A map \(f\colon S^n\to S^n\) with \(n\ge1\) induces multiplication by an integer \(\deg f\) on \(H_n(S^n)\cong\ZZ\). The degree depends only on the homotopy class of \(f\), is multiplicative under composition, and equals \((-1)^{n+1}\) for the antipodal map; if \(y\in S^n\) has finite preimage, \(\deg f\) is the sum of the local degrees of \(f\) at the points of \(f^{-1}(y)\). A map \(S^n\to S^n\) without fixed points is homotopic to the antipodal map, which gives Brouwer’s fixed point theorem, that every continuous map \(D^n\to D^n\) has a fixed point, and the hairy ball theorem, that \(S^n\) has a continuous nowhere-vanishing tangent vector field if and only if \(n\) is odd. The Borsuk–Ulam theorem: every continuous map \(S^n\to\RR^n\) sends some pair of antipodal points to the same point. For a retract \(X\) of a finite simplicial complex and continuous \(f\colon X\to X\), if the Lefschetz number \(\tau(f)=\sum_k(-1)^k\operatorname{tr}\big(f_*\colon H_k(X;\QQ)\to H_k(X;\QQ)\big)\) is nonzero, then \(f\) has a fixed point.
Degree of a map of spheres (Tag D-XC53X)
Let \(n\geq 1\) and let \(f\colon S^n \to S^n\) be a continuous map. Since \(H^n(S^n;\ZZ) \cong \ZZ\), the induced homomorphism \(f^*\colon H^n(S^n;\ZZ) \to H^n(S^n;\ZZ)\) is multiplication by a unique integer \(d\), that is, \(f^*(u) = d\,u\) for every \(u\in H^n(S^n;\ZZ)\). The degree of \(f\) is \(\deg f \coloneqq d\).
Let \(n\geq 1\) and \(f, g\colon S^n\to S^n\) be continuous.
If \(f\) and \(g\) are homotopic, then \(\deg f = \deg g\).
\(\deg f\) equals the integer \(d'\) with \(f_*(v) = d'\,v\) for every \(v\in H_n(S^n;\ZZ)\).
[@Hat02].
Local degree (Tag D-ZVY6X)
Let \(n\geq 1\), let \(f\colon S^n\to S^n\) be a continuous map, and let \(y\in S^n\) be a point whose preimage \(\inverseof{f}(y) = \ts{x_1, \ldots, x_m}\) is finite. Homology is taken with coefficients in \(\ZZ\). Choose an open neighborhood \(V\) of \(y\) and disjoint open neighborhoods \(U_i\) of \(x_i\) with \(f(U_i)\subseteq V\), and fix a generator of \(H_n(S^n)\). Then \(f\) restricts to maps of pairs \(f_i\colon (U_i, U_i\sm\ts{x_i})\to (V, V\sm\ts{y})\), and under the isomorphisms \[ H_n(U_i, U_i\sm\ts{x_i})\cong H_n(S^n, S^n\sm\ts{x_i})\cong H_n(S^n)\cong\ZZ, \qquad H_n(V, V\sm\ts{y})\cong H_n(S^n, S^n\sm\ts{y})\cong H_n(S^n)\cong\ZZ \] given by excision and the long exact sequence of the pair, the induced homomorphism \(f_{i*}\) is multiplication by an integer. This integer is the local degree \(\deg f\vert_{x_i}\) of \(f\) at \(x_i\).
Let \(n\geq 1\), let \(f\colon S^n\to S^n\) be a continuous map, and let \(y\in S^n\) be a point with finite preimage \(\inverseof{f}(y) = \ts{x_1, \ldots, x_m}\). Then \(\deg f = \sum_{i=1}^m \deg f\vert_{x_i}\), where \(\deg f\) is the degree of \(f\).
[@Hat02].
Lefschetz number (Tag D-3UY5O)
Let \(X\) be a finite simplicial complex, or a retract of one, and let \(f\colon X\to X\) be a continuous map. The Lefschetz number of \(f\) is \[ \tau(f) \coloneqq \sum_{n\geq 0} (-1)^n \tr\qty{ f_*\colon H_n(X;\QQ) \to H_n(X;\QQ) } .\]
Since \(\tr\qty{\id_{H_n(X;\QQ)}} = \dim_\QQ H_n(X;\QQ)\), the identity has \(\tau(\id_X) = \chi(X)\), the Euler characteristic. By the Lefschetz fixed point theorem, if \(\tau(f)\neq 0\), then \(f\) has a fixed point.
[@Hat02].
Lefschetz fixed point theorem (Tag T-BX4LD)
Let \(X\) be a finite simplicial complex, or a retract of one, and let \(f\colon X\to X\) be continuous with Lefschetz number \[ \tau(f) = \sum_{k \geq 0} (-1)^k \tr\qty{f_*\colon H_k(X; \QQ) \to H_k(X; \QQ)} .\] If \(\tau(f) \neq 0\), then \(f\) has a fixed point [@Hat02].
Brouwer fixed-point theorem (Tag T-S2OLJ)
Every continuous map \(f\colon D^n \to D^n\) has a fixed point [@Hat02].
Hairy ball theorem (Tag T-VQTR6)
For \(n\geq 1\), the sphere \(S^n\) has a continuous field of nonzero tangent vectors if and only if \(n\) is odd. In particular, every continuous tangent vector field on \(S^{2k}\) vanishes somewhere [@Hat02].
Borsuk--Ulam theorem (Tag T-WNOWY)
For every continuous map \(f\colon S^n \to \RR^n\) there is a point \(x\in S^n\) with \(f(x) = f(-x)\) [@Hat02].