Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces

TL;DR

This paper investigates the structure of range decreasing homomorphisms and holomorphic maps between generalized loop spaces, showing they are often induced by pullback maps, with applications to automorphisms of these spaces.

Contribution

It characterizes range decreasing group homomorphisms and holomorphic maps between generalized loop spaces as pullback operators, extending understanding of their structure and automorphisms.

Findings

Range decreasing homomorphisms are pullbacks by a map between base manifolds.

Provides conditions under which holomorphic maps are pullback operators.

Identifies classes of automorphisms of spaces of holomorphic maps.

Abstract

Let $\mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{\infty}$ compact connected manifold, possibly with boundary. Fix a smoothness class $\mathcal{F}=C^{\infty}$, H\"older $C^{k, \alpha}$ or Sobolev $W^{k, p}$. The space $\mathcal{F}(V, \mathcal{G})$ resp. $\mathcal{F}(V, M)$ of all $\mathcal{F}$ maps $V \to \mathcal{G}$ resp. $V \to M$ is a Banach/Fr\'echet Lie group resp. complex manifold. Let $\mathcal{F}^0(V, \mathcal{G})$ resp. $\mathcal{F}^{0}(V, M)$ be the component of $\mathcal{F}(V, \mathcal{G})$ resp. $\mathcal{F}(V, M)$ containing the identity resp. constants. A map $f$ from a domain $\Omega \subset \mathcal{F}_1(V, M)$ to $\mathcal{F}_2(W, M)$ is called range decreasing if $f(x)(W) \subset x(V)$, $x \in \Omega$. We prove that if $\dim_{\mathbb{R}} \mathcal{G} \ge 2$, then any range decreasing group homomorphism $f: \mathcal{F}_1^0(V, \mathcal{G}) \to \mathcal{F}_2(W, \mathcal{G})$ is the pullback by a map $\phi: W \to V$. We also provide several sufficient conditions for a range decreasing holomorphic map $\Omega$ $\to$ $\mathcal{F}_2(W, M)$ to be a pullback operator. Then we apply these results to study certain decomposition of holomorphic maps $\mathcal{F}_1(V, N) \supset \Omega \to \mathcal{F}_2(W, M)$. In particular, we identify some classes of holomorphic maps $\mathcal{F}_1^{0}(V, \mathbb{P}^n) \to \mathcal{F}_2(W, \mathbb{P}^m)$, including all automorphisms of $\mathcal{F}^{0}(V, \mathbb{P}^n)$.