Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 18 / 5

For closed connected oriented $d$-manifolds, the degree of a map between oriented manifolds is the integer defined by $f_{\ast }[M]=(\mathrm{deg}f)[N]$ in $H_d(N;\mathbb{Z})$. If $y\in N$ has finite fiber $\{x_1,\dots ,x_r\}$, choose disjoint small neighbourhoods $U_i$ containing no other points of that fiber. The map of local relative homology groupssends the local orientation generator at $x_i$ to $d_i$ times the one at $y$. This integer is its local degree of a continuous map, independent of the small neighbourhood by excision. Excision also identifies $H_d(M,M\setminus f^{-1}(y))$ with the direct sum of the local groups, and the global fundamental class restricts to their orientation generators. Naturality of the maps to relative homology therefore proves the degree as a sum of local degrees:The empty fiber gives degree zero. For smooth maps at a regular value, a local degree is the sign of the derivative determinant, but general continuous maps can have larger local multiplicities.

Using complex coordinates near zero, $F_k(z)=z^k$ for $k>0$ induces winding number $k$ on a small circle, so its local relative homology multiplier is $k$. For $k<0$, $F_k(z)={\overline{z}}^{|k|}$ has local degree $k$, since conjugation reverses the winding. For zero, take $F_0(z)=|z{|}^2\in \mathbb{C}$: its zero fiber is the single point zero, while the image of a small circle is a single positive real point, so its local degree is zero. Equivalently a nearby nonreal target has no preimages.

Each of these maps is proper and hence extends continuously over the one-point compactifications to a map $S^2\to S^2$ with zero as an isolated preimage of zero. ThusThese are local degrees of arbitrary integer value. Taking $F_k$ times the identity on ${\mathbb{R}}^{d-2}$ gives the same local degrees for every $d\ge 2$. In dimension one, an isolated local degree is restricted to $-1,0,1$, so the all-integers assertion is realized in dimensions at least two.

Identify $T^2$ with the product of two unit circles and use the unit sphere $S^2\subset {\mathbb{R}}^3$. The surjective degree-zero sphere-to-torus map isGiven any torus point, choose its two angle representatives with $x,y\in [-1/2,1/2]$. Then $x^2+y^2\le 1/2$, so $z=\sqrt{1-x^2-y^2}$ completes a preimage on the sphere. Thus the map is surjective. The explicit homotopy $F_t(x,y,z)=(e^{2\pi itx},e^{2\pi ity})$, $0\le t\le 1$, contracts it to a constant map. By homotopy invariance of mapping degree, $\boxed{\mathrm{deg}F=0}$, despite surjectivity.

Choose an oriented embedded closed $d$-disc $D$ in $M$ and collapse $M\setminus \mathrm{int}D$ to a point. The quotient is $D/\partial D\cong S^d$, with orientation chosen to agree on the disc interior. The resulting collapse map of degree one onto a sphere isIndeed, any target point other than the collapsed point has exactly one preimage, where the map is an orientation-preserving homeomorphism and has local degree $+1$. The degree as a sum of local degrees therefore gives global degree one. This is the usual connected, positive-dimensional setting for mapping degree; extra components, if allowed, may all be collapsed.

For $k>0$, the surface presentationhas an epimorphism to $\mathbb{Z}/k$ sending $a_1$ to $1$ and the other generators to zero. Its kernel has index $k$. By the classification of connected covering spaces, it determines a connected $k$-sheeted cover $p:\tilde{\Sigma }\to {\Sigma }_2$. Pull back the orientation, so each local sheet has degree $+1$ and $\mathrm{deg}p=k$. Counting lifts of cells gives $\chi (\tilde{\Sigma })=k\chi ({\Sigma }_2)=-2k$, hence $2-2g=-2k$ and $g=k+1$.

For $k<0$, use the $|k|$-sheeted cover and compose with an orientation-reversing diffeomorphism of ${\Sigma }_2$, for instance reflection in a symmetric embedded genus-two surface. By multiplicativity of mapping degree, the degree becomes $-|k|=k$. For $k=0$ take a constant map from the torus. Thus the arbitrary-degree maps to a surface of genus two can be chosen with