MATH 528, DIFFERENTIABLE MANIFOLDS
-
Syllabus
- Texts:
- [B] Somnath Basu
Cohomology of Lie groups and Lie algebras
.
- [BC] Richard Bishop, Richard Crittenden
Geometry of manifolds.
- [BG] Richard Bishop, Samuel Goldberg
Tensor analysis on manifolds.
- [EM] Yakov Eliashberg, Nikolai Mishachev
Introduction to the h-principle.
- [K] Alexander Kupers,
Lectures on diffeomorphism groups of manifolds.
- [L] John Lee,
Introduction to smooth manifolds.
(Second edition)
- [M] John Milnor,
Topology from the Differentiable Viewpoint.
- [P] Leonid Polterovich,
The geometry of the group of symplectic diffeomorphisms.
- [SM] Bruno Scardua, Carlos Morales,
Geometry, dynamics and topology of foliations.
- [W] Andrew Wallace,
Differential topology: First steps.
-
Extra 4 lectures.
- HWA-11, due Mon, Dec 6.
- Read [P, chapter 1].
- Let \((M, \omega)\) be a symplectic manifold and let \(f \in \mathrm{Ham}(M, \omega)\) be generated by a time-independent Hamiltonian function \(H\colon M \to \mathbb{R}\). Show that every isolated fixed point \(p \in M\) of \(f\) is a critical point of \(H\).
- Let \((M, \omega)\) be a symplectic manifold and \(f\colon M \to M\) be a diffeomorphism.
Show that \(f\) is a symplectomorphism if and only if its graph \(\Gamma_f = \{\,(x, f(x))\in M\times M\,\}\) is a Lagrangian submanifold of \(M\times M\) with symplectic form \(\Omega=\mathrm{pr}_2^\ast \omega - \mathrm{pr}_1^\ast \omega\), where \(\mathrm{pr}_i\colon M\times M\to M\) are the projections to the factors.
- Let \(f_t\) and \(g_t\) be a Hamiltonian flows generated by \(F_t\) and \(G_t\) respectively. Show that \(f_t \circ g_t\) is the Hamiltonian flow generated by \(H_t= F_t + G_t \circ f_t\).
- Let \(\alpha\) be a 1-form on on a manifold \(N\). Show that the graph of \(\alpha\) in \(T^\ast N\) is a Lagrangian
submanifold if and only if \(\alpha\) is closed.
- Let \(H\colon M \to \mathbb{R}\) be a smooth function on a symplectic manifold \(M\).
Consider a closed submanifold \(L\) that lies in a level set of \(H\).
Suppose that \(L\) is Lagrangian.
Show that \(L\) is invariant under the Hamiltonian flow of \(H\).
- HWA-10, due Mon, Nov 29.
- Read Chapters 1--5 in [SM] + Chapter 7 in [K].
- Exercises from [SM]: 1.2.3, 1.2.4, 1.2.6, 2.4.1, 4.3.1
- HWA-09, due Mon, Nov 15.
Presentations:
- (Zelong) Sard's theorem: proof and the tranversality homotopy theorem, intersection number.
- (Nicholas)
Degree: definitions and their equivalence, Brouwer fixed-point theorem.
- (Zelong)
Vector fields: integral curves, flows, Moser's trick, straightening lemma [L 9.22], definition of Lie derivative of tensor fields.
- (Safaan)
Lie bracket and Lie derivative, straightening lemma for commuting fields [L 9.46] OR [BG 3.7.1].
- (Hannah)
Morse lemma:
degenerate and nondegenerate critical points,
product structure at noncritical levels,
approximation by a Morse function, proof of the Morse lemma.
- (Nicholas)
Handle decomposition: rearrangement of handles, handle body decomposition of 3-manifolds (Heegaard splitting).
- (Safaan)
Cartan's calculus.
- (Nicolas)
De Rham cohomology: definitions, homotopy invariance, cup-product, Poincaré's lemma.
- (Hannah)
Calculation of De Rham cohomology via symmetry: manifolds with an action of compact Lie group; bi-invarinat forms on compact Lie group. [B, up to Lemma 2.4]
- (Zelong)
Mayer--Vietoris theorem: formulation + an application.
- (Nicolas)
Top cohomology and Poincaré lemma with compact support.
- HWA-08, due Mon, Oct 18.
- Let \(\alpha = xdx + ydy + zdz\) and \(\omega = dx \wedge dy \wedge dz\) be differential forms on \(\mathbb{R}^3\).
(a) Find a differential form \(\beta\) on \(\mathbb{R}^3 \backslash \{0\}\) such that \(\omega= \alpha\wedge \beta\).
(b) Show that there is no differential form \(\gamma\) on \(\mathbb{R}^3\) such that \(\omega= \alpha\wedge \gamma\).
- Let \(f\colon \mathbb{S}^{2{\cdot}n-1}\to \mathbb{S}^n\) for \(n \ge 2\), be a smooth map, and let \(\omega\) be an \(n\)-form on \(\mathbb{S}^n\) such that \(\int_{S^n}\omega = 1\). Show that \(f^* \omega\) is exact, and if \(f^* \omega= d\alpha\), then the number
\(\int_{S^{2n-1}}\alpha\wedge d \alpha\)
is independent of the choice of \(\omega\) and \(\alpha\).
- Let \(\omega\) be a nowhere zero 1-form on a smooth compact manifold \(M\).
Show that if \(\omega\wedge d\omega=0\), then there exists a 1-form \(\alpha\) such that \(d\omega= \alpha\wedge \omega\).
Hint: First do it locally and then use a partition of unity.
- Describe a homomorphism \(h:H^2_{dR}(\mathbb{S}^2\times \mathbb{S}^2)\to H^2_{dR}(\mathbb{T}^4)\) that cannot be realized as an induced homomorphism for a smooth map \(\mathbb{T}^4\to \mathbb{S}^2\times \mathbb{S}^2\) (here \(\mathbb{T}^4\) denotes 4-dimensional torus).
-
Construct a closed 2-form \(\omega\) on \(M = (\mathbb{S}^2 )^n\) such that \(\omega^{\wedge n} \not= 0\) at any point of \(M\).
- HWA-07, due Mon, Oct 11.
- Read [L, Chapter 17].
- Problems from [L]: 14-5, 14-9, 17-10, 17-13.
- Let \(\alpha\) be a 2-covector on \(\mathbb{R}^4\). Show that \(\alpha\) is decomposabe if and only if \(\alpha\wedge\alpha=0\).
- HWA-06, due Mon, Oct 04.
- Read [BG, 3.2 + 3.6]; skim thru the rest of Chapter 3; prepare one question.
- Problems from [BG]: 3.2.2, 3.6.1, 3.7.1, 3.7.2, 3.8.1.
- HWA-05, due Mon, Sep 27.
- Skim thru [W], prepare one question.
- Read and think [BC, Chapter 2].
- Construct a Morse function on \(\mathbb{R}\mathrm{P}^2\) with 3 critical points. Describe the sublevel sets of this function. Try to do the same for \(\mathbb{C}\mathrm{P}^2\).
- Let \(E\to B\) be a smooth locally trivial fiber bundle with fiber \(F\).
Suppose that the base \(B\) admits a Morse function with \(m\) critical points and fiber \(F\) admits a Morse function with \(m\) critical points.
Show that the total space \(E\) admits a Morse function with \(m{\cdot}n\) critical points.
- Suppose that a compact connected manifold \(M\) admits a Morse function with \(n\) citical points of index 1 and \(m\) citical points of index 2.
Show that the fundamental group of \(M\) admits a presentation with \(n\) generators and \(m\) relations.
- Problems from [BC, Chapter 2]: 10, 11.
- HWA-04, due Mon, Sep 20.
- Read [BC, Chapter 1];
- Read and think [W] starting from Chapter 4.
- Problems from [BC, Chapter 1]: 14, 19, 22.
- Prove the Jacobi identity \([[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0\) for any vector fields \(X,Y\), and \(Z\).
-
Let \(f\) be a smooth function defined on a smooth manifold \(M\).
Suppose \(p\) is a critical point of \(f\), and \(X,Y\) are vector fields on \(M\).
Show that \((X(Yf))(p)\) depends only on \(X(p),Y(p)\in\mathrm{T}_pM\).
Furthermore, the map \((X(p),Y(p))\mapsto X(Y(f))(p)\) is a symmetric bilinear form on \(\mathrm{T}_pM\).
- HWA-03, due Mon, Sep 13.
- Read and think [BC, Chapter 1].
- Exercises from [M, §8]: 14, 15.
- Suppose that \(f\colon M\to N\) is a smooth map between compact smooth oriented \(n\)-manifolds.
Show that \(\mathrm{deg}f\) is the intersection number of the graph of \(f\) with a horizontal submanifold \(M\times x\subset M\times N\).
- Given \(n\in\mathbb{Z}\), construct a smooth 4-manifold with an embedded 2-sphere that has self-intersection number \(n\).
- Let \(X\) and \(Y\) be submanifolds of \(M\).
Show that \(X\) is transversal to \(Y\) if and only if \(X\times Y\subset M\times M\) is transversal to the diagonal \(\Delta=\{\,(x,x)\in M\times M\,\}\).
(Scan to pdf and upload to CANVAS.)
- HWA-02, due Wed, Sep 8.
- Read and understand [M, §§4--5].
- Read and think [M, §§6--7].
- Exercises from [M, §8]: 3, 6, 7, 13.
- Let \(\iota\) be a smooth involution of a smooth manifold \(M\).
Show that each connected component of the fixed point set \(S\subset M\) of \(\iota\) is a submanifold.
(Scan to pdf and upload to CANVAS.)
- HWA-01, due Mon, Aug 30.
- Read and understand [M, §§1--3].
- Read and think [M, §§4--5].
- Exercises from [M, §8]: 9, 10, 11, 12.
- Show that any two open star-shaped sets in \(\mathbb{R}^n\) are diffeomorphic.
(Scan to pdf and upload to CANVAS.)