On the involutions of closed surfaces

Collapsing three-dimensional closed Alexandrov spaces with a lower curvature bound

“…By Theorem 2.51, there is an involution on K 2 having only four fixed points. This is a contradiction (see [N,Lemma 2]). Therefore, ∂B(S 2 ; 4) ≈ T 2 .…”

ℤ2-equivariant hierarchies of irreducible 3-manifolds containing two-sided projective planes

“…Therefore P } = P k and (kgk~1)(P t ) = h(P t ) for all i. If h(P ( ) = P { , then by lemma 1 of [20], h\ P . is equivalent to (kgk~1)\ p .…”

“…equivariant hierarchies 121 follows by lemma 1 of Natsheh[20] that fix (h\ P ) consists of two components: a point, and a one-sided loop. This loop is contained in a 2-dimensional component of the fixed point set.…”

Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric

“…We now describe three different closed Alexandrov 3-spaces as quotients of certain involutions (cf. [19])…”

“…Moreover, the map induces an involution on the core Klein bottle K 2 × {0} of K 2 ×I with the preceding two points as fixed points. By the classification of involutions on K 2 (see [19]), K 2 × {0}/ϕ is isometric to an RP 2 with a flat metric with two metric singularities. Comparing this construction with the construction in [17,Corollary 2.56], one sees that the double branched cover of B(RP 2 ) is K 2 ×I.…”