= Solution
The <topological degree> measures an oriented net number of sheets of a map. Its simplest integer-valued setting is a map $f:M\to N$ between connected oriented <closed manifolds> of the same positive dimension $n$. The hypotheses matter: an integer sign count needs orientations; compactness prevents preimages from escaping; and the same dimension makes a regular fiber discrete. A <fundamental class> gives the definition for continuous maps:
$$
\boxed{f_*[M]=(\deg f)[N],\qquad \deg f\in\mathbb Z.}
$$
Here $H_n(M;\mathbb Z)$ and $H_n(N;\mathbb Z)$ are generated by their chosen <fundamental classes>. Reversing either orientation changes the sign of the <degree of a map between oriented manifolds>. Disconnected sources contribute a sum over components; disconnected targets require a degree for each target component rather than one universal integer.
For a smooth map, choose a <regular value> $y$. The <Sard theorem> guarantees plentiful choices. Each point of $f^{-1}(y)$ has an invertible derivative, so the <inverse function theorem> makes it an isolated point. The fiber is finite by compactness of $M$. An oriented coordinate chart defines $\operatorname{sgn}\det df_x$, and the <degree as a sum of local degrees> becomes
$$
\boxed{\deg f=\sum_{x\in f^{-1}(y)}\operatorname{sgn}\det df_x.}
$$
An empty fiber contributes zero. The sign is independent of the chosen oriented charts. Thus the <mapping degree> counts sheets with signs, not simply the number of preimages: orientation-preserving and orientation-reversing sheets cancel.
A differential-form argument both proves that this count is independent of $y$ and relates it to the topological definition. Around a <regular value>, take a small neighborhood $V$ whose full preimage is a disjoint union of neighborhoods mapped diffeomorphically onto $V$. Compactness rules out additional preimages approaching $V$ from elsewhere. Choose a smooth top-degree form $\eta$ supported in $V$, with $\int_N\eta=1$. The <change of variables formula> gives
$$
\int_M f^*\eta=\sum_{x\in f^{-1}(y)}\operatorname{sgn}\det df_x.
$$
Although $\eta$ need not be everywhere positive, it represents the same normalized top cohomology class as any normalized <volume form>. The <top de Rham cohomology of a compact connected oriented manifold> says that two top forms with equal integrals differ by $d\beta$. Their pullback integrals agree, because the <Generalized Stokes theorem> gives $\int_Md(f^*\beta)=0$. Therefore \b[the <degree by integration of a pullback volume form> is]
$$
\boxed{\int_Mf^*\eta=(\deg f)\int_N\eta}
$$
for every smooth top form $\eta$. Pairing <de Rham cohomology> with <fundamental classes> identifies this integer with the homological definition. This proof also shows that every <regular value> gives the same signed sum.
The <homotopy invariance of mapping degree> makes it stable under continuous deformation. Homologically this follows from homotopic maps inducing the same map on <homology>. For a smooth homotopy $F:M\times[0,1]\to N$, a direct proof uses $d\eta=0$ and the <Generalized Stokes theorem> on the cylinder to get
$$
\int_M f_1^*\eta-\int_M f_0^*\eta=\int_{M\times[0,1]}d(F^*\eta)=0,
$$
with the cylinder orientation chosen to give the indicated boundary signs. Smooth approximation extends the geometric description to continuous maps, while the <fundamental class> definition already applies without differentiability. The <multiplicativity of mapping degree> follows from functoriality and gives
$$
\deg(g\circ f)=\deg g\,\deg f,\qquad \deg\operatorname{id}=1.
$$
A constant map has degree zero when $n>0$. If the <mapping degree> is nonzero, the map is onto: a point outside its image is a <regular value> with empty preimage. An orientation-preserving <diffeomorphism> has degree $1$, and an orientation-reversing one has degree $-1$. An orientation-preserving $k$-sheeted <covering map> has degree $k$; the signs of sheets must be included for other orientation conventions.
Examples show both the strength and the limits of the invariant. On the <circle>, $e^{i\theta}\mapsto e^{ik\theta}$ has degree $k$, including negative integers and zero; it is the <winding number>. On the <sphere> $S^n$, an ambient reflection has degree $-1$, and the <antipodal map> has degree $(-1)^{n+1}$, because the determinant of $-I$ on the ambient $\mathbb R^{n+1}$ is $(-1)^{n+1}$. On the <torus>, an integer matrix $A$ induces a map of degree $\det A$, obtained by pulling back the constant top form. However, degree does not classify general manifold maps: the identity and an integer shear on the two-torus both have degree one but induce different maps on $H_1$, so are not homotopic.
The <mapping degree> gives useful obstruction and existence arguments. There is no <retraction> of a closed ball onto its boundary sphere: such a retraction would make the degree-one identity of the boundary null-homotopic, since it would extend across a contractible ball, contradicting homotopy invariance and the degree-zero constant map. This is the degree obstruction underlying the <Brouwer fixed-point theorem>. Also an everywhere nonzero tangent <vector field> on an even-dimensional sphere would, after normalization, produce the homotopy $\cos(\pi t)x+\sin(\pi t)v(x)$ from the identity to the antipodal map. Orthogonality makes every value a unit vector, but the endpoint degrees are $1$ and $-1$, impossible. This proves the corresponding <Hairy ball theorem> obstruction.
There are appropriate extensions rather than an unrestricted integer degree for every pair of manifolds. For a <proper map> between connected oriented manifolds without boundary of equal dimension, the regular-value count remains finite and is invariant; compactly supported top forms or locally finite <homology> replace the compact fundamental-class description. Without orientations, <degree modulo two> counts preimages modulo two and uses mod-two fundamental classes. For manifolds with boundary one uses <relative mapping degree> for maps of pairs, keeping the boundary away from the chosen target value. Maps of unequal dimensions, or nonproper maps where preimages can escape to infinity, do not in general admit this same signed sheet-count invariant. \b[The central principle is an integer signed count stable under homotopy, with its hypotheses explicitly preserved.]
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