On Study of Geometric-Topological Properties of Manifolds From Differential Viewpoint
In this report, some elementary topics of Differential Topology will be presented. Differential Topology is the study of properties of a subset of a Euclidean space which remain invariant under diffeomorphism. We will see some famous theorems which are usually pop out as a consequence of homology and cohomology functors, but here we will revive those theorems by giving smooth structures to manifolds and studying immersion, submersion and properties like Transversality. A proof of Sard's theorem and consequently existence of Morse function will be presented. Using partition of unity an outline of 'Whitney Embedding Theorem' will be given. We will see the classification of compact one dimensional manifold and study four useful theorems on Transversality and using that will show some beautiful invariant of intersecting manifolds i.e. mod 2 degree of a map, mod 2 winding number which have some nice application in proving no-retraction theorem, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem etc. Next we will study so called Oriented Intersection Theory which is in some sense stronger than the mod 2 intersection theory. we will learn about the degree of a map between manifolds of same dimension and will be able to prove the fundamental theorem of algebra. Then we will move to Lefschetz fixed point theory and learn about lefschetz maps and lefschetz number and their relation to Euler characteristic. Next we will learn about vector fields, index of a vector field and see the beautiful restriction on behavior of a smooth vector field on a manifold, the famous Poincare-Hopf index theorem. Then we will learn about framed cobordism and Pontraygin manifold, which is the generalization of the concept of degree of a map and study three theorems implying that the classes of homotopic maps between manifolds are in one-one correspondence with the framed cobordism classes of submanifolds and that will eventually prove the Hopf degree theorem. Later we will see exterior algebra and differential forms on manifolds and learn to differentiate and integrate them on manifold and learn how these are related to degree of a map. We will learn some famous theorem of integral calculus like stoke’s theorem and prove the famous Gauss-Bonnet theorem. Finally we see some basic concepts of De Rham cohomology like co-chain maps and co-chain homotopy, Poincare lemma, Mayer-Vietoris sequence and will compute De Rham groups of some familiar manifolds.
Table of Symbols
| dim||dimension|| ∂||boundary|| Lx||local Lefschetz number|| ⊗||tensor product operation||rectangular solid|
| Sn||n-sphere|| Hx(X)||upper half space in Tx(X)||vector field|| Sp||group of permutation of (1,.....,p)||"is transversal to"|
| Δ||diagonal|| N(Y)||normal bundle|| ind||index of a vector field|| Tπ|| T||volume form|
| dfx||derivative of f|| N(Z;Y)||relative normal bundle of Z|| ϕ∗⇀v||pull-back of a vector field|| Alt||alternation operation||neighborhood|
| O(n)||orthogonal group|| I2||mod 2 intersection number|| ⇀grad||gradient|| Λp||space of alternating p-tensors|| f#||induced map on cohomology|
| Tx(X)||tangent space to X|| deg2||mod 2 degree|| Vp||p-fold Cartesian product of V|| ⋀||wedge product|| P||operator|
| T(X)||tangent bundle|| W2||mod 2 winding number|| ℑp||space of all p-tensors|| A∗||pull back of a tensor|| dθ||angle form|
| Hk||upper half space in Rk|| χ||Euler characteristic|| #||number of points|| f∗ω||pull-back of a form|| κ(x)||curvature at x|
| ∫X||integral over X|| dω||exterior derivative|| ∼||"is homotopic to" / "is cohomologous to"|| Hp(X)||p-th cohomology group|
“[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.”―Paul R. Halmos
I would like to start my report by discussing some basic definition and properties of manifolds and smooth maps between them and then move on to our main theme of this report like Transversality, Intersection Theory and Integration on Manifolds.
Manifolds and Smooth Maps
A mapping of an open set into is called smooth if it has continuous partial derivatives of all orders. In general a map defined on an arbitary subset in is called smooth if it may be locally extended to a smooth map on open sets; that is, if around each point there is open set and a smooth map such that equals on .
A smooth map of subsets of two Euclidean space is a diffeomorphism if it is a bijective map, and if the inverse map is also smooth. and are called diffeomorphic if such a map exists.
is a k-dimensional manifold if it is locally diffeomorphic to , meaning that each point possesses a neighborhood in which is diffeomorphic to an open set of . A diffeomorphism is called a parametrization of the neighborhood .
The inverse diffeomorphism is called a coordinate system on . , the k smooth functions on are called coordinate functions.
Derivatives and Tangents
Suppose that is smooth and is any point in its domain. Then for any vector , the derivative of in the direction taken at the point , is defined on all of by
The derivative map is linear and can be represented as a matrix in terms of standard bases. if , then this matrix is just the jacobian matrix of at :
Suppose that be a local parametrization of at where , then define the tangent space of at to be the image of of the map ,which we denote
Now we want to know about the derivative of a smooth map between two manifolds . Suppose that parametrizes about and parametrizes about such that . If is small enough, then we can draw the following commutative diagram,
We know what must be, and the chain rule specifies that taking derivatives converts the preceding diagram into a commutative square of linear transformation:
Chain Rule: If are smooth maps of manifolds, then
The Inverse function theorem and Immersions
The Inverse Function Theorem. Suppose that is a smooth map whose derivative at the point is an isomorphism. Then is a local diffeomorphism at . In other words, if is isomorphism, one can choose local coordinates around and so that appears to be the identiy,
Two maps are said to be equivalent if there exists diffeomorphisms completing a commutative diagram:
So, Inverse function theorem says that if is an isomorphism then is locally equivalent,at ,to the identity.
Immersion. if and, if is injective,then is said to be an immersion at .
The canonical immersion is the standard inclusion map of
Local Immersion Theorem. Suppose that is an immersion at . Then is locally equivalent to the canonical immersion.
Proof: Lets look at the following commutative diagram
As is injective, by a change of basis in we may assume that it has an matrix where is an identity matrix. Now define a map .
is a local diffeomorphism and . Notice that can be used a local parametrization of ,so the following diagram commutes upon shrinking and sufficiently.
Embedding. A map is called proper if every preimage of compact set is compact and a proper and injecvtive immersion map is called embedding.
Theorem. An embedding maps diffeomorphically onto a submanifold of .
If then like injectivity of the map in case of immersion, now the best we can hope for is surjectivity. A map that is a submersion at every point is called a submersion.
A canonical submersion is just the projection map of for , where
Local Submersion Theorem. Every submersion is locally equivalent to a canonical submersion.
proof. Similar kind to the previous one.
For a smooth map , a point is called a regular value of if is surjective at every preimage point of .
Preimage Theorem. If is a regular value of ,then the preimage is a submanifold of ,with .
Proof. By local submersion theorem,select a local coordinates around and such that and corresponds to . So every point in locally looks like .If denote the neighborhood on which those coordinate functions are defined then form a coordinate system on the set .
Note. Any point that is not a regular value, is called a critical value. Any point which is in outside of the image of is a regular value by definition.
Proposition. If the smooth, real-valued functions g1,...,gl on X are independent at each point where they all vanish, then the set Z of common zeros is a submanifold of X with dimension equal to dim(X)−l
Define, codim(Z):=dim(X)−dim(Z) where is a submanifold of X.
Proposition. Let be the preimage of a regular value y∈Y
The map is said to be transversal with the submanifold if the above equation holds and denoted by
Theorem. If the smooth map f:X→Y
Two submanifolds X and of Y
Theorem. The intersection of two transversal submanifolds of Y
Homotopy and Stability
Two maps are homotopic if there exist a smooth map such that
A property is stable provided that whenever posseses the property and is a homotopy of ,then, for some , each with also possesses the property.
Stability Theorem. The following classes of smooth maps of a compact manifold X into a manifold Y
Sard's Theorem and Morse function
Sard's Theorem. If f:X→Y
Fubini theorem. Let be a closed subset of such that has measure zero for all . Then has measure zero in .
Proof of Sard's Theorem. By the second axiom of countability, we can find a countable collection of open sets such that cover X and and the 's and 's are diffeomorphic to open sets in then it suffices to prove that if is open in and is smooth then is of measure zero where is the set of critical points of .
The theorem is trivial for n=0, assume that it is true for n−1 and will prove it for n. Partition C into a sequence of nested subsets where is the set of all such that the partial derivatives of f of order vanish at
lemma 1 has measure zero
Around each , we will find an open set such that has measure zero. Since , there is some parial derivative, say , that is not zero at . Consider the map defined by
is nonsingular, so it maps a neighborhood of diffeomorphically onto an open set . Then map onto with the same critical value as f restricted to and it maps points of the form . So maps onto and
a point of is critical for iff it is critical for . So by induction, the set of critical values of has measure zero. Consequently, by Fubini's theorem, the set of critical values of has measure zero.
lemma 2. has measure zero for
lemma 3. For is of measure zero.
Let be a cube whose sides are of length . If is sufficiently large then, we will prove that has measure zero, since can be covered by finitely many of them, this will prove has measure zero.
From Taylor's theorem, compactness of and definition of , we see that where for . Now subdivide into cubes of side . Let be any of the cubes that contains x then any point of can be written like with so it follows that lies in a cube with sides of length . Hence is contained in the union of at most cubes having total volume then if so has measure zero.
This proves Sard's theorem.
Let where is a manifold. Then the Hessian matrix of f is
If the Hessian matrix is nonsingular at a critical point , then is said to be a nondegenerate critical point of f.
Morse Lemma. Suppose that a point is a nondegenerate critical point of the function f, and is the Hessian of fat . Then there exist a local coordinate around such that near .
lemma. Suppose that f is a function on with a nondegenerate critical point at , and, is a diffeomorphism with .Then also has a nondegenerate critical point at .
A function whose all critical points are nondegenerate is called a Morse function.
Suppose that a manifold sits in Rn, and let be the usual coordinate functions on Rn. If fis a function on and , we define a new function on b by
Theorem. No matter what the function is, for almost every the function is a Morse function.
lemma. Let fbe a smooth function on an open set . Then for almost all , the function is a Morse function.
Embedding of manifolds in Euclidean space
A tangent bundle of a manifold is a subset of , defined by, . contains a natural copy of consisting of points
Proposition. The tangent bundle of a manifold is another manifold of dimension twice of the previous one.
Theorem. Every dimensional manifolds admits a one-to-one immersion in .
Define a map by and by and . Choose an and let be the projection of onto orthogonal complement of
Then is our required injective immersion.
Theorem. Let be an arbitary subset of . For any covering of by open subsets , there exists a sequence smooth functions on , called a partition of unity subordinate to the open cover , with the following properties:
1) for all
2) Each has a neighborhood on which all but finitely many functions are identically zero
3) Each is identically zero except on some closed set contained in one of the
4) For each ,
Corollary. On any manifold there exist a proper map
Whitney Embedding Theorem. Every dimensional manifold embeds in
Let be a proper map then define a map where is a injective immersion. Then will be an embedding where is the projection defined like previously. it can be shown that the set of points where is contained in the set where .
Transversality and Intersection
Manifolds with Boundary
Definition. is a k-dimensional manifold with boundary if it is locally diffeomorphic to . The boundary of , denoted by is the image of the boundary of .
Proposition. The product of a manifold without boundary and a manifold with boundary is another manifold with boundary.
Proposition. If is a k-dimensional manifold with boundary, then is a dimensional manifold without boundary.
Theorem. Let be a smooth map from a manifold with boundary to a manifold without boundary, and suppose that both are transversal to a boundaryless submanifold of . Then the preimage is a manifold with boundary, and .
The classification of one dimensional manifold. Every compact, connected, one dimensional manifold with boundary is diffeomorphic to or
Corollary. The boundary of any compact one dimensional manifold with boundary consists of an even number of points.
Theorem. If is any compact manifold with boundary, then there exists no smooth map such that is the identity
The Transversality Theorem. If is a smooth map where only has boundary and be a boundaryless submanifold of such that both is transversal to , then for almost every , both is transversal to .
It can be shown that whenever is a regular value of the natural projection map restricted to , and same goes for when is a regular value of the map .
Neighborhood Theorem. For a compact boundaryless manifold and a positive number , let be the open set of points in with distance less than from . If is sufficiently small, then each point possesses a unique closest point in , denoted by . Moreover the map is a submersion.
Corollary. Let f:X→Y
Transversality Homotopy Theorem. For any smooth map f:X→Y
The normal bundle is defined to be the set
Proposition. If , then is a manifold of dimension and the projection is a submersion.
Extension Theorem. Suppose that is a closed submanifold of , both boundaryless, and is a closed subset of . Let f:X→Y
Corollary. If , for f:X→Y
Intersection Theory Mod 2
Two submanifolds and inside have complementary dimension if , If , then the dimension condition makes their intersection a zero dimensional manifold. If both and is closed and one of them, say is compact , then will be finite also.
Then for a smooth map f:X→Y
Theorem. If are homotopic and both transversal to , then .
Corollary. If are arbitary homotopic maps, then we have .
Define the mod 2 intersection number of with , where is the inclusion.
Boundary Theorem. Suppose that is a boundary of some compact manifold and is a smooth map. If may be extended to all of then for any closed submanifold of with complemenrtary dimension..
Theorem. If f:X→Y
Proposition. If is a smooth complex function and is a smooth compact region in the plane. Assume that has no zero on . Then if mod 2 degree of is non zero then the function has a zero inside .
Winding Number and Jordan-Brouwer Separation Theorem
Let be a compact connected manifold of dimension