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Summer Research Fellowship Programme of India's Science Academies

On Study of Geometric-Topological Properties of Manifolds From Differential Viewpoint

Nilendu Das

Indian Institute of Science Education And Research, Mohali, Knowledge City, Sector 81, S.A.S Nagar, 140306.

Guided by:

Dr. Siddhartha Gadgil

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Abstract

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.

Keywords: Oriented Intersection Number, Degree of map,Vector Fields, Differential Forms,De Rham Cohomology.

Table of Symbols

Symbols
dimdimdimension  boundary  LxLxlocal Lefschetz number  tensor product operation  S(a,b)S(a,b) rectangular solid
SnSnn-sphere  Hx(X)Hx(X)upper half space in  Tx(X)Tx(X) v\overrightharpoon{v} vector field SpSpgroup of permutation of  (1,.....,p) (1,.....,p) \pitchfork"is transversal to" 
ΔΔ diagonal N(Y)N(Y)normal bundle  indindindex of a vector field  TπTπ TTpermuted by ππ υX\upsilon_Xvolume form 
dfxdfxderivative of  ff at  xx N(Z;Y)N(Z;Y)relative normal bundle of  ZZin YY ϕvϕvpull-back of a vector field  AltAltalternation operation  YϵY^\epsilon ϵ \epsilon neighborhood 
O(n)O(n)orthogonal group  I2I2 mod 2 intersection number gradgradgradient  ΛpΛpspace of alternating p-tensors  f#f#induced map on cohomology 
Tx(X)Tx(X)tangent space to  XX at  xx deg2deg2mod 2 degree  VpVpp-fold Cartesian product of   VV  wedge product PPoperator 
T(X)T(X)tangent bundle  W2W2mod 2 winding number  pIp space of all p-tensors AA pull back of a tensor dθdθangle form 
HkHkupper half space in  RkRk χχEuler characteristic  ##number of points  fωfωpull-back of a form  κ(x)κ(x)curvature at  xx
XXintegral over XX dωdωexterior derivative  "is homotopic to" / "is cohomologous to"  Hp(X)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

Smooth Maps

A mapping f f of an open set URn U\subset R^n into Rm R^m is called smooth if it has continuous partial derivatives of all orders. In general a map f:XRmf:X\rightarrow R^m defined on an arbitary subset XX in RnR^n is called smooth if it may be locally extended to a smooth map on open sets; that is, if around each point xXx\in Xthere is open set URnU\subset R^n and a smooth map F:URmF:U\rightarrow R^m such that FFequals ffon UXU\cap X.

A smooth map f:XYf:X\rightarrow Y of subsets of two Euclidean space is a diffeomorphism if it is a bijective map, and if the inverse map f1:YXf^{-1}:Y\rightarrow X is also smooth. XX and YY are called diffeomorphic if such a map exists.

Manifolds

XRnX\subset R^n is a k-dimensional manifold if it is locally diffeomorphic to RkR^k, meaning that each point xx possesses a neighborhood VVin XX which is diffeomorphic to an open set UU of​ RkR^k. A diffeomorphism ϕ:UV\phi :U\rightarrow V is called a parametrization of the neighborhood VV​.

The inverse diffeomorphism ϕ1:VU\phi^{-1}:V\rightarrow U is called a coordinate system on VV. ϕ1=(x1,........,xk)\phi^{-1}=(x_1,........,x_k), the k smooth functions x1,.......,xkx_1,.......,x_k on VV are called coordinate functions.

Derivatives and Tangents

Suppose that f:RnRmf:R^n\rightarrow R^m is smooth and xx is any point in its domain. Then for any vector h Rnh\in  R^n, the derivative of ff in the direction hh taken at the point xx, is defined on all of RnR^n by

dfx(h)=limt0f(x+th)f(x)tdf_x(h)=lim_{t\rightarrow 0}\frac{f(x+th)-f(x)}{t}

The derivative map is linear and can be represented as a matrix in terms of standard bases. if f(x)=(f1(x),.....,fm(x))f(x)=(f_1(x),.....,f_m(x)), then this matrix is just the jacobian matrix of ff at​ xx:

(f1x1(x)...f1xn(x)....fmx1(x)...fmxn(x))\begin{pmatrix}\frac{\partial f_1}{\partial x_1}(x)&.&.&.&\frac{\partial f_1}{\partial x_n}(x)\\.&&&&.\\.&&&&.\\\frac{\partial f_m}{\partial x_1}(x)&.&.&.&\frac{\partial f_m}{\partial x_n}(x)\end{pmatrix}

Suppose that XRn,ϕ:UXX\subset R^n, \phi:U\rightarrow X be a local parametrization of XX at xx where URkU\subset R^k, then define the tangent space of XX at xx to be the image of of the map dϕ0:RkRnd\phi _0:R^k\rightarrow R^n,which we denote Tx(X)T_x(X)

Now we want to know about the derivative of a smooth map between two manifolds f:XYf:X\rightarrow Y. Suppose that ϕ:UX\phi:U\rightarrow X parametrizes XX about xx and ψ:VY\psi:V\rightarrow Y parametrizes YY about yy such that ϕ(0)=x,ψ(0)=y\phi(0)=x, \psi(0)=y . If UU is small enough, then we can draw the following commutative diagram,

fig 1.PNG
    Induced map

    We know what dϕ0,dψ0,dh0d\phi_0,d\psi_0,dh_0 must be, and the chain rule specifies that taking derivatives converts the preceding diagram into a commutative square of linear transformation:

    fig 2.PNG
      Induced map between tangent spaces

      Chain Rule: If f:XY and g:YZf:X\rightarrow Y  \text{and}  g:Y\rightarrow Z are smooth maps of manifolds, then d(gf)x=dgf(x)dfxd(g\circ f)_x=dg_{f(x)}\circ df_x

      The Inverse function theorem and Immersions

      The Inverse Function Theorem. Suppose that f:XYf:X\rightarrow Y is a smooth map whose derivative dfxdf_x at the point xx is an isomorphism. Then ff is a local diffeomorphism at xx. In other words, if dfxdf_x is isomorphism, one can choose local coordinates around xx and yy so that ff appears to be the identiy,​ f(x1,...,xk)=(x1,...,xk)f(x_1,...,x_k)=(x_1,...,x_k)

      Two maps f:XY and f:XYf:X\rightarrow Y  \text{and}  f':X'\rightarrow Y' are said to be equivalent if there exists diffeomorphisms α and β\alpha  \text{and}  \beta completing a commutative diagram:

      fig 3.PNG
        Two equivalent maps

        So, Inverse function theorem says that if dfxdf_x is an isomorphism then ff is locally equivalent,at​​ xx,to the identity.

        Immersion. if dim(X)<dim(Y)dim(X)< dim(Y) and, if dfx:Tx(X)Ty(Y)df_x:T_x(X)\rightarrow T_y(Y) is injective,then ff is said to be an immersion at​ xx.

        The canonical immersion is the standard inclusion map of Rk into Rl for lk, where (a1,....,ak)(a1,..,ak,0,...,0)R^k  \text{into}  R^l  \text{for}  l\ge k,  \text{where}  (a_1,....,a_k)\rightarrow (a_1,..,a_k,0,...,0)

        Local Immersion Theorem. Suppose that f:XYf:X\rightarrow Y is an immersion at​ xx. Then ff is locally equivalent to the canonical immersion.​

        Proof: Lets look at the following commutative diagram

        fig 4.PNG

          As dg0:RkRldg_0:R^k\rightarrow R^l is injective, by a change of basis in RlR^l we may assume that it has an l×k  l\times k   matrix (Ik0)\begin{pmatrix}\frac{I_k}{0}\end{pmatrix} where Ik  I_k   is an k×kk\times k identity matrix. Now define a map G:U×RlkRl by G(x,z)=g(x)+(0,z) G:U\times R^{l-k}\rightarrow R^l  \text{by}  G(x,z)=g(x)+(0,z) ​.

          GG is a local diffeomorphism and g=G(canonical immersion)g=G\circ (\text{canonical immersion}). Notice that ψG\psi\circ G can be used a local parametrization of YY,so the following diagram commutes upon shrinking UU and VV sufficiently.

          fig 6.PNG
            Commutative diagram

            Embedding. A map f:XYf:X\rightarrow Y is called proper if every preimage of compact set is compact and a proper and injecvtive immersion map is called embedding.

            Theorem. An embedding f:XYf:X\rightarrow Y maps​ XX diffeomorphically onto a submanifold of YY.

            Submersion

            If dim(X)dim(Y)dim(X)\ge dim(Y) then like injectivity of the map dfxdf_x 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 RkRlR^k\rightarrow R^l for klk\ge l, where (a1,....,ak)(a1,...,al)(a_1,....,a_k)\rightarrow (a_1,...,a_l)

            Local Submersion Theorem. Every submersion is locally equivalent to a canonical submersion.

            proof. Similar kind to the previous one.

            For a smooth map f:XYf:X\rightarrow Y, a point yYy\in Y is called a regular value of ff if dfx:Tx(X)Ty(Y)df_x:T_x(X)\rightarrow T_y(Y) is surjective at every preimage point xx of yy.

            Preimage Theorem. If yy is a regular value of​ f:XYf:X\rightarrow Y,then the preimage f1(y)f^{-1}(y) is a submanifold of XX,with dimf1(y)=dim(X)dim(Y)dimf^{-1}(y)=dim(X)-dim(Y).

            Proof. By local submersion theorem,select a local coordinates around xx and yy such that f(x1,....,xk)=(x1,...,xl)f(x_1,....,x_k)=(x_1,...,x_l) and yy corresponds to (0,....,0)(0,....,0). So every point​ xx in f1(y)f^{-1}(y) locally looks like (0,...,0,xl+1,...,xk)(0,...,0,x_{l+1},...,x_k).If VV denote the neighborhood on which those coordinate functions are defined then xl+1,...,xkx_{l+1},...,x_k form a coordinate system on the set f1(y)Vf^{-1}(y)\cap V.

            Note. Any point yYy\in Y that is not a regular value, is called a critical value. Any point which is in outside of the image of ff 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 ZZ is a submanifold of X.

            Proposition. Let ZZ be the preimage of a regular value yYyY under the smooth map f:XYf:XY .Then the kernel of the derivative map dfx:Tx(X)Ty(Y)dfx:Tx(X)Ty(Y) at any point xZx\in Z is precisely the tangent space to Z,Tx(Z)Z, T_x(Z).

            Transversality

            Let f:XYf:XY be a smooth map of manifold​. One natural question to ask is if ZZ is a submanifold of​ YY,under what circumstances f1(Z)f^{-1}(Z) will be a submanifold of X. Notice that ZZ can be written as a common zero set of some functions g1,...,glg_1,...,g_l where l=codim(Z)l=codim(Z). Now f1(Z)=(gf)1(0)f^{-1}(Z)=(g\circ f)^{-1}(0) will be a manifold if 00 is a regular value. As d(gf)x=dgydfxd(g\circ f)_x=dg_y\circ df_x, the condition of gfg\circ f being a submersion deduce to Image(dfx)+Ty(Z)=Ty(Y)Image(df_x)+T_y(Z)=T_y(Y)

            The map​ ff is said to be transversal with the submanifold ZZ if the above equation holds and denoted by​ fZf\pitchfork Z

            Theorem. If the smooth map f:XYf:XY is transversal to a submanifold ZYZ\subset Y,then the preimage f1(Z)f^{-1}(Z) is a submanifold of X

            Two submanifolds X and ZZ of YY are said to be transversal if the inclusion map i:XYi:X\rightarrow Y is transversal to ZZ and denoted by XZX\pitchfork Z

            Theorem. The ​intersection of two transversal submanifolds of YY is again a submanifold and codim(XZ)=codim(X)+codim(Z)codim(X\cap Z)=codim(X)+codim(Z)

            Homotopy and Stability

            Two maps f0,f1:XYf_0,f_1:X\rightarrow Y are homotopic if there exist a smooth map F:X×IYF:X\times I\rightarrow Y such that F(x,0)=f0,F(x,1)=f1F(x,0)=f_0,F(x,1)=f_1

            A property is stable provided that whenever f0:XYf_0:X\rightarrow Y posseses the property and ft:XYf_t:X\rightarrow Y is a homotopy of f0f_0,then, for some ϵ>0\epsilon >0, each ftf_t with t<ϵt<\epsilon also possesses the property.

            Stability Theorem. The following classes of smooth maps of a compact manifold X into a manifold YY are stable classes: Local diffeomorphism, immersion, submersion, maps transversal to any specified submanifold​ ZYZ\subset Y, embeddings, diffeomorphism.

            Sard's Theorem and Morse function

            Sard's Theorem. If f:XYf:XY is any smooth map of manifolds, then almost every point in​ YY is a regular value of ff or the set of critical values of ff​ has measure zero.

            Fubini theorem. Let AA be a closed subset of RnR^n such that AVcA\cap V_c has measure zero for all cRkc\in R^k. Then AA has measure zero in​ RnR^n.

            Proof of Sard's Theorem. By the second axiom of countability, we can find a countable collection of open sets (Ui,Vi)(U_i,V_i) such that UiU_i cover X and f(Ui)Vif(U_i)\subset V_i and the​ UiU_i's and ViV_i's are diffeomorphic to open sets in​ RnR^n​ then it suffices to prove that if UU is open in RnR^n and f:URpf:U\rightarrow R^p is smooth then f(C)f(C) is of measure zero where CC is the set of critical points of​ ff​.

            The theorem is trivial for n=0, assume that it is true for n1 and will prove it for n. Partition C into a sequence of nested subsets CC1C2C3.......C\supset C_1\supset C_2\supset C_3\supset ....... where CiC_i is the set of all xx such that the partial derivatives of​ f of order i\le i vanish at xx

            lemma 1 f(CC1)f(C-C_1) has measure zero

            Around each xCC1x\in C-C_1, we will find an open set VV such that f(V(CC1))f(V\cap (C-C_1)) has measure zero. Since xC1x\notin C_1, there is some parial derivative, say fx1\frac{\partial f}{\partial x_1}, that is not zero at xx. Consider the map h:URnh:U\rightarrow R^n defined by h(x)=(f1(x),x2,...,xn)h(x)=(f_1(x),x_2,...,x_n)

            dhxdh_x is nonsingular, so it maps a neighborhood​ VV of xx diffeomorphically onto an open set VV'. Then g=fh1g=f\circ h^{-1} map VV' onto RpR^p with the same critical value as f restricted to VV and it maps points of the form (t,x2,.....,xn)(t,y1,......,yn)(t,x_2,.....,x_n)\rightarrow (t,y_1,......,y_n). So g(t×Rn1)V=gtg|_{(t\times R^{n-1})\cap V'}=g^t maps onto t×Rp1t\times R^{p-1} and​​

            (gixj) = (1 0 gitxj)\begin{pmatrix} \frac{\partial g_i}{\partial x_j}\end{pmatrix}  =  \begin{pmatrix} 1  & |& 0 \\ \hline & | & \\ * & |&  \frac{\partial g^t_i}{\partial x_j} \end{pmatrix}

            a point of t×Rn1t\times R^{n-1} is critical for gtg^t iff it is critical for gg. So by induction, the set of critical values of gtg^t has measure zero. Consequently, by Fubini's theorem, the set of critical values of gg has measure zero.

            lemma 2. f(CkCk1)f(C_k-C_{k-1}) has measure zero for k1k\ge 1

            Similar proof.

            lemma 3. For k>np1,f(Ck)k>\frac{n}{p} -1, f(C_k) is of measure zero.

            Let SUS\subset U be a cube whose sides are of length δ\delta. If kk is sufficiently large then, we will prove that f(CkS)f(C_k\cap S) has measure zero, since CkC_k can be covered by finitely many of them, this will prove f(Ck)f(C_k) has measure zero.

            From Taylor's theorem, compactness of SS and definition of CkC_k, we see that f(x+h)=f(x)+R(x,h)f(x+h)=f(x)+R(x,h) where R(x,h)\|R(x,h)\| for xCkS,x+hSx\in C_k\cap S, x+h\in S. Now subdivide SS into rnr^n cubes of side δ/r\delta /r. Let S1S_1 be any of the cubes that contains x then any point of​ S1S_1 can be written like x+hx+h with h<n(δ/r)|h|<\sqrt{n} (\delta /r) so it follows that f(S1)f(S_1) lies in a cube with sides of length b/rk+1,b=2a(nδ)k+1b/r^{k+1}, b=2a(\sqrt{n} \delta)^{k+1}. Hence f(CkS)f(C_k\cap S) is contained in the union of at most rnr^n cubes having total volume vbprn(k+1)pv\le b^pr^{n-(k+1)p} then if k+1>n/p,v0 as rk+1> n/p, v\rightarrow 0  \text{as}  r\rightarrow \infty so f(CkS)f(C_k\cap S) has measure zero.

            This proves Sard's theorem.​

            \bulletLet f:XRf:X\rightarrow R where XX is a manifold. Then the Hessian matrix of f is

            H=(2fxixj)H= \begin{pmatrix} \frac{\partial ^2f}{\partial x_i\partial x_j} \end{pmatrix}

            If the Hessian matrix is nonsingular at a critical point xx, then​ xxis said to be a nondegenerate critical point of f.

            Morse Lemma. Suppose that a point aRka\in R^kis a nondegenerate critical point of the function f, and (hij)=(2fxixj(a))(h_{ij})= \begin{pmatrix} \frac{\partial ^2f}{\partial x_i\partial x_j}(a) \end{pmatrix}is the Hessian of fat aa. Then there exist a local coordinate around aasuch that f=f(a)+Σhijxixjf=f(a)+\Sigma h_{ij}x_ix_jnear aa.

            lemma. Suppose that f is a function on RkR^kwith a nondegenerate critical point at 00, and, ψ\psiis a diffeomorphism with ψ(0)=0\psi (0)=0.Then fψf\circ \psi also has a nondegenerate critical point at 00.

            \bulletA function whose all critical points are nondegenerate is called a Morse function. ​

            \bulletSuppose that a manifold XXsits in​ Rn, and let x1,....,xnx_1,....,x_nbe the usual coordinate functions on Rn​. If fis a function on​ XXand a=(a1,....,an)a=(a_1,....,a_n), we define a new function on XXb by fa=f+a1x1+....+anxnf_a=f+a_1x_1+....+a_nx_n

            Theorem. No matter what the function f:XRf:X\rightarrow R is, for almost every​ aRna\in R^n​the function fa=f+a1x1+....+anxnf_a=f+a_1x_1+....+a_nx_nis a Morse function​.

            lemma. Let fbe a smooth function on an open set URkU\subset R^k. Then for almost all​ aRka\in R^k, the function fa=f+a1x1+....+akxkf_a=f+a_1x_1+....+a_kx_k is a Morse function.

            Embedding of manifolds in Euclidean space

            \bulletA tangent bundle of a manifold XRnX\subset R^nis a subset of X×RnX\times R^n, defined by, T(X)={(x,v)X×Rn:vTx(X)}T(X)=\{(x,v)\in X\times R^n:v\in T_x(X)\}. T(X)T(X)contains a natural copy of XXconsisting of points (x,0)(x,0)

            Proposition. The tangent bundle of a manifold is another manifold of dimension twice of the previous one.

            Theorem. Every kk dimensional manifolds admits a one-to-one immersion in R2k+1R^{2k+1}.

            Define a map h:X×X×RRMh:X\times X\times R\rightarrow R^M by h(x,y,t)=t[f(x)f(y)]h(x,y,t)=t[f(x)-f(y)] and g:T(X)RMg:T(X)\rightarrow R^M by g(x,v)=dfx(v)g(x,v)=df_x(v) and M>2k+1M>2k+1. Choose an aIm(f),Im(g)a\notin Im(f),Im(g) and let π\pi be the projection of RMR^Monto orthogonal complement HHof aa

            Then πf\pi\circ f is our required injective immersion.

            Theorem. Let XX be an arbitary subset of​ RMR^M. For any covering of XX by open subsets​ {Uα}\{U_\alpha\}, there exists a sequence smooth functions {θi}\{\theta_i\}on XX, called a partition of unity subordinate to the open cover​ {Uα}\{U_\alpha\}, with the following properties:

            1) 0θi1 0\le\theta_i \le 1 for all xXx\in X

            2) Each xXx\in X has a neighborhood on which all but finitely many functions θi\theta_iare identically zero

            3) Each​ θi\theta_iis identically zero except on some closed set contained in one of the UαU_\alpha

            4) For each xXx\in X, Σiθi(x)=1\Sigma_i\theta_i(x)=1

            Corollary.​ On any manifold XXthere exist a proper map ρ:XR\rho:X\rightarrow R

            Whitney Embedding Theorem. Every kk dimensional manifold embeds in R2k+1R^{2k+1}

            Let ρ:XR\rho:X\rightarrow R be a proper map then define a map F(x)=(f(x),ρ(x))F(x)=(f(x),\rho(x))where ffis a injective immersion. Then πF\pi\circ Fwill be an embedding where​ π\piis the projection defined like previously. it can be shown that the set of points xXx\in Xwhere πF(x)c|\pi\circ F(x)|\le c is contained in the set where ρ(x)d|\rho(x)|\le d.

            Transversality and Intersection

            Manifolds with Boundary

            Definition. XRnX\subset R^nis a k-dimensional manifold with boundary if it is locally diffeomorphic to HkH^k. The boundary of​ XX, denoted by X\partial Xis the image of the boundary of HkH^k.

            Proposition. The product of a manifold without boundary XXand a manifold with boundary YYis another manifold with boundary. (X×Y)=X×Y\partial(X\times Y)=X\times \partial Y

            Proposition. If XRnX\subset R^n is a k-dimensional manifold with boundary, then X\partial Xis a (k1)(k-1)dimensional manifold without boundary.

            Theorem. Let ff be a smooth map from a manifold with boundary to a manifold without boundary, and suppose that both f:XY,f:XYf:X\rightarrow Y,\partial f:\partial X\rightarrow Yare transversal to a boundaryless submanifold of YY. Then the preimage f1(Z)f^{-1}(Z) is a manifold with boundary, {f1(Z)}=f1(Z)X\partial\{f^{-1}(Z)\}=f^{-1}(Z)\cap\partial X and codim(f1(Z))=codim(Z)codim(f^{-1}(Z))=codim(Z).

            The classification of one dimensional manifold. Every compact, connected, one dimensional manifold with boundary is diffeomorphic to [0,1][0,1]or S1S^1

            Corollary. The boundary of any compact one dimensional manifold with boundary consists of an even number of points.

            Theorem. If XXis any compact manifold with boundary, then there exists no smooth map g:XXg:X\rightarrow \partial X such that g:XX\partial g:\partial X\rightarrow \partial Xis the identity

            Transversality

            The Transversality Theorem. If F:X×SYF:X\times S\rightarrow Yis a smooth map where only XX has boundary and ZZ be a boundaryless submanifold of YYsuch that both F,FF,\partial F is transversal to​ ZZ, then for almost every sSs\in S, both fs,fsf_s,\partial f_sis transversal to ZZ.

            It can be shown that whenever sSs\in Sis a regular value of the natural projection map​ π:X×SS\pi:X\times S\rightarrow S restricted to W=F1(Z)W=F^{-1}(Z), fsZf_s\pitchfork Z and same goes for fs\partial f_swhen ss is a regular value of the map π:WS\partial\pi:\partial W\rightarrow S.

            ϵ\epsilonNeighborhood Theorem. For a compact boundaryless manifold YRMY\subset R^Mand a positive number ϵ\epsilon, let YϵY^\epsilonbe the open set of points in RMR^Mwith distance less than ϵ\epsilon from​ YY. If ϵ\epsilon is sufficiently small, then each point wYϵw\in Y^\epsilonpossesses a unique closest point in YY, denoted by π(w)\pi(w). Moreover the map π:YϵY\pi:Y^\epsilon\rightarrow Yis a submersion.​

            Corollary. Let f:XYf:XYbe a smoth map,​ YY being boundaryless. Then there is an open ball SS in some Euclidean space and a smooth map F:X×SYF:X\times S\rightarrow Ysuch that F(x,0)=f(x)F(x,0)=f(x) and for any fixed xXx\in Xthe map sF(x,s)s\rightarrow F(x,s) is a submersion SYS\rightarrow Y. In particular both F,FF,\partial Fare submersions.

            Transversality Homotopy Theorem. For any smooth map f:XYf:XYand any submanifold​ ZZ of YY(both boundaryless), there exists a smooth map​ g:XYg:X\rightarrow Yhomotopic to ff such that both g,gZg, \partial g \pitchfork Z

            \bulletThe normal bundle N(Y)N(Y) is defined to be the set {(y,v)Y×RM:vNy(Y)}\{(y,v)\in Y\times R^M:v\in N_y(Y)\}

            Proposition. If YRMY\subset R^M, then N(Y)N(Y)is a manifold of dimension MM and the projection σ:N(Y)Y\sigma:N(Y)\rightarrow Yis a submersion.

            Extension Theorem. Suppose that ZZ is a closed submanifold of YY, both boundaryless, and CC is a closed subset of XX. Let f:XYf:XYbe a smooth map with fZf\pitchfork Z on CCand fZ\partial f\pitchfork Z on CXC\cap \partial X. Then there exists a smooth map​ g:XYg:X\rightarrow Yhomotopic to ff such that both g,gZg, \partial g \pitchfork Z​, and on a neighborhood of CC we have f=gf=g

            Corollary. If , for f:XYf:XY, the boundary map f:XY\partial f : \partial X\rightarrow Yis transversal to ZZ, then there exists a smooth map​​ g:XYg:X\rightarrow Yhomotopic to ff such that f=g\partial f= \partial g and gZg\pitchfork Z

            Intersection Theory Mod 2

            Two submanifolds XX and​ ZZ inside YYhave complementary dimension if dim(X)+dim(Z)=dim(Y)dim(X)+dim(Z)=dim(Y), If XZX\pitchfork Z, then the dimension condition makes their intersection​ XZX\cap Z a zero dimensional manifold. If both XXand​ ZZis closed and one of them, say ​ XX is compact , then XZX\cap Z will be finite also.

            Then for a smooth map f:XYf:XY, define the mod 2 intersection number of the map ff with ZZ​​, I2(f,Z)I_2(f,Z), to be the number of points in f1(Z)f^{-1}(Z) modulo 2.

            Theorem. If f0,f1:XYf_0,f_1:X\rightarrow Yare homotopic and both transversal to ZZ, then I2(f0,Z)=I2(f1,Z)I_2(f_0,Z)=I_2(f_1,Z)​.

            Corollary. If g0,g1:XYg_0,g_1:X\rightarrow Yare arbitary homotopic maps, then we have I2(g0,Z)=I2(g1,Z)I_2(g_0,Z)=I_2(g_1,Z).

            \bulletDefine the mod 2 intersection number of XXwith​ ZZ, I2(X,Z)=I2(i,Z)I_2(X,Z)=I_2(i,Z)where i:XYi:X\hookrightarrow Yis the inclusion.

            Boundary Theorem. Suppose that XXis a boundary of some compact manifold WWand g:XYg:X\rightarrow Yis a smooth map. If gg may be extended to all of​​ WWthen I2(g,Z)=0I_2(g,Z)=0 for any closed submanifold ZZ of YYwith complemenrtary dimension..

            Theorem. If f:XYf:XYis a smooth map of a compact manifold XXinto a connected manifold YYwith same dimension as XX, then I2(f,{y})I_2(f,\{y\})is the same for all yYy\in Y. This common value is called the mod 2 degree of ff, denoted deg2(f)deg_2(f)

            Proposition. If p:CCp:\mathbb{C}\rightarrow \mathbb{C} is a smooth complex function and WWis a smooth compact region in the plane. Assume that pp has no zero on W\partial W. Then if mod 2 degree of pp:WS1\frac{p}{|p|}:\partial W\rightarrow S^1 is non zero then the function​ pp has a zero inside WW.

            Winding Number and Jordan-Brouwer Separation Theorem

            Let XX be a compact connected manifold of dimension n1n-1