ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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Klein bottle as two crosscaps

Example

The one-polygon word $a a b b$ of Polygonal schemas and paired boundary edges realizes the Klein bottle $K$ of The Klein bottle as a square quotient, equivalently the standard square word $a b a^{- 1} b$ under the polygonal cut-and-paste of Homeomorphism-preserving polygonal schema moves. It is nonorientable and has $V = 1$ , $E = 2$ , $F = 1$ , hence $χ (K) = 1 - 2 + 1 = 0$ (Euler characteristic of a finite CW complex). The two equal-exponent pairs are the two crosscap blocks, and the one-letter block $a a$ realizes ${R P}^{2}$ by Projective plane crosscap polygon. This finite cut-and-paste computation uses no choice axiom.

Facts & Assumptions

Given: The square $Q = [0, 1]^{2}$ with corners $v_{0} = (0, 0)$ , $v_{1} = (1, 0)$ , $v_{2} = (1, 1)$ , $v_{3} = (0, 1)$ , the one-polygon schema whose boundary sides read $a$ (bottom $v_{0} v_{1}$ ), $a$ (right $v_{1} v_{2}$ ), $b$ (top $v_{2} v_{3}$ ), $b$ (left $v_{3} v_{0}$ ) with equal-exponent pairs glued with matching parameters, and the quotient $Y$ of $Q$ by these pairings.

[L1]

Schema conventions: sides of a schema are paired by specified corner-preserving homeomorphisms, affine when both sides are straight, the boundary word records for each side whether the boundary traversal agrees with a reference direction, and renaming letters, cyclically rotating the word or reversing the polygon orientation give homeomorphic quotients; the quotient vertices, edges and faces are the corner classes, the paired side classes and the disk interiors, giving a finite cell structure with counts $(V, E, F)$ ; in a one-polygon word a pair with opposite exponents is orientation compatible while a pair with equal exponents is twisted (Polygonal schemas and paired boundary edges).

[L2]

Cut-and-paste moves: cutting a polygon along an embedded polygonal diagonal whose interior lies in the polygon interior and regluing the two new boundary sides preserves the quotient homeomorphism type, as does the inverse gluing of two faces along a paired pair of sides; and the resulting finite identities move two same-direction occurrences of a letter together as a crosscap block $b b$ , extract an interlaced pair pattern as a commutator handle block, and replace one handle plus one crosscap by three crosscaps (Homeomorphism-preserving polygonal schema moves).

[L3]

The Klein bottle $K$ is the quotient of $Q$ by $(x, 0) \sim (x, 1)$ and $(0, y) \sim (1, 1 - y)$ , and the one-polygon word of that presentation is $a b a^{- 1} b$ (The Klein bottle as a square quotient).

[L4]

The one-polygon word $a a$ , with its two sides paired in the same boundary direction, realizes ${R P}^{2}$ and is nonorientable (Projective plane crosscap polygon).

[L5]

An integral orientation is a continuous generating section of the local homology system; over a coordinate ball the system is trivialized and a continuous generator section is locally constant, so a continuous orientation is unchanged by transport along any loop (R-orientation of a topological manifold, Orientation local system and orientation cover).

[L6]

The Euler characteristic of a space with finitely many cells is $χ (X) = \sum_{n} (- 1)^{n} c_{n} (X)$ (Euler characteristic of a finite CW complex).

Verification

technique · direct

1.1L1

Both pairs of the word $a a b b$ carry equal exponents: the bottom side $a$ is glued to the right side $a$ with matching parameters, $(t, 0) \sim (1, t)$ , and the top side $b$ is glued to the left side $b$ with matching parameters, $(1 - t, 1) \sim (0, 1 - t)$ . The a-pair identifies the corners $v_{0} \sim v_{1}$ and $v_{1} \sim v_{2}$ , and the b-pair identifies $v_{2} \sim v_{3}$ and $v_{3} \sim v_{0}$ ; hence all four corners form one vertex class, so the quotient $Y$ has $V = 1$ vertex class, $E = 2$ paired side classes and $F = 1$ face, and these are the cells of its finite CW structure.

1.2L1L2L3

Read the word of the Klein-bottle presentation of [L3] cyclically from its middle letter, $b a^{- 1} b a$ ; in the crosscap identity of [L2] with the letter $a$ renamed $b$ , the separating arc $X = a^{- 1}$ and the complementary arc $Y = a$ give a quotient homeomorphic to that of $b b Y^{- 1} X = b b a^{- 1} a^{- 1}$ . Cyclically rotating this word and renaming the letter $a^{- 1}$ to $a$ turns it into $a a b b$ ; since the presentation of [L3] realizes the Klein bottle $K$ , the quotient $Y$ of the word $a a b b$ is homeomorphic to $K$ .

2.1L1L4L5step 1.1step 1.2

Both equal-exponent pairings preserve boundary direction: their formulas in step 1.1 use matching boundary parameters. Choose collars around paired interior points of the $a$ -sides, with $t$ increasing along the boundary and $r \geq 0$ pointing inward. These coordinates give the same face orientation on the two collars. A chart across the paired edge uses $(t, r)$ on the first collar and $(t, - r)$ on the second, so its generator has opposite signs relative to the face generators on the two sides. The sign is the local-homology sign of a reflection, which reverses the cyclic orientation of the boundary of a small disk, exactly as in the seam calculation for [L4]. A path through the open face joining the collar interiors, closed by one crossing of the seam, therefore transports its generator to its negative. A continuous orientation is unchanged by loop transport [L5], which is impossible for a generator of an infinite cyclic group. Hence $Y ≅ K$ is nonorientable. The two equal-exponent pairs are crosscap blocks, and [L4] identifies the single block with ${R P}^{2}$ .

3.1L1L6step 1.1step 1.2step 2.1∎

The cell counts of step 1.1 give $χ (Y) = V - E + F = 1 - 2 + 1 = 0$ by [L6], and this value is that of the Klein bottle by step 1.2. Together with steps 1.2 and 2.1 the word $a a b b$ realizes the Klein bottle, is nonorientable, and has Euler characteristic $0$ . Every construction used is a finite explicit pairing, rotation or crosscap move of polygons, so no choice axiom is used.

Remarks

The word $a a b b$ is the two-crosscap normal form, the case $h = 2$ of the corresponding block word, and the standard word $a b a^{- 1} b$ of The Klein bottle as a square quotient is recovered from it by the same crosscap identity read backwards. The argument uses only the finite cut-and-paste moves of Homeomorphism-preserving polygonal schema moves and never the classification theorem or the Axiom of Choice.

Depends on

Used by

Equal Euler characteristic without homeomorphism Counterexample

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gallier and Xu, A Guide to the Classification Theorem for Compact Surfaces (standard reference, not scraped)
Koch, Classification of Surfaces (standard reference, not scraped)