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For a continuous self-map $f$ of a finite CW complex, its Lefschetz number over $\mathbb{Q}$ isThe Lefschetz fixed-point theorem states that $L(f)\ne 0$ implies that $f$ has a fixed point. Consequently every fixed-point-free self-map has Lefschetz number zero.

The space in the question is a cyclic point-gluing of two-spheres. Before gluing, the $n$ spheres have total Euler characteristic $2n$. Each of the $n$ point identifications lowers the Euler characteristic by one, soEquivalently, $X\simeq S^1\vee {\bigvee }_{j=1}^n{S}^2$, with rational homology dimensions $1,1,n$ in degrees zero, one, and two.

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

The Lefschetz fixed-point theorem says that a continuous self-map $f$ of a compact triangulable space has a fixed point whenever its Lefschetz numberis nonzero. In the smooth nondegenerate case, the Lefschetz-Hopf fixed-point theorem expresses this number as the sum of the local fixed-point indices.

Identify $H_1({\mathbb{R}}^2/{\mathbb{Z}}^2;\mathbb{Z})$ with ${\mathbb{Z}}^2$ by sending $v\in {\mathbb{Z}}^2$ to the class of the loop $t\mapsto tv\mathrm{mod}{\mathbb{Z}}^2$. The image of this loop under $f$ lifts from $0$ to the path $t\mapsto \tilde{f}(tv)$, whose endpoint is $\tilde{f}(v)=A(v)$. Its homology class is therefore $A(v)$. Thus $A=f_{\ast }$ under this identification, and in particular $A$ is a homomorphism represented by an integer matrix.

The expansion inequality implies, after taking a derivative, thatfor every tangent vector. The supplied linear-algebra fact shows that every eigenvalue of $D{\tilde{f}}_x$ has modulus at least $\mu >1$, so $1$ is not an eigenvalue. Every fixed point of $f$ is consequently nondegenerate and contributes local index $+1$ or $-1$.

On the torus, the induced maps on $H_0,H_1,H_2$ have traces $1,\mathrm{tr}A,\det A$. HenceThe original expansion inequality applied to lattice vectors gives $|Av|\ge \mu |v|$ for $v\in {\mathbb{Z}}^2$, and homogeneity and density extend it to all $v\in {\mathbb{R}}^2$. If ${\lambda }_1,{\lambda }_2$ are the possibly complex eigenvalues of $A$, then $|{\lambda }_j|\ge \mu$, soSince every fixed point contributes an index of absolute value one, the Lefschetz-Hopf formula gives