the differential df of the map f : R2 → R3 given by f(x, y) = (x2 + 3y, xy + 2x, y3 + x2)

at any point (x, y), and ﬁnd df(∂x). 14 DIFFERENTIAL GEOMETRY. PROBLEM LIST.

Midterm II

(1) Let X be an n × n matrix. d tX tX • Show that dt e = Xe . • What is the velocity vector of the curve c(t) = etX on GL(n, R) at t = 0. (2) Give the definition of a differential k-form on a smooth manifold M. Give a representation of a differential k-form in local coordinates. Show that a k-form ω is smooth if and only if for any k smooth vector fields X1, ..., Xk, the function ω(X1, ..., Xk) is smooth. 2 2 3 2 (3) Let F : Rx,y → Ru,v be a map given by (x, y) 7→ (x + 2xy − y , xy). Find F ∗(udu − dv). DIFFERENTIAL GEOMETRY. PROBLEM LIST. 15