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

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## 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.

## Table of Symbols

$dim$ | dimension | $∂$ | boundary | $L_{x}$ | local Lefschetz number | $⊗$ | tensor product operation | $S(a,b)$ | rectangular solid |

$S_{n}$ | n-sphere | $H_{x}(X)$ | upper half space in $T_{x}(X)$ | $\overrightharpoon{v}$ | vector field | $S_{p}$ | group of permutation of $(1,.....,p)$ | $\pitchfork$ | "is transversal to" |

$Δ$ | diagonal | $N(Y)$ | normal bundle | $ind$ | index of a vector field | $T_{π}$ | $T$permuted by $π$ | $\upsilon_X$ | volume form |

$df_{x}$ | derivative of $f$ at $x$ | $N(Z;Y)$ | relative normal bundle of $Z$in $Y$ | $ϕ_{∗}v$ | pull-back of a vector field | $Alt$ | alternation operation | $Y^\epsilon$ | $\epsilon$neighborhood |

$O(n)$ | orthogonal group | $I_{2}$ | mod 2 intersection number | $grad $ | gradient | $Λ_{p}$ | space of alternating p-tensors | $f_{#}$ | induced map on cohomology |

$T_{x}(X)$ | tangent space to $X$ at $x$ | $deg_{2}$ | mod 2 degree | $V_{p}$ | p-fold Cartesian product of $V$ | $⋀$ | wedge product | $P$ | $P$operator |

$T(X)$ | tangent bundle | $W_{2}$ | mod 2 winding number | $ℑ_{p}$ | space of all p-tensors | $A_{∗}$ | pull back of a tensor | $dθ$ | angle form |

$H_{k}$ | upper half space in $R_{k}$ | $χ$ | 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" | $H_{p}(X)$ | 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 $f$ of an open set $U\subset R^n$ into $R^m$ is called *smooth* if it has continuous partial derivatives of all orders. In general a map $f:X\rightarrow R^m$ defined on an arbitary subset $X$ in $R^n$ is called smooth if it may be locally extended to a smooth map on open sets; that is, if around each point $x\in X$there is open set $U\subset R^n$ and a smooth map $F:U\rightarrow R^m$ such that $F$equals $f$on $U\cap X$.

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

## Manifolds

$X\subset R^n$ is a k-dimensional manifold if it is locally diffeomorphic to $R^k$, meaning that each point $x$ possesses a neighborhood $V$in $X$ which is diffeomorphic to an open set $U$ of $R^k$. A diffeomorphism $\phi :U\rightarrow V$ is called a parametrization of the neighborhood $V$.

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

## Derivatives and Tangents

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

$df_x(h)=lim_{t\rightarrow 0}\frac{f(x+th)-f(x)}{t}$

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

$\begin{pmatrix}\frac{\partial f_1}{\partial x_1}(x)&.&.&.&\frac{\partial f_1}{\partial x_n}(x)\\.&&&&.\\.&&&&.\\\frac{\partial f_m}{\partial x_1}(x)&.&.&.&\frac{\partial f_m}{\partial x_n}(x)\end{pmatrix}$

Suppose that $X\subset R^n, \phi:U\rightarrow X$ be a local parametrization of $X$ at $x$ where $U\subset R^k$, then define the tangent space of $X$ at $x$ to be the image of of the map $d\phi _0:R^k\rightarrow R^n$,which we denote $T_x(X)$

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

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

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

## The Inverse function theorem and Immersions

**The Inverse Function Theorem. **Suppose that $f:X\rightarrow Y$ is a smooth map whose derivative $df_x$ at the point $x$ is an isomorphism. Then $f$ is a local diffeomorphism at $x$. In other words, if $df_x$ is isomorphism, one can choose local coordinates around $x$ and $y$ so that $f$ appears to be the identiy, $f(x_1,...,x_k)=(x_1,...,x_k)$

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

So, Inverse function theorem says that if $df_x$ is an isomorphism then $f$ is locally equivalent,at $x$,to the identity.

**Immersion. **if $dim(X)< dim(Y)$ and, if $df_x:T_x(X)\rightarrow T_y(Y)$ is injective,then $f$ is said to be an* immersion* at $x$.

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

**Local Immersion Theorem. **Suppose that $f:X\rightarrow Y$ is an immersion at $x$. Then $f$ is locally equivalent to the canonical immersion.

*Proof: *Lets look at the following commutative diagram

As $dg_0:R^k\rightarrow R^l$ is injective, by a change of basis in $R^l$ we may assume that it has an $l\times k$ matrix $\begin{pmatrix}\frac{I_k}{0}\end{pmatrix}$ where $I_k$ is an $k\times k$ identity matrix. Now define a map $G:U\times R^{l-k}\rightarrow R^l \text{by} G(x,z)=g(x)+(0,z)$.

$G$ is a local diffeomorphism and $g=G\circ (\text{canonical immersion})$. Notice that $\psi\circ G$ can be used a local parametrization of $Y$,so the following diagram commutes upon shrinking $U$ and $V$ sufficiently.

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

**Theorem.** An embedding $f:X\rightarrow Y$ maps $X$ diffeomorphically onto a submanifold of $Y$.

## Submersion

If $dim(X)\ge dim(Y)$ then like injectivity of the map $df_x$ 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 $R^k\rightarrow R^l$ for $k\ge l$, where $(a_1,....,a_k)\rightarrow (a_1,...,a_l)$

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

*proof.* Similar kind to the previous one.

For a smooth map $f:X\rightarrow Y$, a point $y\in Y$ is called a regular value of $f$ if $df_x:T_x(X)\rightarrow T_y(Y)$ is surjective at every preimage point $x$ of $y$.

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

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

Note.****** **Any point $y\in Y$ that is not a regular value, is called a critical value. Any point which is in outside of the image of $f$ is a regular value by definition.

**Proposition.** If the smooth, real-valued functions $g_{1},...,g_{l}$ 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 $Z$ is a submanifold of $X$.

**Proposition.** Let $Z$ be the preimage of a regular value $y∈Y$ under the smooth map $f:X→Y$ .Then the kernel of the derivative map $df_{x}:T_{x}(X)→T_{y}(Y)$ at any point $x\in Z$ is precisely the tangent space to $Z, T_x(Z)$.

## Transversality

Let $f:X→Y$ be a smooth map of manifold. One natural question to ask is if $Z$ is a submanifold of $Y$,under what circumstances $f^{-1}(Z)$ will be a submanifold of $X$. Notice that $Z$ can be written as a common zero set of some functions $g_1,...,g_l$ where $l=codim(Z)$. Now $f^{-1}(Z)=(g\circ f)^{-1}(0)$ will be a manifold if $0$ is a regular value. As $d(g\circ f)_x=dg_y\circ df_x$, the condition of $g\circ f$ being a submersion deduce to $Image(df_x)+T_y(Z)=T_y(Y)$

The map $f$ is said to be transversal with the submanifold $Z$ if the above equation holds and denoted by $f\pitchfork Z$

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

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

**Theorem. **The** **intersection of two transversal submanifolds of $Y$ is again a submanifold and $codim(X\cap Z)=codim(X)+codim(Z)$

## Homotopy and Stability

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

A property is stable provided that whenever $f_0:X\rightarrow Y$ posseses the property and $f_t:X\rightarrow Y$ is a homotopy of $f_0$,then, for some $\epsilon >0$, each $f_t$ with $t<\epsilon$ also possesses the property.

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

## Sard's Theorem and Morse function

**Sard's Theorem.** If $f:X→Y$ is any smooth map of manifolds, then almost every point in $Y$ is a regular value of $f$ or the set of critical values of $f$ has measure zero.

**Fubini theorem.** Let $A$ be a closed subset of $R^n$ such that $A\cap V_c$ has measure zero for all $c\in R^k$. Then $A$ has measure zero in $R^n$.

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

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 $C\supset C_1\supset C_2\supset C_3\supset .......$ where $C_i$ is the set of all $x$ such that the partial derivatives of $f$ of order $\le i$ vanish at $x$

**lemma 1 $f(C-C_1)$** has measure zero

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

$dh_x$ is nonsingular, so it maps a neighborhood $V$ of $x$ diffeomorphically onto an open set $V'$. Then $g=f\circ h^{-1}$ map $V'$ onto $R^p$ with the same critical value as $f$ restricted to $V$ and it maps points of the form $(t,x_2,.....,x_n)\rightarrow (t,y_1,......,y_n)$. So $g|_{(t\times R^{n-1})\cap V'}=g^t$ maps onto $t\times R^{p-1}$ and

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

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

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

Similar proof.

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

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

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

This proves Sard's theorem.

$\bullet$Let $f:X\rightarrow R$ where $X$ is a manifold. Then the Hessian matrix of $f$ is

$H= \begin{pmatrix} \frac{\partial ^2f}{\partial x_i\partial x_j} \end{pmatrix}$

If the Hessian matrix is nonsingular at a critical point $x$, then $x$is said to be a nondegenerate critical point of $f$.

Morse Lemma.**** Suppose that a point $a\in R^k$is a nondegenerate critical point of the function $f$, and $(h_{ij})= \begin{pmatrix} \frac{\partial ^2f}{\partial x_i\partial x_j}(a) \end{pmatrix}$is the Hessian of $f$at $a$. Then there exist a local coordinate around $a$such that $f=f(a)+\Sigma h_{ij}x_ix_j$near $a$.

**lemma. **Suppose that $f$ is a function on $R^k$with a nondegenerate critical point at $0$, and, $\psi$is a diffeomorphism with $\psi (0)=0$.Then $f\circ \psi$ also has a nondegenerate critical point at $0$.

$\bullet$A function whose all critical points are nondegenerate is called a Morse function.

$\bullet$Suppose that a manifold $X$sits in $R_{n}$, and let $x_1,....,x_n$be the usual coordinate functions on $R_{n}$. If $f$is a function on $X$and $a=(a_1,....,a_n)$, we define a new function on $X$b by $f_a=f+a_1x_1+....+a_nx_n$

Theorem.****** **No matter what the function $f:X\rightarrow R$ is, for almost every $a\in R^n$the function $f_a=f+a_1x_1+....+a_nx_n$is a Morse function.

**lemma. **Let $f$be a smooth function on an open set $U\subset R^k$. Then for almost all $a\in R^k$, the function $f_a=f+a_1x_1+....+a_kx_k$ is a Morse function.

## Embedding of manifolds in Euclidean space

$\bullet$A *tangent bundle* of a manifold $X\subset R^n$is a subset of $X\times R^n$, defined by, $T(X)=\{(x,v)\in X\times R^n:v\in T_x(X)\}$. $T(X)$contains a natural copy of $X$consisting of points $(x,0)$

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

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

Define a map $h:X\times X\times R\rightarrow R^M$ by $h(x,y,t)=t[f(x)-f(y)]$ and $g:T(X)\rightarrow R^M$ by $g(x,v)=df_x(v)$ and $M>2k+1$. Choose an $a\notin Im(f),Im(g)$ and let $\pi$ be the projection of $R^M$onto orthogonal complement $H$of $a$

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

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

**1)** $0\le\theta_i \le 1$for all $x\in X$

**2)** Each $x\in X$ has a neighborhood on which all but finitely many functions $\theta_i$are identically zero

**3)** Each $\theta_i$is identically zero except on some closed set contained in one of the $U_\alpha$

**4)** For each $x\in X$, $\Sigma_i\theta_i(x)=1$

**Corollary.** On any manifold $X$there exist a proper map $\rho:X\rightarrow R$

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

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

## Transversality and Intersection

## Manifolds with Boundary

**Definition.** $X\subset R^n$is a k-dimensional *manifold with boundary* if it is locally diffeomorphic to $H^k$. The boundary of $X$, denoted by $\partial X$is the image of the boundary of $H^k$.

Proposition.****** **The** **product of a manifold without boundary $X$and a manifold with boundary $Y$is another manifold with boundary. $\partial(X\times Y)=X\times \partial Y$

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

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

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

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

**Theorem.** If $X$is any compact manifold with boundary, then there exists no smooth map $g:X\rightarrow \partial X$ such that $\partial g:\partial X\rightarrow \partial X$is the identity

## Transversality

**The Transversality Theorem. **If $F:X\times S\rightarrow Y$is a smooth map where only $X$ has boundary and $Z$ be a boundaryless submanifold of $Y$such that both $F,\partial F$ is transversal to $Z$, then for almost every $s\in S$, both $f_s,\partial f_s$is transversal to $Z$.

It can be shown that whenever $s\in S$is a regular value of the natural projection map $\pi:X\times S\rightarrow S$ restricted to $W=F^{-1}(Z)$, $f_s\pitchfork Z$ and same goes for $\partial f_s$when $s$ is a regular value of the map $\partial\pi:\partial W\rightarrow S$.

$\epsilon$**Neighborhood Theorem.** For a compact boundaryless manifold $Y\subset R^M$and a positive number $\epsilon$, let $Y^\epsilon$be the open set of points in $R^M$with distance less than $\epsilon$ from $Y$. If $\epsilon$ is sufficiently small, then each point $w\in Y^\epsilon$possesses a unique closest point in $Y$, denoted by $\pi(w)$. Moreover the map $\pi:Y^\epsilon\rightarrow Y$is a submersion.

**Corollary.** Let $f:X→Y$be a smoth map, $Y$ being boundaryless. Then there is an open ball $S$ in some Euclidean space and a smooth map $F:X\times S\rightarrow Y$such that $F(x,0)=f(x)$ and for any fixed $x\in X$the map $s\rightarrow F(x,s)$ is a submersion $S\rightarrow Y$. In particular both $F,\partial F$are submersions.

**Transversality Homotopy Theorem.** For any smooth map $f:X→Y$and any submanifold $Z$ of $Y$(both boundaryless), there exists a smooth map $g:X\rightarrow Y$homotopic to $f$ such that both $g, \partial g \pitchfork Z$

$\bullet$The normal bundle $N(Y)$ is defined to be the set $\{(y,v)\in Y\times R^M:v\in N_y(Y)\}$

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

Extension Theorem.****** **Suppose that $Z$ is a closed submanifold of $Y$, both boundaryless, and $C$ is a closed subset of $X$. Let $f:X→Y$be a smooth map with $f\pitchfork Z$ on $C$and $\partial f\pitchfork Z$ on $C\cap \partial X$. Then there exists a smooth map $g:X\rightarrow Y$homotopic to $f$ such that both $g, \partial g \pitchfork Z$, and on a neighborhood of $C$ we have $f=g$

**Corollary.** If , for $f:X→Y$, the boundary map $\partial f : \partial X\rightarrow Y$is transversal to $Z$, then there exists a smooth map $g:X\rightarrow Y$homotopic to $f$ such that $\partial f= \partial g$ and $g\pitchfork Z$

## Intersection Theory Mod 2

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

Then for a smooth map $f:X→Y$, define the *mod 2 intersection number* of the map $f$ with $Z$, $I_2(f,Z)$, to be the number of points in $f^{-1}(Z)$ modulo 2.

**Theorem.** If $f_0,f_1:X\rightarrow Y$are homotopic and both transversal to $Z$, then $I_2(f_0,Z)=I_2(f_1,Z)$.

**Corollary.** If $g_0,g_1:X\rightarrow Y$are arbitary homotopic maps, then we have $I_2(g_0,Z)=I_2(g_1,Z)$.

$\bullet$Define the mod 2 intersection number of $X$with $Z$, $I_2(X,Z)=I_2(i,Z)$where $i:X\hookrightarrow Y$is the inclusion.

**Boundary Theorem.** Suppose that $X$is a boundary of some compact manifold $W$and $g:X\rightarrow Y$is a smooth map. If $g$ may be extended to all of $W$then $I_2(g,Z)=0$ for any closed submanifold $Z$ of $Y$with complemenrtary dimension..

Theorem.****** **If $f:X→Y$is a smooth map of a compact manifold $X$into a connected manifold $Y$with same dimension as $X$, then $I_2(f,\{y\})$is the same for all $y\in Y$. This common value is called the *mod 2 degree of* $f$, denoted $deg_2(f)$

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

## Winding Number and Jordan-Brouwer Separation Theorem

Let $X$ be a compact connected manifold of dimension $n-1$