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I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable mathematical consequences of this theory so far:
Do any others come to mind?
Degree theory is a very interesting topic in differential topology/geometry. I think you should take a look at chapters 11 and 12 of the book by Madsen - "From Calculus to Cohomology" where most of the theory is developed. In particular you have very interesting concepts relating the degree coming from cohomology to the index of a vector field, Gauss maps and linking numbers and results like the Poincaré-Hopf index theorem which is related also to the Gauss-Bonnet theorem. Much valuable and readable information and some easy proofs about some of those are also done in chapters 4, 5 and 6 of the fundamental book by Milnor - "Topology from the Differentiable Viewpoint". A very complete theory of degree of a map with almost all its important applications can be found in chapter 3 of Dubrovin/Fomenko/Novikov - "Modern Geometry: Methods and Applications Vol. II". There is an exciting new book focused completely on degree theory: Outerelo/Ruiz - Mapping Degree Theory, where most of this theory is developped in detail.
The degree of a map $f:M\rightarrow N$, between connected oriented closed manifolds of the same dimension, at a regular value $y\in N$ can easily be defined as the integer given by the signs of the orientations induced by the pushforward map $f_{\ast}|_{x_i}:T_{x_i}M\rightarrow T_{y}N$ at the preimage points (finite in amount for being isolated ($f$ is a local diffeomorphism by the inverse function theorem) and by compactness of $M$): $$ \deg f :=\sum_{x_i\in f^{-1}(y)}\text{sgn}(\det f_{\ast}|_{x_i}) $$
The degree is independent of the regular value $y$ and only dependent of the homotopy class of $f$. Conversely, for the case of two smooth maps $f,g:M\rightarrow\mathbb{S}^n$, they are homotopic if $\deg f= \deg g$.
For complex manifolds $M,N$ and holomorphic $f$, if $\deg f=q$ then $q\geqslant 0$ and any regular value $y\in N$ has exactly $q$ preimages.
This definition of degree can be proved to agree with the one given by the number relating the isomorphisms that the n-de Rham cohomologies have with $\mathbb{R}$ by integration and which are related by $H^n(f)$, the induced map between the cohomology classes given by the pull-back $f^{\ast}:\Omega^n(N)\rightarrow\Omega^n(M)$:
In particular, the index of a vector field with only isolated zeros, at a point $p_0$ on the manifold is defined in terms of the degree of the Gauss map $G_{p_0}:\partial D_{\epsilon}(p_0)\rightarrow S^m$ which takes a neighborhood of that point and maps each unit vector of the field on the boundary of the neighborhood into the unit sphere of the appropriate dimension (and is shown not to depend on the particular $D$). This allows to define a total index of a vector field taking into account all the isolated zeros: $$ \text{Index}(X,p_0):=\deg G_{p_0}\Rightarrow \text{Index}(X):=\sum_{p_i\in \text{Zero}(X)}\deg G_{p_i}. $$
This index is related to the Euler-Poincaré characteristics given for example in terms of de Rham cohomology $\chi(M)=\sum_{i=0}^n (-1)^i h^i(M,\mathbb{R})$ or singular homology $\chi(M)=\sum_{i=0}^n (-1)^i b_i(M)$ by the Poincaré-Hopf theorem $$\chi(M)=\text{Index}(X)=\deg \mathcal{G}_X$$ where this total index is further related to the degree of the Gauss map $\mathcal{G}_X$ over a compact that surrounds all the isolated zeros of $X$. Therefore it is independent of the chosen vector field $X$ and only depends on the topology of the manifold. This kind of results have amusing consequences like the hairy ball theorem.
Besides, those indices are also related to Morse functions which give the famous result $\chi(M)=2-2g$ for closed oriented 2-manifolds of genus $g$.
A similar concept is defined for the index on a general hypersurface which leads to the celebrated Gauss-Bonnet theorem, a wonderful result relating the total Gaussian curvature of a closed oriented 2-manifold $M^2_g$ with all the previous concepts: $$ 2\deg G=\frac{1}{2\pi}\int_{M^2_g} K d\sigma=\text{Index}(X)=\chi(M^2_g)=2-2g $$ where here the $G$ is the Gauss map $G:M^2_g\rightarrow S^2$ and the g-torus $M^2_g$ can be thought as embedded in $\mathbb{R^3}$. This result culminates with the general Chern's theorem which relates all the theory to characteristic classes and higher dimensions.
The total index can also be shown to agree with what is called the intersection index $M(0)\circ M(X)=\text{Index(X)}$ of the two submanifolds of the tangent bundle $TM$ given by the embeddings of $M$ into $TM$ with the given vector field as section. Using this concept of intersection index one can arrive by a different path than homology to the Lefschetz number $\Lambda_f:=\sum_i (-1)^i\text{Tr}(H_i(f))$ and thus to the Lefschetz fixed point theorem which generalizes the Brower fixed point theorem:
And all this fixed point theory is related to geometry by Gauss-Bonnet, to topology by Poincaré-Hopf and all the previous mentioned degree applications by the following fundamental result: $$ \Lambda_{\text{Id}}=\chi(M). $$
Hence, I think degree theory with all its particular interconnections is an omnipresent concept fundamental within some of the most important and beautiful results in differential topology and geometry. It is worth studying and being able to relate everything with everything else.
Working with plane curves you get the notion of Whitney numbers (signed self-intersections) and you can even prove the fundamental theorem of algebra using this. Also, linking numbers which measure some knotting features of submanifolds of complementary codimension are related to degrees of maps similar to Gauss'.
For all this and more, consult the chapters I mentioned from the books at the beginning.
I hope you enjoyed this brief summary because it took very long for me when I studied it to understand how everything was put together. :-D
ADDITIONAL DIGRESSION for the fun of it:
There is even more interesting and deep connections in degree theory. What I have mentioned above is all the theory of the Brower degree of a map between manifolds which leads to the index of vector fields, Gauss maps and the important theorems I talked about.
But! There is even more to it! Gauss-Bonnet and Poincaré-Hopf are the interesting case of compact orientable 2-dimensional manifolds of topological genus $g$, $M^2_g$, that by the classification theorem of surfaces all are spheres with $g$ handles. But a holomorphic complex structure can be given precisely for these manifolds, making them 1-dimensional complex manifolds and therefore we arrive at the wonderful subject of Riemann surfaces and complex analysis. But these are all embeddable in projective complex space by Chow's theorem and therefore can be shown to be equivalent to the study of complex projective algebraic curves.
In this setting these great results expressed by degree theory can be related to another extremely important result from algebraic geometry which is also an index theorem: Riemann-Roch theorem for curves: $$ \dim\mathcal{L}(D)-\dim\mathcal{L}(\mathcal{K}-D)=\deg (D) + 1- g $$ This relates the dimension of the vector space $\mathcal{L}(D)$ of meromorphic functions on the Riemann surface with prescribed zeros and poles of orders (bounded by) $n_i\in\mathbb{Z}$ at points $P_i$ given by a divisor $D=\sum n_i P_i$, with the degree of the divisor $\deg (D)=\sum n_i$ and the genus $g$. Actually there is an extra correction term involving the canonical divisor $\mathcal{K}$ which represents the divisor given by the zeros and poles of a meromorphic 1-form on the algebraic curve/Riemann surface making all this very intertwined with line bundles (in whose language Riemann-Roch can be reformulated).
The remarkable result is that since projective complex algebraic curves are compact Riemann surfaces, our Gauss-Bonnet and Poincaré-Hopf theorems can be applied considering them with the underlying smooth real structure. BUT! Gauss-Bonnet theorem is the 2-dimensional case of Chern's theorem written in the language of characteristic classes, using the Euler class of the tangent bundle $e(TM)$ of the surface or the first Chern form as well $$ \chi(M^2_g)=\int_{M^2_g}c_1(TM^+)=\int_{M^2_g}e(TM^2_g)=\frac{1}{2\pi}\int_{M^2_g}R_{1212}d\theta^1\wedge\theta^2=\frac{1}{2\pi}\int_{M^2_g}Kd\sigma $$ where $R_{1212}$ is the component of the Riemann curvature form and $K$ the gaussian curvature and $d\sigma$ the surface area. Each divisor $D$ on our algebraic curve/Riemann surface can be used to build up a line bundle $[D]$ over the manifold for which the first Chern class gives us a definition of linde bundle degree, since it satisfies that $$ \deg ([D]):=\int_{M^2_g}c_1([D])=\deg (D). $$ From this it can be shown that the gaussian curvature of $M^2_g$ is given by the first Chern class of the inverse bundle of the canonical bundle $[\mathcal{K}]^{-1}$ which is a holomorphic tangent bundle: $c_1([\mathcal{K}]^{-1})=\frac{i}{4\pi}K\cdot h dz\wedge d\bar{z}$, with $h$ a Hermitian metric on the line bundle. But it is an important corollary of Riemann-Roch to get the degree of the canonical divisor in terms of the genus by $\deg (\mathcal{K})=2g-2$. Now, our old concept of index of a (real 2-dim) vector field $X$ is in this case what sometimes is called the degree of the field, $\deg (X):=\deg\mathcal{G}_X$, because we mentioned above that the degree of the Gauss map over a compact within $M^2_g$ which surrounds all the zeros of $X$ gives the same result. Therefore, we have arrived at our final connection of all index theorems for compact oriented 2-manifolds; this is the reunion of all the previous concepts of degree relating together topology, geometry and algebra! $$ \begin{aligned} \chi (M^2_g)&=\sum_{j=0}^2 (-1)^j\dim H_{j}(M)=\sum_{j=0}^2 (-1)^j\dim H^j_{dR}(M)= \sum_{j=0}^2 (-1)^j\dim \check{H}^j(M,\underline{\mathbb{R}}) \\ &= V-E+F =2-2g= 2\deg\mathcal{G}_X=\sum_{P\in\text{Zero(X)}}\deg G_{P}=\text{Index}(X)= \\ & =\int_{M^2_g}c_1(TM^+)=\int_{M^2_g}e(TM^2_g)=\int_{M^2_g}c_1([D])=\frac{1}{2\pi} \int_{M^2_g}Kd\sigma= \\ &= \deg ([\mathcal{K}]^{-1})=-\deg (\mathcal{K})=\Lambda_{\text{Id}} \end{aligned}, $$
where we have made explicit the Euler characteristics in terms of singular homology, de Rham cohomology and Čech cohomology over the constant sheaf $\underline{\mathbb{R}}$ which are all three equivalent in this case but arise conceptually from very different constructions (simplicial triangulations, differential forms, sheaf cohomology). Thus, $V,E,F$ is the number of vertices, edges and faces of any good triangulation of $M^2_g$, $g$ the topological no. of donut holes and all the aforementioned characteristic classes, curvatures, degrees of Gauss maps, degree of the canonical inverse line bundle and divisor, and the Lefschetz number of the identity map.
You can find this kind of results in chapter 5 of the book Jost - "Compact Riemann Surfaces".
Moreover, the concept of degree of a divisor $deg (D)$ is in this case the total sum of the algebraic multiplicities of zeros and poles imposed at the points of $D$. Locally, our complex manifold $M^2_g$ can be charted out biholomorphically by open sets of the complex plane $\mathbb{C}$ where the points of $D$ can thus be thought as algebraic varieties of dimension $0$ with multiplicities given by the degree of the appropriate defining polynomials. In general for higher dimensional varieties, like curves and surfaces embedded in projective space, their degree is defined to be the number of intersection points counted with multiplicities that they have in common with a generic hyperplane of dimension the codimension of our variety. For example, the degree of an algebraic curve in the plane is the number of intersections it has with a generic straight line and can easily be seen by Bézout's theorem to be the algebraic degree of its defining polynomial (as our intuition would suggest thinking of the fundamental theorem of algebra for a polynomial function intersecting the $x$-axis). But algebraic curves are divisors of algebraic surfaces and similar ideas follow and so on. Therefore all the concepts of degree seen so far seem to be deeply related somehow!! Of course, the generalization of Chern-Gauss-Bonnet-Poincaré-Hopf and Riemann-Roch is done by some of the most important, general and powerful results in abstract geometry of the XXth century: the Grothendieck-Hirzebruch-Riemann-Roch theorem in the algebraic geometry setting and the Atiyah-Singer index theorem in the differential setting.
I have added this as an extension to my final digression of my other answer to this question, and as a complement to my answer to other question concerning the different definitions of degree for the case of complex algebraic geometry (so all the consequences below apply to the specified smooth projective algebraic varieties over the complex numbers).
Degree characterization of linear varieties:
$X_k\subseteq\mathbb{CP}^n$ is a linear space, i.e. $X_k\cong\mathbb{CP}^k,\;\Leftrightarrow \; \deg X_k = 1$.
Degree-Dimension bound inequality:
If $X_k\subseteq\mathbb{CP}^n$ is a projective nondegenerate variety (i.e. $X\nsubseteq$ hyperplane), then $$\deg X_k\geq \operatorname{codim} X_k +1.$$
Minimal Degree characterization of projective varieties:
Let $X\subset\mathbb{CP}^n$ be any nondegenerate variety having minimal degree $\deg X=n-\dim X+1$ (cf. above). Then $X$ is one of the following varieties:
- a quadric hyersurface,
- a projecting cone over the Veronese surface $\nu_2(\mathbb{CP}^2)\subset\mathbb{CP}^5$, or
- a rational normal scroll.
Weak Bézout's Theorem:
If varieties $X_k, Y_s\subset\mathbb{CP}^n$ such that $k+s\geq n$ and intersect transversely, then $$\deg (X\cap Y)=(\deg X)\cdot (\deg Y),$$ in particular if $r+s=n$ the intersection consists exactly of $\deg X\cdot\deg Y$ points.
Strong Bézout's Theorem:
If varieties $X, Y\subset\mathbb{CP}^n$ intersect properly, then $$(\deg X)\cdot (\deg Y)=\sum_{\substack{X\cap Y=\cup_i Z_i \\ \text{irred. comp.}}} \operatorname{mult}_{Z_i}(X\cap Y)\cdot\deg Z_i,$$ where $\operatorname{mult}_{Z_i}(X\cap Y)$ is the multiplicity at the irreducible component $Z_i$ of the intersection. Thus, for varieties meeting properly $\deg (X\cap Y)\leq\deg X\cdot\deg Y$.
Wirtinger's Theorem:
If $M\subset\mathbb{CP}^n$ is a compact oriented $2k$-dimensional submanifold (if it does not admit a complex structure, it is not algebraic so $\deg M$ is meant in the sense of (co)homology, cf. H. above), with the induced metric from the Hermitian metric of projective space, and $L_k$ a linear $k$-variety, then $$(2k)\text{-vol}(M)\geq |\deg M|\cdot (2k)\text{-vol}(L_k)$$ with equality if and only if $M$ is an algebraic subvariety $X$ (thus having minimal volume in its homology cycle). In particular, rescaling the metric so that the linear $k$-subspaces have unit volume, the degree of the variety is just its volume, i.e. $$\deg X = \int_{X}\imath^\ast\omega$$ where $\omega\in H^{n}(\mathbb{CP}^n, \mathbb{C})$ is a volume form for such a metric, and $\imath:X_k\hookrightarrow\mathbb{CP}^n$ is the inclusion. E.g. the area of any closed orientable 2-manifold (so for any compact Riemann surface) in this metric is just its degree as projective algebraic curve.
Degree-Arithmetic Genus Formula:
The arithmetic genus of a variety $X_k\subset\mathbb{CP}^n$ defined from its Hilbert's polynomial constant term as $p_a(X):=(-1)^{\dim X}\cdot (P_X(0)-1)$, or equivalently by the Euler characteristic of its structure sheaf as $p_a(X):=(-1)^{\dim X}\cdot (\chi (\mathcal{O}_X)-1)$, is independent of the embedding into $\mathbb{CP}^n$. For hypersurfaces $k=n-1$ is related to the degree: $$p_a(X_{n-1})={\deg X - 1\choose n},$$ in particular for plane curves $C\subset\mathbb{CP}^2$ $$p_a(C )=\frac{(\deg C -1)(\deg C -2)}{2}$$ and for $C$ a complete intersection of surfaces $S_1, S_2$ in $\mathbb{CP}^3$ $$p_a(S_1\cap S_2)=\frac{1}{2}(\deg S_1\cdot \deg S_2)(\deg S_1+\deg S_2 -4) +1.$$ Note: for nonsingular complex projective algebraic curves, thus closed orientable 2-manifolds = compact Riemann surfaces, it is a fundamental theorem that the arithmetic genus coincides with the topological genus $g$ (#doughnut holes) and the geometric genus $p_g(X ):=\dim_{\mathbb{C}} H^0(X,\bigwedge^n\Omega_{X})$. Interestingly, $p_a(C )+1$ is also the upper bound in Harnack's theorem.
Hirzebruch-Riemann-Roch Theorem:
For a locally free sheaf (i.e. vector bundle) $\mathcal{E}$ of rank $r$ on a nonsingular variety $X_k\subset\mathbb{CP}^n$, with Chern character $\operatorname{ch}(\mathcal{E})$, Todd class of its tangent sheaf $\operatorname{td} (\mathcal{T}_X)$, and [X] the rational equivalence class in the Chow ring. Then its Euler characteristic is given by $$\chi(\mathcal{E})=\deg\, (\operatorname{ch}(\mathcal{E})\cdot\operatorname{td} (\mathcal{T}_X)|_k\frown [X])=\int_X\, \operatorname{ch}(E)\wedge\operatorname{td} (TX)$$ where $(\;)|_k$ denotes the component of order $k=\dim X$ in Chow's ring $A(X)\otimes\mathbb{Q}$, and $E, TX$ are the vector bundles corresponding to the locally free sheaves, so that integration is done on the top order differential forms of the Chern character and Todd class given in terms of the curvature form of any covariant derivative/bundle connection on $X$. The product of both classes gives a finite sum of Chern classes $c_i$ powers, whose higest-order pairing $c_k(\mathcal{E})\frown [X]$ gives (when $E$ is generated by its global sections) the generic class of the $(n-r)$-dimensional zero-locus $Z(s)$ of a generic global section, $s\in H^0(X, \mathcal{E})$. Therefore the theorem gives the Euler characteristic in terms of the degree of the generic cycle obtained from the product.
Riemann-Roch Theorem for Curves:
For a divisor $D$ on an algebraic curve $C$ of genus g, Hirzebruch-Riemann-Roch along with Serre's duality theorem, reduce to: $$\chi(\mathcal{O}_C(D))=\dim H^0(\mathcal{O}_C(D))-\dim H^0(\mathcal{O}_C(K_C-D))=\deg D + 1- g,$$ which characterizes the number of independent sections of the associated line bundles from the canonical divisor $K_C$, $D$, its degree and the genus of the curve. An associated invertible sheave $\mathcal{O}_C(D)$ is the subset of rational functions on $C$ with lower bound multiplicity of zeros and poles at the components of $-D$. As fundamental corollary we obtain the degree of any canonical divisor, which is just the Gauß-Bonnet theorem for the curve seen as a real closed orientable 2-manifold $M_C$ (or compact Riemann surface): $$\deg K_C = 2g-2=-\chi_{\text{top.}}(M_C ).$$
Hurwitz Theorem for Curves:
Let $f:X\rightarrow Y$ be a finite separable regular map of curves, with degree as in F. above, and $R$ the ramification divisor of $f$, which corrects the canonical divisor pullback by $f$, $K_X\sim^{\text{lin.}}f^\ast K_Y+R$, then $$2g_X-2=\deg f\cdot (2g_Y-2)+\deg R,$$ so if $f$ has only tame ramification then $$\deg R = \sum_{P\in X}(e_P-1),$$ where $e_P$ is the ramification index at point $P\in X$.
Atiyah-Singer Theorem for Real Surfaces, or Chern-Gauß-Bonnet-Poincaré-Hopf-Lefschetz Theorems:
For a nonsingular complex projective curve = closed orientable real 2-manifold $M^g_2$= compact Riemann surface, the Atiyah-Singer theorem reduces to Riemann-Roch so its Euler-Poincaré characteristic (of singular, de Rham and Cech cohomology) is related to other fundamental invariants in differential topology by the formulas:$$ \begin{aligned} \chi (M_2^g)&=\sum_{j=0}^2 (-1)^j\dim H_{j}(M)=\sum_{j=0}^2 (-1)^j\dim H^j_{dR}(M)= \sum_{j=0}^2 (-1)^j\dim \check{H}^j(M,\underline{\mathbb{R}}) \\ &= V-E+F =2-2g= 2\deg\mathcal{G}_X=\sum_{P\in\text{Zero(X)}}\deg G_{P}=\text{Index}(X)= \\ & =\int_{M}c_1(TM^+)=\int_{M}e(TM^2_g)=\int_{M}c_1([D])=\frac{1}{2\pi} \int_{M}Kd\sigma= \\ &= \deg ([\mathcal{K}]^{-1})=-\deg \mathcal{K}=\Lambda_{\text{Id}}, \end{aligned}, $$ where $V, E, F$ is the number of vertex, edges and faces of any triangulation of $M$, $\mathcal{G}_X$ is the Gauß map over the boundary of a region in the surface surrounding all the ceros of a generic vector field, $G_P$ is the Gauß map over a neighbourhood boundary of a zero point of a generic vector field, $c_1$ is the first Chern class, $e$ is the Euler characteristic class, $[D]$ is the homology cycle of the [divisor] of zeros of a generic vector field, $K$ is the total Gaussian curvature, $\mathcal{K}$ is a canonical divisor and $[\mathcal{K}]^{-1}$ the inverse line bundle isomophism-class asociated to it, and $\Lambda_{\text{Id}}$ is the Lefschetz number of the identity map.
Some Historical Remarks Concerning Degree Theory by Siegberg seems interesting.
There is an interesting proof of the fundamental theorem of algebra using degree theory. One starts by first assuming a polynomial with no roots exist. Then given such a polynomial $f$, one constructs a vector field on the Riemann sphere by taking the pushforward of the vector field on the plane defined by $v(x)=1/x$ (and defining it to be 0 at infinity). This gives a continuous vector field if $f$ is not constant.
One can calculate the degree of the singularity at infinity to be the degree of $f$. Since the degree of a vector field is the Euler characteristic of the underlying space, the degree of $f$ must be 2. This implies the only nonconstant polynomials with no roots have to be degree 2. However, we have the quadratic formula so this case is ruled out as well.
One can prove the "hairy ball theorem" readily using degree theory.
Let $S^n$ be the $n$-sphere, $x \mapsto \vec{v}(x)$ a vector field, and $f$ be the antipodal map. We may assume that the vector field is normalized. Then we can define $\cos(t)\vec{x}+\sin(t)\vec{v}(x)$ for $t \in [0,\pi]$, which is a homotopy from $i$ to $-i$. Because the induced homomorphisms are the same on the level of homology , we then know that $1=deg(i)=deg(-i)=(-1)^{n+1}$, which is possible only if n is odd.
