Don't Just Read It; Fight

Home , Connectedness, Real projective space, Topological manifold, Topological space

Don’t just read it; ﬁght it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

1 2 Course Information

♣ About the course: • Instructor: Zuoqin Wang (Email: [email protected]) • Lecture time/room: MF 2:00-3:35 pm @ 5107 • Webpage: http://staff.ustc.edu.cn/∼wangzuoq/Courses/16F-Manifolds/ • PSets: Every two weeks. Will be posted on the course webpage. • Exams: There will be one midterm and one final exam. • Language: We will use English in all lectures, PSets and Exams. ♦ Reference books: • Introduction to Smooth Manifolds, 2nd ed, by John Lee • An Introduction to Manifolds, 2nd ed, by Loring W. Tu ♥ Prerequisites: • Basic Analysis: Ck maps, multiple integrals, the inverse and implicit function theorems, existence and uniqueness theory for ODEs ——c.f. Appendix C and Appendix D in Lee’s book • Basic Topology: Topological spaces, quotient spaces, connectedness, compactness, Haus- dorff, second countable, continuity, proper ——c.f. Appendix A in Lee’s book • Basic Algebra: Linear spaces, direct sums, inner products, linear transformations, matrices, groups, quotient groups ——c.f. Appendix B in Lee’s book ♠ Contents: Smooth manifolds are nice geometric objects on which one can do analysis: they are higher dimensional generalizations of smooth curves and smooth surfaces; they appear as the solution sets of systems of equations, the phase spaces of many physics system, etc. They are among the most important objects in modern mathematics and physics. In this course we will cover • Basic theory: definitions, examples, structural theorems etc. – Smooth manifolds and submanifolds – Smooth maps and differentials – Vector fields and flows – Lie groups and Lie group actions – Vector bundles and tensor bundles • Geometry of differential forms – Differential forms and integration – de Rham cohomology – Riemannian and symplectic structures – Other topics (e.g. Chern-Weil) if time permit LECTURE 1: TOPOLOGICAL MANIFOLDS

1. Topology; Topological Manifolds Recall that a topology on a set X is a collection O of subsets of X whose elements are called open sets, such that • The set X and the empty set ∅ are open sets. • Any union of open sets is an open set. • Any ﬁnite intersection of open sets is an open set. As usual, the complement of an open set is called a closed set. It is easy to see that if O is a topology on X, and Y ⊂ X, then OY = {O ∩ Y | O ⊂ O} is a topology on Y . This is called the induced subspace topology. Topological spaces are spaces on which one can deﬁne continuous maps. Recall that A map f : X → Y between topological spaces is called continuous if for any open set V in Y , the pre- image f −1(V ) is an open set in X. Two topological spaces X and Y are homeomorphic, if there is a continuous map f : X → Y which is one-to-one and onto, so that f −1 is also continuous. Such a map f is called a homeomorphism. Of course this gives us an equivalence relation in the category of all topological spaces. The following theorem is elementary, but the proof is very complicated: Theorem 1.1 (Invariance of Domain). If U is an open set in Rn and V is an open set in Rm, and f : U → V is a homeomorphism, then m = n.

We will only study “nice” topological spaces. Recall that a topological space X is • Hausdorff if for any x 6= y ∈ X, there exist open sets U 3 x and V 3 y so that U ∩ V = ∅. • second-countable if there exists a countable sub-collection O0 of O so that any open set is a union of (not necessarily finite) open sets in O0. All spaces we are going to study in this course will be Hausdorff and second countable. Note that if X is Hausdorff or second-countable, and A ⊂ X, then A is also Hausdorff or second-countable. There are two more topological conceptions that we will be frequently used in this course: the compactness and connectedness. Recall that a subset D in a topological space X is compact if for any collection of open sets Uα satisfying D ⊂ ∪αUα, there exists a finite sub-collection

Uα1 , ··· ,Uαk so that D ⊂ Uα1 ∪ · · · ∪ Uαk . Usually the compactness will make our life much easier. Finally we recall the conception of connectedness. A topological space X is said to be disconnected if there exists two non-empty open sets U1 and U2 in X so that U1 ∪U2 = X and U1 ∩U2 = ∅. It is called connected if it is not disconnected. (If X is disconnected, then any maximal connected subset of X is called a connected component of X.) Also we call X path-connected if for any p, q ∈ X, there is a continuous map f : [0, 1] → X so that f(0) = p, f(1) = q. Such a map is called a path from p to q.

3 4 LECTURE 1: TOPOLOGICAL MANIFOLDS

Exercise. (1) Any path connected topological space is connected, but the converse is not true. (2) Use connectedness to prove theorem 1.1 for n = 1.

Now we are ready to define topological manifolds. Roughly speaking, topological manifolds are nice topological spaces that locally looks like Rn. Definition 1.2. An n dimensional topological manifold M is a topological space so that (1) M is Hausdorff. (2) M is second-countable. (3) M is locally an Euclidean space of dimension n, i.e. for every p ∈ M, there exists a triple {ϕ, U, V }, called a chart (around p), where U is an open neighborhood of p in M, V an open subset of Rn, and ϕ : U → V a homeomorphism. Remark. The three conditions in the definition of topological manifold are independent of each other. For example, two crossing lines form a topological space which is Hausdorff and second countable but not locally Euclidean; an uncountable disjoint union of real lines form a Hausdorff and locally Euclidean topological space which is not second countable. In exercise, we will see an topological space which is locally Euclidean (and second countable) but not Hausdorff. Remark. Both the Hausdorff and the second-countable conditions are important in defining a reasonably nice geometric object. For example, according to the Hausdorff property, the limit of a convergent sequence is unique. We will prove later that the Hausdorff property together with the second countability property imply the existence of partitions of unity, which is a fundamental tool in studying manifolds.

As we mentioned, any path connected topological space is connected, but the converse is not true in general. However, for topological manifolds we have Theorem 1.3. A topological manifold M is connected if and only if it is path-connected. Proof. It is enough to show that if a topological manifold M is connected, then it is also path- connected. We ﬁx a point p ∈ M and let A be the set of points in M that can be connected to p by a path. We will show that A is both open and closed. Once this is done, A must be M itself: we can write the connected topological space M as M = A ∪ (M \ A), where both A and M \ A are open, and A ∩ (M \ A) = ∅. So A 6= ∅ (since p ∈ A) implies M \ A = ∅, i.e. A = M. To prove A is open: For any q ∈ A, we take a chart {ϕ, U, V } around q. Since V is an open set n in R containing ϕ(q), one can ﬁnd a small open ball Bϕ(q) in V containing ϕ(q). Each point in Bϕ(q) can be connected to ϕ(q) by a line-segment path. As a consequence, any point in the open −1 −1 set ϕ (Bq) can be connected to q, and thus to p, by a path. So q ∈ ϕ (Bq) ⊂ A, i.e. A is open. To prove A is closed: For any q 6∈ A, we can repeat the same argument above to get an open set around q that is not in A. So M \ A is open, i.e. A is closed. Note that a topological manifold has at most countable many connected components, each of which is a topological manifold. This generalize the well-known fact that any open set in Rn is a countable union of connected open domains. LECTURE 1: TOPOLOGICAL MANIFOLDS 5

2. Examples of Topological Manifolds The simplest examples of topological manifolds include the empty set, a countable set of points, Rn itself, and open sets in Rn. Here are some more interesting examples of manifolds. Example. (Graphs). For any open set U ⊂ Rm and any continuous map f : U → Rn, the graph of f is the subset in Rm+n = Rm × Rn defined by m+n Γ(f) = {(x, y) | x ∈ U, y = f(x)} ⊂ R . With the subspace topology inherited from Rm+n, Γ(f) is Hausdorff and second-countable. It is locally Euclidean since it has a global chart {ϕ, Γ(f),U}, where ϕ(x, y) = x is the projection onto the first factor map. Obviously ϕ is continuous, invertible, and its inverse ϕ−1(x) = (x, f(x)) is continuous. So Γ(f) is a topological manifold of dimension m. Example. (Spheres). For each n ≥ 0, the unit n-sphere

n 1 n n+1 1 2 n 2 n+1 2 n+1 S = {(x , ··· , x , x ) | (x ) + ··· + (x ) + (x ) = 1} ⊂ R with the subspace topology is Hausdorﬀ and second-countable. To show that it is locally Euclidean, we can cover Sn by two open subsets

n n U+ = S \{(0, ··· , 0, −1)},U− = S \{(0, ··· , 0, 1)} n n and deﬁne two charts {ϕ+,U+, R } and {ϕ−,U−, R } by the stereographic projections 1 ϕ (x1, ··· , xn, xn+1) = (x1, ··· , xn). ± 1 ± xn+1

Then ϕ± are continuous, invertible, and the inverse 1 ϕ−1(y1, ··· , yn) = 2y1, ··· , 2yn, ±(1 − (y1)2 − · · · − (yn)2) ± 1 + (y1)2 + ··· + (yn)2 is also continuous.

n Example. (Projective Spaces). The n dimensional real projective space RP is by deﬁnition the set of 1-dimensional linear subspaces in Rn+1, endowed with the quotient topology as the quotient space n n+1 RP = R − {(0, ··· , 0)} ∼, where the equivalent relation ∼ is given by (x1, ··· , xn+1) ∼ (tx1, ··· , txn+1), ∀t 6= 0. n One can also regard RP as the quotient of Sn by gluing the antipodal points. From this description n n it is not hard to see that RP is Hausdorﬀ and second-countable. To prove that RP is locally n Euclidean, we will denote the element in RP containing the point (x1, ··· , xn+1) by [x1 : ··· : xn+1], and consider the open sets

1 n+1 i Ui = {[x : ··· : x ] | x 6= 0}. 6 LECTURE 1: TOPOLOGICAL MANIFOLDS

n The charts {ϕi,Ui, R }, 1 ≤ i ≤ n + 1, are given by x1 xi−1 xi+1 xn+1 ϕ ([x1 : ··· : xn+1]) = , ··· , , , ··· , . i xi xi xi xi It is not hard to check that this map is well-defined and is continuous, and has a continuous inverse 1 1 n 1 i−1 i n ϕi (y , ··· , y ) = [y : ··· : y : 1 : y : ··· : y ]. n (By a similar way one can define the n dimensional complex projective space CP and verify that it is a topological manifold.) Example. (The Grassmannians). For any k < n, we can define the Grassmannian Gr(k, n) as the space of all k-dimensional subspaces in Rn. We will see later that Gr(k, n) is a manifold of dimension k(n − k). To construct local charts on Gr(k, n), c.f. page 22-24 of Lee’s book. Some systematic ways to create new manifolds Method 1: Open Subsets. Any open subset of Rn is a topological manifold, with the chart map ϕ the identity map. More generally, any open subset of a topological manifold, with the induced topology, is automatically a topological manifold. Example. (The General Linear Group). Let M(n, R) be the set of all n × n real matrices. 2 Then M(n, R) is a linear space that is isomorphic to Rn . So M(n, R) is a topological manifold in the natural way. A more interesting example is the general linear group GL(n, R) = {A ∈ M(n, R) | det(A) 6= 0}. It is an open subset in M(n, R), and thus is a topological manifold of dimension n2. Later we will construct many other manifolds consists of matrices. These are important examples of Lie groups.

Method 2: Product Manifolds. If M1 and M2 are topological manifolds of dimension n1 and n2 respectively, then the product M1 × M2, endowed with product topology, is a topological manifold of dimension n1 + n2. In fact, if {ϕ1,U1,V1} and {ϕ2,U2,V2} are charts on M1 and M2 around p and q respectively, then {ϕ1 × ϕ2,U1 × U2,V1 × V2} is a chart around (p, q) in M1 × M2. Example. (Tori). In particular, the n-torus n 1 1 T = S × · · · × S is a topological manifold of dimension n.

Method 3: Connected Sums. Let M1 and M2 be n dimensional topological manifolds and p ∈ M1, q ∈ M2 be points. Let {ϕ1,U1,V1} and {ϕ2,U2,V2} be charts around p and q respectively. For simplicity, we assume ϕ1(p) = 0 and ϕ2(q) = 0. Choose ε small enough so that B(0, 2ε) ⊂ V1 ∩ V2. Deﬁne a map of annuli ψ : B(0, 2ε) \ B(0, ε) → B(0, 2ε) \ B(0, ε), exchanging the boundaries, by 2ε2 ψ(x1, ··· , xn) = (x1, ··· , xn). (x1)2 + ··· + (xn)2 LECTURE 1: TOPOLOGICAL MANIFOLDS 7

The connected sum of M1 and M2, denoted by M1#M2, is deﬁned to be the quotient −1 G −1 . M1#M2 = (M1 \ ϕ1 (B(0, ε)) (M2 \ ϕ2 (B(0, ε)) ∼, −1 where ∼ is the identiﬁcation ϕ2(x) ∼ ψ(ϕ1(x)) for x ∈ ϕ1 (B(0, 2ε)). Geometrically, connected sum looks like

A B

A # B

Example. (Closed Surfaces). A closed surface is by definition a compact 2-dimensional manifold 2 (without boundary). Examples includes S2, T2, RP , and connected sums of them. The classi- fication theorem of closed surfaces claims that any connected closed surface is homeomorphic to one of them! Note that • For any surface S, S2#S ' S. 2 2 2 2 • RP #RP #RP ' RP #T2. 2 2 – But RP #RP is the Klein bottle, which is not homeomorphic to T2. So we actually have three families of connected closed surfaces: (1) S2. (2) T2# ··· #T2.(Oriented closed surface of genus k) 2 2 (3) RP # ··· #RP .(Non-oriented closed surfaces) Later we will construct more manifolds by regarding them as • Nice level sets of a smooth map; • Integral manifolds of nice vector fields; • Manifolds related to nice Lie group actions.