Degree Theory

Abstract

Two ordered bases of a finite dimensional vector space have the same orientation if the linear transformation mapping the $i^{\text {th}}$ vector of the first basis to the $i^{\text {th}}$ vector of the second basis has a positive determinant. This is an equivalence relation with two equivalence classes, which are the orientations of the vector space. An orientation of a smooth manifold is a continuous assignment of orientations to the tangent spaces at each of the manifold’s points. We provide detailed geometric analysis in preparation for establishing the fundamental properties of orientation, which is a matter of showing that an orientation at one point of the manifold can be transported continuously along a curve that has that point as one of its endpoints. An orientation of a $\partial$-manifold induces an orientation of the manifold’s boundary. A smooth map between two oriented manifolds of the same dimension is orientation preserving (reversing) at a point if the derivative at the point maps positively oriented ordered bases to positively (negatively) oriented ordered bases. If the domain is compact, the degree over a regular value is the number of preimages at which the map is orientation preserving minus the number of preimages at which the map is orientation reversing. This concept satisfies three axioms, and we show that there is a unique extension to continuous functions that satisfies these axioms. The degree is well behaved with respect to composition and cartesian products of functions.

Exercises

12.1

For $(v_1, \ldots , v_q) \in F^q$ and $(w_1, \ldots , w_q) \in O^q$, prove that

$$(w_1, \ldots , w_q) = \varGamma ^q(v_1, \ldots , v_q)$$

if and only if there is a $q \times q$ lower triangular matrix $M = (a_{ij})$ with positive entries on the diagonal such that

$$M\begin{pmatrix} w_1 \\ \vdots \\ w_q \end{pmatrix} = \begin{pmatrix} v_1 \\ \vdots \\ v_q \end{pmatrix} \;.$$

Prove that $\varGamma ^q$ is an open map.

12.2

If X is m-dimensional, endow the Grassman manifold $G^q$ with the structure of an $q(m - q)$-dimensional smooth manifold.

12.3

Let $\alpha : S^m \rightarrow S^m$ be the antipodal map $\alpha (p) := -p$ where, as usual, $S^m := \{\, p = (p_0, \ldots , p_m) \in \mathbb {R}^{m+1} : \Vert p\Vert = 1 \,\}$ is the m-dimensional unit sphere.

• (a) Prove that $\alpha$ is orientation preserving if and only if m is odd.

Real m-dimensional projective space is

$$P^m := \{\, \{p,\alpha (p)\} : p \in S^m \,\}.$$

For $i = 0,\ldots , m$ let $U_i := \{\, p \in S^m : p_i > 0 \,\}$, and let $\varphi _i : U_i \rightarrow P^m$ be the map $\varphi _i(p) := \{p,\alpha (p)\}$. In the obvious sense these maps may be regarded as a $C^\infty$ atlas of parameterizations for $P^m$.

• (b) Prove that $P^m$ is orientable if and only if m is odd.

• (c) When m is odd, what is $\deg (\alpha )$?

12.4

The Riemann sphere is the space $S := \mathbb {C}\cup \{\infty \}$ with a differentiable structure defined by the two parameterizations $\mathrm {Id}_\mathbb {C}$ and the map $z \mapsto 1/z$ (with $0 \mapsto \infty$). One can define a category of complex manifolds and holomorphic maps between them, for which this is the most elementary example, but we will ignore the complex structure (except that we use complex arithmetic in defining maps) and regard this as 2-dimensional manifold. A rational polynomial is a ratio $r(z) = p(z)/q(z)$ of two polynomials. Show that r may be regarded as a $C^\infty$ function $r : S \rightarrow S$. What is its degree?

12.5

Let $\mathbb {Z}_2 := \{0,1\}$ be the integers mod 2. Let M and N be (not necessarily orientable) smooth m-dimensional manifolds. If $(f,q) \in \mathscr {D}^\infty (M, N)$ and q is a regular value of f, let

$$\deg ^\infty _2(f, q) := |f^{-1}(q)| \; \mathrm {mod} \; 2.$$

• (a) Prove that if $h : C \times [0,1] \rightarrow N$ is a smooth homotopy that is smoothly degree admissible over q, then $\deg ^\infty _2(h_0,q) = \deg ^\infty _2(h_1,q)$. (Prove this first with the additional assumption that q is a regular value of h.)

• (b) Prove that there is a unique function $\deg ^2 : \mathscr {D}(M, N) \rightarrow \mathbb {Z}_2$, taking (f, q) to $\deg ^2_q(f)$ that satisfies (D1) (modified by removing the requirement that f be orientation preserving) (D2), and (D3).