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This theory compares the area, boundary length and Euler characteristic of a surface mapped holomorphically to another surface, using the pulled-back metric. It extends covering-degree and Riemann-Hurwitz formula estimates when some boundary maps into the interior of the target. The Ahlfors second fundamental theorem and Ahlfors five islands theorem are important consequences.

For a finite simply connected covering surface over the Riemann sphere and $q\ge 3$ fixed separated Jordan domains, let $n_j$ count its simply connected islands over the $j$th domain once each, regardless of mapping degree. Let $S$ be its average sheet number and $L$ its spherical boundary length. The displayed inequality holds with $h$ depending only on the target domains. For regular bordered surfaces $f:X\to Y$, a general formulation is $\max \{-\chi (X),0\}\ge -S\chi (Y)-hL$, where $L$ counts only the boundary mapped into the interior of $Y$ and $\chi$ is the modern Euler characteristic. The boundary error disappears for an actual branched covering. The covering-surface formulation is explained in www.math.purdue.edu/~eremenko/dvi/ahl.pdf ; the simply connected island estimate is also recorded in Lemma 2 of pdfs.semanticscholar.org/d247/562607e237af7c8fc6e683b77be6b94829fa.pdf .

For a finite group generated by $a,b$, take the regular covering of a thrice-punctured sphere with monodromy $a,b,(ab{)}^{-1}$ and fill each lifted puncture by a disk. Locally the projection is $z\mapsto z^m$, where $m$ is the order of its monodromy. The result is a closed oriented topological surface with an orientation-preserving group action. The nontrivial stabiliser subgroups are the conjugates of $⟨a⟩$, $⟨b⟩$ and $⟨ab⟩$ at the three types of ramification points. Counting lifted cells gives the displayed Euler characteristic. A subgroup acts freely exactly when it meets every conjugate of these three cyclic groups trivially. Its quotient is then a regular covering of degree equal to the subgroup order, and quotient Euler characteristic is divided by that order.

Let $X$ be a finite CW complex, with $c_i$ topological cells of dimension $i$ and none above $d$. Its Euler characteristic is the displayed alternating sum. For a finite simplicial complex, $c_i$ counts its $i$-dimensional simplexes, each counted once rather than once for each containing simplex. In dimension two this becomes $V-E+F$. A topological cell's interior is counted; identified parts of its boundary are counted in their own lower-dimensional topological cells.^[ref]

The cellular homology theorem identifies cellular homology with singular homology, so the count equals the alternating rational Betti numbers. These are the free ranks of integral homology; torsion subgroups do not contribute to them. Over any field, the alternating homology dimensions still give the same Euler characteristic, although individual Betti numbers can depend on the field.

A connected compact surface is determined by its orientability, number $b$ of boundary circles and genus. An orientable surface of genus $g$ has Euler characteristic $2-2g-b$; a surface which is not orientable with $r$ crosscaps has Euler characteristic $2-r-b$. In particular, a closed connected surface with vanishing first mod-two homology is a sphere, and a connected closed orientable surface of Euler characteristic zero is a torus.

Every closed combinatorial surface has geometric realization homeomorphic to exactly one member of the following list: the sphere $S^2$; the connected sum of surfaces of $g\ge 1$ tori; or the connected sum of surfaces of $k\ge 1$ real projective planes. Every member admits a finite triangulation. The orientable family has Euler characteristic $2-2g$, including $g=0$ for the sphere; the nonorientable surface family has Euler characteristic $2-k$. Thus orientability and Euler characteristic determine the homeomorphism type. The first two nonorientable cases are the real projective plane and the Klein bottle. For a finite disconnected simplicial complex satisfying the same local conditions, the unordered list of its connected components' types determines its type.

A compact two-dimensional manifold without boundary. A connected oriented surface is classified by its genus; for genus $g$ its Euler characteristic is $2-2g$.

There are exactly four types: the two-sphere, real projective plane, torus, and Klein bottle. Four sides admit three pairings and four direction choices for each pairing. Up to cyclic relabelling, labels are adjacent, $a{a}^{s}b{b}^t$, or alternating, $ab{a}^s{b}^t$. For adjacent labels, the vertex counts for $(s,t)=(-1,-1)$, a mixed-sign pair, and $(1,1)$ are $3,2,1$. For alternating labels they are $1,1,2$. There are two edge classes and one face, so Euler characteristic is $V-1$. Orientability occurs exactly when $s=t=-1$. The classification theorem for surfaces identifies every case, leaving no additional closed surfaces from a square. Examples are $a{a}^{-1}b{b}^{-1}$ for the sphere, $aab{b}^{-1}$ for the projective plane, $ab{a}^{-1}{b}^{-1}$ for the torus and $ab{a}^{-1}b$ for the Klein bottle.

Take a hyperbolic torus with a single cone point of angle $2\pi /7$. Its orbifold fundamental group is $⟨a,d\mid [d,a{]}^7=1⟩$. An epimorphism to $\mathrm{PSL}(3,2)$ whose commutator has exact order $7$ has torsion-free kernel. The point and line stabilizers have index $7$ and order $24$, hence avoid all nontrivial cone stabilizers. Their quotient covers are smooth surfaces with Euler characteristic $7(-6/7)=-6$, so have genus $4$. Their equal coset permutation characters give equal heat traces by the Sunada theorem. Nonisometry still requires a geometric distinction, such as the intersection test for nonisometric finite covers.

For two connected closed surfaces, remove an open disc from each and identify the resulting boundary circles by a homeomorphism. The resulting closed surface is their connected sum, defined up to homeomorphism independently of these choices. A triangulation version removes one triangular face from each and glues their boundary three-cycles by a simplicial homeomorphism.

At a seam vertex the two vertex links of a simplex are paths; gluing their matching endpoints produces a cycle graph, proving the local surface condition. Removing a triangular face subtracts one from each Euler characteristic, while boundary gluing reduces both the vertex count and the edge count by three. Therefore $\chi (M\#N)=\chi (M)+\chi (N)-2$. If the result were orientable, either punctured summand would inherit consistent choices of orientation of a simplex, and these would extend over its capping face. Thus a nonorientable summand forces a nonorientable surface result. The classification theorem for closed combinatorial surfaces then gives $T^2\#{\mathbb{RP}}^2\cong {\mathbb{RP}}^2\#{\mathbb{RP}}^2\#{\mathbb{RP}}^2$.

For a polygonal schema with all its sides paired and two same-direction occurrences of an edge label $x$, the words $xAxB$, $xx{A}^{-1}B$, and $A^{-1}xxB$ describe homeomorphic quotient spaces. Here inversion of a string reverses its order and inverts each letter. Cut from the start of the first $x$ side to the start of the second, using a diagonal $d$. The pieces have boundary words $xA{d}^{-1}$ and $dxB$. Gluing the old $x$ sides gives outside word $A{d}^{-1}{d}^{-1}{B}^{-1}$; reversing the whole boundary and choosing a new starting point gives $dd{A}^{-1}B$. Alternatively cut from the end of the first $x$ side to the end of the second. The pieces are $xdB$ and $Ax{d}^{-1}$, and gluing their old $x$ sides gives $dB{A}^{-1}d$, cyclically $A^{-1}ddB$. Renaming the new paired side gives both asserted forms. These are actual cutting and gluing homeomorphisms, so they preserve all the other side identifications as well as the Euler characteristic.

For a closed connected nonorientable surface, the crosscap number is the number of real projective plane summands in its connected sum classification. It is at least one, and its Euler characteristic is $2-h$. The projective plane has one crosscap and the Klein bottle has two.

A constant dilaton contributes ${\Phi }_0\chi (\Sigma )$ to the Euclidean worldsheet action by the Gauss-Bonnet theorem. With string coupling $g_s=e^{{\Phi }_0}$ and Euler characteristic $\chi =2-2g$, the path-integral weight is $g_s^{2g-2}$. Canonically normalized external closed-string vertices add one power of $g_s$ each.

The Dirac-Born-Infeld action describes the leading derivative dynamics of one D-brane, including nonlinear dependence on slowly varying worldvolume field strengths. Its induced tensors are D-brane worldvolume pullbacks. The combination $P[B]+2\pi {\alpha }^{\prime }F$ is invariant under the combined two-form and boundary gauge transformation. The dilaton factor reflects a disk's Euler characteristic. Expanding about flat space with small field strengths and embedding gradients gives the worldvolume gauge and scalar kinetic terms. A coincident stack has a non-Abelian low-energy Yang-Mills action; this does not imply that a naive traced determinant is exact at every order.

Suppose a nonconstant holomorphic map $f:X\to {\mathbb{P}}^1$ from a connected compact Riemann surface has degree of a holomorphic map two and exactly four distinct branch values of a holomorphic map. The valency theorem forces each of their fibres to contain one ramification point with $e_p=2$, a simple ramification point of a holomorphic map. HenceThis determines the genus without recognizing a torus in a drawing.

A concrete realization is the compact Riemann surface obtained by adding the two points at infinity to the even-degree hyperelliptic model given by the algebraic curveProjection to $z$ has degree of a holomorphic map two. At a simple zero $a$ of $P$, $w$ is a local coordinate and $z-a=w^2/P^{\prime }(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm 1$, each unramified since $u$ is a local coordinate. The four finite roots are therefore all the branch values of a holomorphic map.^[ref]

Choose a triangulation of a surface on the Riemann sphere with these four branch values of a holomorphic map as vertices, combinatorially the boundary of a tetrahedron. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving Euler characteristic $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The Triangulation proof of the Riemann-Hurwitz formula applies the same counting idea after refining a general triangulation of a surface around the branch values of a holomorphic map.^[ref]

Take eight congruent hexagonal tiles with three pairwise nonadjacent distinguished faces. On tile indices modulo eight let $\sigma (j)=1-j$, $t(j)=-j$, and use $u_1(j)=3j$ for one assembly and $u_2(j)=3j+4$ for the other. Pair faces by reflection and leave fixed faces, and the three remaining faces, with Neumann boundary conditions. If $R{e}_j=e_{j+1}$, the displayed matrix intertwines all three face actions: its circulant coefficients satisfy $c_r=c_{-r}=c_{3r+4}$. Its eigenvalues are $1+(-1{)}^k+4\cos (\pi k/4)$, which are nonzero for $0\le k\le 7$. The transplantation theorem gives equal spectra. The second gluing graph is bipartite, with parts $\{0,5,6,7\}$ and $\{1,2,3,4\}$, so it is orientable. The first has the odd cycle $1,3,6,2,7,1$ and is nonorientable. Both graphs are connected with eight vertices and ten edges, and their bordered surfaces retract up to homotopy to the graphs. Both have Euler characteristic $-2$ and fundamental groups free of rank three. Neither is simply connected.

The Euler characteristic compresses a finite CW complex's homology into one integer: count its topological cells with alternating signs, or count its Betti numbers with the same signs. A filled triangle has Euler characteristic one; its cycle graph boundary has Euler characteristic zero. The missing face changes the answer even though the vertices and edges are identical.

For a finite CW complex $X$, including the geometric realization of a finite simplicial complex,Here $b_i={\dim }_{\mathbb{Q}}{H}_i(X;\mathbb{Q})$ is the rational Betti number. More generally this ordinary homology definition makes sense whenever every rational homology group is finite-dimensional and only finitely many are nonzero. An arbitrary infinite CW complex need not satisfy these assumptions; its formal infinite alternating cell count is not a definition of this invariant.

The usefulness is that a finite description can reveal something about a space without computing every homology group. By the cell-count formula for Euler characteristic, count pieces rather than solve all the boundary operators. If two spaces give different answers, topological invariance of Euler characteristic rules out a homeomorphism, and even a homotopy equivalence. Equal answers are less decisive: the Euler characteristic of an annulus and the Euler characteristic of a torus are both zero, while their homology groups differ. This is a compact diagnostic, with a clearly defined limit on what it can tell us.

An $n$-sheeted surface cover of an orbifold of underlying genus $g_0$ with $r$ cone points has the displayed ordinary Euler characteristic, where $c_i$ is the number of cycles of the $i$th cone monodromy permutation. The ramification deficit over that point is $n-c_i$, so the formula follows by counting lifted vertices, edges and faces, or by the Riemann-Hurwitz formula. If every cone cycle has its full prescribed length, this equals $n{\chi }_{\mathrm{orb}}$.

For a compact smooth Riemannian surface with smooth boundary and a pure Dirichlet boundary condition or pure Neumann boundary condition, the constant coefficient in the heat trace isThe equality is the Gauss-Bonnet theorem. Thus such isospectral manifolds have the same Euler characteristic. A simply connected compact connected bordered surface is a disk with $\chi =1$, whereas a nonorientable surface with $c\ge 1$ crosscaps and $b\ge 1$ boundary components has $\chi =2-c-b\le 0$. These cannot be isospectral in this setting. Corners and mixed boundary conditions contribute additional heat coefficients, so the smooth pure-boundary-condition hypothesis is essential.

For closed connected oriented $d$-manifolds with $d\ge 2$,Deleting an open ball changes the Euler characteristic by $-1+\chi (S^{d-1})$, and gluing subtracts $\chi (S^{d-1})$. Since $\chi (S^{d-1})+\chi (S^d)=2$, the formula follows. Thus a connected sum of oriented manifolds subtracts two from the sum of Euler characteristics in even dimension and subtracts zero in odd dimension.