= Solution
The <degree of a map between oriented manifolds> measures how many times the domain covers the target, with signs recording the local <local orientation of a manifold>. Let $M,N$ be connected, oriented <closed manifolds> of the same dimension $n>0$. A continuous map $f:M\to N$ acts on top-dimensional <homology> by
$$
\boxed{f_*[M]=\deg(f)[N],\qquad\deg(f)\in\mathbb Z,}
$$
where $[M],[N]$ are their <fundamental classes>. Connectedness and the choices of <local orientation of a manifold> identify $H_n(N;\mathbb Z)$ with $\mathbb Z$. Reversing the orientation of either manifold changes the sign; reversing both does not.
For a smooth map, <Sard theorem> supplies a <regular value> $y$. Its inverse image is discrete and, by compactness, finite. At each $x\in f^{-1}(y)$ the differential is an isomorphism; let its sign be $+1$ or $-1$ according to whether it preserves or reverses the chosen local orientations. The <degree as a sum of local degrees> is
$$
\boxed{\deg(f)=\sum_{x\in f^{-1}(y)}\operatorname{sgn}\det(df_x).}
$$
The sign is computed in oriented <manifold charts>. The value is independent of the chosen <regular value>, even when inverse images appear or disappear: the signed count is the coefficient of $[N]$ in $f_*[M]$.
There is a useful local-density expression for the same <topological degree>. If $\omega$ is a <volume form> with $\int_N\omega=1$, then
$$
\boxed{\deg(f)=\int_M f^*\omega.}
$$
This <degree by integration of a pullback volume form> follows first by choosing a smooth top-form supported in a small neighborhood of a <regular value>, where the inverse branches contribute their orientation signs. Any other normalized top-form differs from it by an exact form: integration identifies $H^n_{\mathrm{dR}}(N)$ with $\mathbb R$. The integral of its pullback difference vanishes by <Stokes theorem>. In particular, for any top-form $\eta$, $\int_M f^*\eta=\deg(f)\int_N\eta$.
A <homotopy> $H:M\times[0,1]\to N$ preserves this integral, since $d\omega=0$ and <Stokes theorem> gives
$$
\int_M f_1^*\omega-\int_M f_0^*\omega=\int_{M\times[0,1]}d(H^*\omega)=0.
$$
Thus <topological degree> is a <homotopy> invariant. It is multiplicative under composition, because the induced maps on <homology> compose: $\deg(g\circ f)=\deg(g)\deg(f)$. The identity has degree one, a constant map has degree zero for $n>0$, and an orientation-reversing <diffeomorphism> has degree minus one. An orientation-preserving finite <covering map> has degree equal to its number of sheets. Nonzero <topological degree> forces surjectivity, since an omitted point would be a <regular value> with an empty inverse image.
For the <circle>, $e^{i\theta}\mapsto e^{ik\theta}$ has degree $k$, positive or negative. This is its <winding number>, computable as $(2\pi i)^{-1}\int f^{-1}df$. The antipodal map of $S^n$ has degree $(-1)^{n+1}$: its extension $-I$ on the ambient $(n+1)$-dimensional vector space has that determinant sign and respects the outward-normal convention. A holomorphic map $z\mapsto z^k$, $k\geq1$, on the <Riemann sphere> has degree $k$, whereas its complex conjugate has degree $-k$. These examples show how orientation, rather than simply the number of inverse images, determines the integer.
For maps $S^n\to S^n$, <topological degree> gives the complete <homotopy> classification $\pi_n(S^n)\cong\mathbb Z$. The <degree does not classify general manifold maps>: the identity of the <torus> and the map induced by the integer matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ both have degree one, but have different induced maps on $H_1(T^2;\mathbb Z)$ and so are not homotopic. A nonzero-degree map $S^n\to S^n$ cannot extend continuously to $B^{n+1}$, because such an extension would make the boundary map null-homotopic. In the smooth setting, <Stokes theorem> gives the same obstruction by applying it to the pulled-back normalized <volume form>.
The hypotheses can be adjusted, but must be stated. For connected oriented noncompact manifolds, a <proper map> has a degree defined using compactly supported top-forms, and it is invariant under proper <homotopies>. For <manifolds with boundary> one uses relative <fundamental classes> and maps of pairs, or fixes appropriate boundary conditions. Without an integral orientation one can still count inverse images modulo two, obtaining a mod-two degree. The integer integral formula used below assumes the oriented setting.
In <classical field theory>, these ideas turn continuous fields into quantized <topological charges>. Suppose a field on $\mathbb R^d$ approaches one fixed target value at spatial infinity. The <one-point compactification> makes it a map $\phi:S^d\to\mathcal V$. When the target $\mathcal V$ is an oriented closed $d$-manifold, its <topological degree> labels <topological sectors>. More generally the sectors are described by <homotopy groups>; an integer degree is available only when the domain and target have the appropriate dimensions and orientations. Smooth time evolution preserving the boundary condition is a <homotopy>, so it cannot change the integer. A change requires a singular field, escape from the allowed target, or a change at the boundary.
A normalized closed target $d$-form gives the <pullback-volume representation of a topological current>. On spacetime, put $\alpha=\phi^*\omega$. Since $d\alpha=\phi^*(d\omega)=0$, its dual current is identically conserved, and
$$
Q=\int_{\text{space}}\alpha=\deg(\phi)
$$
is independent of time when there is no flux at infinity. This conservation law follows from geometry without using the field equations; it need not arise from a continuous symmetry through <Noether theorem>.
A concrete example is the <O3 nonlinear sigma model> in two spatial dimensions. Its unit-vector field $\mathbf n$ approaches a constant at infinity, defining $S^2\to S^2$. The normalized area form of the target gives the <degree charge of an O3 sigma-model lump>:
$$
\boxed{Q=\frac1{4\pi}\int_{\mathbb R^2}\mathbf n\cdot(\partial_1\mathbf n\times\partial_2\mathbf n)\,dx^1dx^2\in\mathbb Z.}
$$
For the energy normalization $E=\tfrac12\int(|\partial_1\mathbf n|^2+|\partial_2\mathbf n|^2)$, the identities $\mathbf n\cdot\partial_i\mathbf n=0$ give
$$
E=\frac12\int|\partial_1\mathbf n\pm\mathbf n\times\partial_2\mathbf n|^2\,d^2x\ \pm4\pi Q,\qquad\boxed{E\geq4\pi|Q|.}
$$
This is the <Bogomolny degree bound for the O3 sigma model>. Choosing the sign appropriate to $Q$ makes the square nonnegative; vanishing of the square gives first-order <Bogomolny equations> and a <sigma-model lump> saturating the bound. With the oriented <stereographic projection>
$$
\mathbf n=\frac{(2\operatorname{Re}w,2\operatorname{Im}w,1-|w|^2)}{1+|w|^2},\qquad z=x^1+ix^2,
$$
the maps $w=z^k$ have $Q=k$ and $E=4\pi k$. Their conjugates have $Q=-k$ with the same energy. Holomorphic rational maps have positive degree equal to their degree as rational maps; taking a reciprocal does not reverse the orientation. Antiholomorphic dependence reverses it.
The <Skyrme model> supplies a three-dimensional example. A field $U:\mathbb R^3\to\mathrm{SU}(2)$ with $U\to I$ at infinity is a map $S^3\to\mathrm{SU}(2)\cong S^3$. Take $T_i=-i\tau_i$ and $U^{-1}dU=\theta^iT_i$, with $\theta^1\wedge\theta^2\wedge\theta^3$ positive. Since $\operatorname{tr}(T_iT_jT_k)=-2\epsilon_{ijk}$, the normalized target <volume form> is
$$
\omega_3=-\frac1{24\pi^2}\operatorname{tr}(U^{-1}dU)^3=\frac1{2\pi^2}\theta^1\wedge\theta^2\wedge\theta^3.
$$
The integral is one on the unit <three-sphere>. Consequently the <Skyrme baryon number as a mapping degree> is
$$
\boxed{B=-\frac1{24\pi^2}\int_{\mathbb R^3}\operatorname{tr}(U^{-1}dU)^3=\deg(U).}
$$
This is the <topological baryon number in the Skyrme model>; the sign has been fixed by the stated orientation and anti-Hermitian generator convention.
A <topological charge> alone does not guarantee a stable finite-size solution. The <degree and energetic stability of a field configuration> concern different properties. For a three-dimensional configuration of size $R$, the two-derivative energy scales as $R$, so it can decrease by shrinking while the <topological degree> remains fixed for every $R>0$. The limit can be singular. The <Skyrme term>, with four derivatives, scales as $R^{-1}$ and can balance the shrinking tendency. This is the role of <Derrick theorem> in distinguishing topological obstruction from energetic stability.
For defects, the relevant boundary map can instead be the sphere surrounding a core. A <vacuum manifold> equal to $S^1$ gives the integer <winding number> of a <vortex>; a vacuum manifold $S^2$ gives the degree of a surrounding $S^2$ for a <magnetic monopole>. This <vacuum-boundary degree as a defect charge> obstructs extending the normalized vacuum field through the enclosed ball. A nonzero integer therefore forces the field to leave the <vacuum manifold> somewhere in the core. This construction does not require the field to take one constant value in every direction at infinity.
Degree also appears in four-dimensional gauge theory through a boundary transition function. For an anti-Hermitian <SU(2)> gauge connection on $\mathbb R^4$, write $F=dA+A\wedge A$ and assume finite-action boundary behavior $A\to g^{-1}dg$ on the large bounding <three-sphere>. In the second-Chern convention
$$
k=\frac1{8\pi^2}\int_{\mathbb R^4}\operatorname{tr}(F\wedge F),
$$
the identity $d\operatorname{tr}(A\wedge dA+\tfrac23A^3)=\operatorname{tr}(F\wedge F)$ and the <Maurer-Cartan equation> give
$$
\boxed{k=-\frac1{24\pi^2}\int_{S^3}\operatorname{tr}(g^{-1}dg)^3=\deg(g).}
$$
This <boundary winding representation of Yang-Mills topological charge> relates the <Second Chern number> to the degree of $g:S^3\to\mathrm{SU}(2)$. The <Chern-Simons 3-form> turns the bulk integral into the boundary winding integral. Conventions which define the instanton number with the opposite trace sign reverse $k$; the integer quantization is unchanged. A <Yang-Mills theta term> weights a sector by $e^{i\vartheta k}$, giving periodicity $\vartheta\mapsto\vartheta+2\pi$. Thus the same <topological degree> that counts oriented inverse images also labels field sectors and expresses their quantized charges as integrals of local densities.
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