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On the descendibility and extendability of homogeneous quasimorphisms

丸山, 修平 名古屋大学

2021.11.03

概要

本学位論文では斉次擬準同型(homogeneous quasimorphism)の descending problem及び extending problem を扱う. 群 G 上の実数値関数 μ : G → R が(斉次)擬準同型とは, μ(gh) - μ(g) - μ(h) の絶対値が一様に有界であり, さらに G の巡回部分群上で準同型となるときをいう(以下では単に擬準同型と呼ぶ). 定義から準同型は擬準同型であり,よって準同型を自明な擬準同型と呼ぶ. G 上の擬準同型全体のなすベクトル空間をQ(G) で表す. 擬準同型の descending problem とは, 群の間の全射 p : Γ → G に対し,Γ 上の擬準同型が G 上のものの引き戻しとなるかを問うものである. 擬準同型のextending problem とは, 群 Γ とその正規部分群 K との組 (Γ, K) に対し, K 上の擬準同型が Γ 上に拡張できるかを問うものである.

本学位論文は, 準備のための Chapter 1, descending problem を扱う Chapter 2, 及びextending problem を扱う Chapter 3 の三つの Chapter からなる.
Chapter 2 では, 主に連結な位相群 G の普遍被覆 p : G~ → G に対する descendingproblem を扱う. Q(G~) には次の二つの部分空間がある: 自明な擬準同型(つまり準同型)のなす空間 H1(G~), G に descend する擬準同型のなす空間 p*Q(G~). ここで次の商空間 ND を考える:

ND = Q(G~)/(p*Q(G) + H1(G~)).

この商空間 ND において非自明な擬準同型というのはつまり, 準同型で調整しても Gに descend しない G~ 上の擬準同型である.
Chapter 2 の主定理は, この商空間 ND の幾何的な意味を与えるものである. 定理を述べるために記号の準備をする. G に離散位相を入れた離散群を Gδ で表すと, 恒等写像ι : Gδ → G はコホモロジーの間の準同型(Bι)* : H2top(BG) → H2(G) を誘導する.ここで H2top(BG) は分類空間 BG のコホモロジーであり, H2(G) は G の群コホモロジーである. またコホモロジーはすべて実数係数である. H2b(G) で G の 2 次有界コホモロジーを表し, cG : H2b(G) → H2(G) を比較写像とする. このとき, 写像(Bι)* : H2top(BG) → H2(G) の像と, 比較写像の像との共通部分は葉層 G 束の普遍有界特性類の空間とみなすことができる.

定理 商空間 ND と, 写像(Bι)* : H2top(BG) → H2(G) の像と比較写像の像との共通部分とは線形同型である.

この定理の系として例えば, ある symplectic fibration や contact fibration のprimary obstruction class についての(非)有界性が得られる.

Chapter 3 では extending problem を扱う. 群 Γ とその正規部分群 K に対し, 包含写像 i は準同型 i* : Q(Γ) → Q(K)Γ を誘導する. ここで Q(K)Γ は K 上の Γ 共役不変擬準同型のなす空間である. Q(K)Γ には次の二つの部分空間がある: 自明なΓ 共役不変擬準同型(つまり Γ 不変準同型)のなす空間 H1(K)Γ, Γ に extend する擬準同型のなす空間 i*Q(Γ). ここで次の商空間 NE を考える:

NE = Q(K)Γ /(i* Q(Γ) + H1(K)Γ).

有限生成群 Γ とその正規部分群 K に対し, この商空間 NE が非自明となる例はこれまで知られていなかった. Chapter 3 の主定理は, NE が非自明となるような組(Γ, K)をたくさん与える.

定理 種数 2 以上の曲面群 Γ とその交換子部分群に対し, NE の次元は 1 である.定理 種数 2 以上のトレリ群内の擬アノソフ写像類をモノドロミーとするmapping torus の基本群 Γ とその交換子部分群に対し, NE の次元は 2g+1 となる.

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