Most important，perhaps，was that they had all maintained with a certain fidelity a manner of technique and composition consistent with those of America's first popular landscape artist，Thomas Cole，who built a career painting the Catskill Mountain scenery bordering the Hudson River.
(倒装句：Most important…was that…=that…was most important)
也许最重要的是，他们都在某种程度上忠实地保持一种与美国第一位流行山水画家托马斯·科尔相一致的技巧和创作方式，后者是以描绘临近哈得逊河的恺特斯基尔山风景而建立其事业的。
解析：倒装结构Most important…was that
正确顺序：That they had all maintained a manner of technique and composition was most important
主句：That…was most important
从句1：they had all maintained a manner of technique and composition
从句2：who built a career
从句3：painting the Catskill Mountain scenery
从句4：bordering the Hudson River.
修饰结构1：with a certain fidelity
修饰结构2：consistent with those of America's first popular landscape
artist
修饰结构3：Thomas Cole
从句1是主句1后that引导的部分，也就是其中的主语
从句2是修饰结构2中的artist的同位语
从句3是修饰从句2中的career
从句4是修饰从句2中的scenery
修饰结构1修饰从句1的谓语maintain
修饰结构2修饰从句1中的technique and composition
修饰结构3是修饰结构2中artist的同位语
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