= Solution
The <topological degree> assigns an integer to a continuous map $f:M\to N$ between closed, connected, oriented manifolds of the same dimension $d$. Their top <homology groups> are infinite cyclic, generated by their <fundamental classes>. The induced map defines
$$
\boxed{f_*[M]=(\deg f)[N],\qquad \deg f\in\mathbb Z.}
$$
For a smooth map and a <regular value> $y$, its equivalent local description is
$$
\deg f=\sum_{x\in f^{-1}(y)}\operatorname{sgn}\det(df_x),
$$
where the sign records preservation or reversal of orientation. The preimage is a finite set because the manifolds are compact and $y$ is regular. This is a signed count; opposite-orientation preimages cancel. If $\omega$ is a normalized target <volume form> with $\int_N\omega=1$, the <pullback of a differential form> gives the equivalent integral $\deg f=\int_Mf^*\omega$.
The <homotopy invariance of mapping degree> follows because homotopic maps induce the same homology map. In the smooth description, apply <Stokes theorem> to a homotopy $F:M\times[0,1]\to N$:
$$
\int_Mf_1^*\omega-\int_Mf_0^*\omega
=\int_{M\times[0,1]}d(F^*\omega)=0,
$$
since a top-dimensional form on $N$ is closed. Thus degree cannot change during a smooth deformation preserving the relevant boundary conditions. Degrees multiply under composition, and a nonzero-degree map is surjective: a missed target point would give a regular value with an empty preimage and hence degree zero.
For maps $S^d\to S^d$, the <homotopy group> $\pi_d(S^d)=\mathbb Z$ is completely detected by degree. This supplies integer <topological sectors> for many <classical field theory solitons>. It is important to distinguish two ways spheres arise. In a <nonlinear sigma model> with the field approaching one fixed target value at infinity, compactify all of $\mathbb R^d$ to $S^d$ and classify the resulting map into the target. For a defect whose vacuum field varies with direction, retain the sphere $S^{d-1}_\infty$ surrounding the core and map it into the <vacuum manifold>. This is the <vacuum-boundary degree as a defect charge>. An asymptotic limit is a boundary hypothesis; finite energy alone in every possible model need not imply one.
For an <Abelian Higgs vortex>, the phase at infinity defines $S^1\to U(1)\cong S^1$. Its degree is the <winding number>
$$
n=\frac1{2\pi}\oint d\chi
=\frac e{2\pi}\int B\,d^2x.
$$
Nonzero $n$ forces a zero of the Higgs field inside: a nowhere-vanishing field would extend its normalized phase through the disk and make the boundary loop contractible. This is also the flux classification of the <Bogomolny vortex equations>.
For a <'t Hooft-Polyakov monopole>, the normalized adjoint <Higgs field> $\hat\Phi$ gives $S^2_\infty\to SU(2)/U(1)\cong S^2$. With outward orientation its degree is
$$
n=\frac1{8\pi}\int_{S^2_\infty}
\epsilon_{abc}\hat\Phi^a\,d\hat\Phi^b\wedge d\hat\Phi^c.
$$
The integrand is the normalized target area form, so the <hedgehog ansatz for a monopole> has degree one. Nonzero degree obstructs extending the normalized Higgs direction through the ball; the full Higgs field must leave the vacuum manifold, normally vanishing in its core. The asymptotic unbroken Abelian <magnetic flux> measures the same integer with a convention-dependent sign.
For the <Skyrme model>, a field $U:\mathbb R^3\to SU(2)$ with $U\to1$ at infinity gives $S^3\to S^3$, because <SU(2) as the three-sphere> identifies the target. A choice of orientations gives its <topological baryon number in the Skyrme model> as the winding integral $\pm(24\pi^2)^{-1}\int\operatorname{tr}(U^{-1}dU)^3$; specifying the orientation fixes the sign. Similarly, the transition map at infinity of a four-dimensional <Yang-Mills instanton> has degree in $\pi_3(SU(2))$, measured by its Chern charge.
A <scalar-field kink> illustrates the discrete-vacuum case: its endpoint data lie in $\pi_0(\{\pm1\})$, rather than being a map between connected positive-dimensional spheres. For the double-well field its <topological charge> is $Q=[\phi(+\infty)-\phi(-\infty)]/2$. A kink has $Q=1$, an antikink $Q=-1$; they cannot individually deform to vacuum while their endpoints are fixed.
Degree classifies topology, rather than every solution. Smooth motion or local perturbations with fixed boundary data preserve it, and a nonzero degree obstructs decay to the vacuum within that class. It does not guarantee existence of a smooth energy minimizer, a fixed size, uniqueness, or dynamical stability. The <degree and energetic stability of a field configuration> are distinct: a field can shrink towards a singular limit at fixed degree, as <Derrick's theorem> diagnoses in many two-derivative theories. Moreover degree need not classify all homotopy classes for an arbitrary target; other <homotopy groups> or invariants may be needed. Within one degree there can be continuous <moduli spaces> and several stationary solutions.
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