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Chain maps $f,g:C_{\ast }\to C_{\ast }^{\prime }$ are chain homotopic when maps $h_n:C_n\to C_{n+1}^{\prime }$ satisfyFor a cycle $z$, $f(z)-g(z)=\partial h(z)$ is a boundary, so $f_{\ast }=g_{\ast }$ on homology.

Fix a vertex $e_0$ of ${\Delta }^n$ and define the cone operator by adjoining $e_0$ to an oriented simplex, with zero when it is already present. The simplicial boundary formula giveson the reduced chain complex. Thus its identity is null-homotopic andfor every $k$; equivalently, $H_0({\Delta }^n)\cong \mathbb{Z}$ and higher homology vanishes.

For the 2-skeleton $K$ of ${\Delta }^6$,It is connected and simply connected, so $H_0\cong \mathbb{Z}$ and $H_1=0$. Euler characteristic then givesFor $\sigma =(0123)(456)$, no vertex or edge is fixed. The only invariant 2-simplex is $[456]$, on which the 3-cycle preserves orientation, so the chain traces are $0,0,1$. The Lefschetz trace formula givestherefore

If $m$ and $n$ are coprime, every divisor of $mn$ is uniquely $ab$ with $a\mid m$ and $b\mid n$. Hence, for a multiplicative arithmetic function $f$,

The Möbius function is $\mu (n)=0$ if a prime square divides $n$, and $\mu (n)=(-1{)}^r$ if $n$ is a product of $r$ distinct primes. The Euler totient function $\varphi (n)$ counts residues modulo $n$ coprime to $n$. From the prime factorizations,Both sides of the second identity are multiplicative, and at a prime power $p^a$,Therefore

Maschke's theorem says that every invariant subspace of a finite-dimensional complex representation of a finite group has an invariant complement. Average any projection over the group to make it equivariant. Induction on dimension therefore decomposes every representation into irreducibles.

The character of a representation is ${\chi }_V(g)=\mathrm{tr}(g{|}_V)$. For a $G$-set $X$, the permutation representation $\mathbb{C}X$ has basis $X$, and its character is the number of fixed points:For the regular action,The multiplicity of an irreducible $V_i$ is the character inner productso

If $V\cong {⨁}_i{n}_i{V}_i$, Schur lemma givesFor $V=\mathbb{C}G$, this is ${\sum }_i(\dim V_i{)}^2=|G|$.

Now suppose $\chi (g)=0$ off the identity. The multiplicity of $V_i$ in $V$ isThus $\dim V/|G|$ is a nonnegative integer and $V$ is that many copies of the regular representation. Finally,so