海因茨·霍普夫Heinz Hopf

海因茨·霍普夫的父亲是Wilhelm Hopf，母亲是Elizabeth Kirchner。Wilhelm Hopf来自一个犹太家庭。他于1887年在布雷斯劳加入了Heinrich Kirchner的酿酒厂。Wilhelm于1892年与Heinrich Kirchner的长女Elizabeth结婚，那时他拥有这家酿酒公司。他们有两个孩子，长女Hedwig生于1893年，而霍普夫生于次年。Elizabeth Hopf是新教徒，1895年，Wilhelm皈依了他妻子的宗教。

霍普夫 从1901年到1904年就读于卡尔·米特尔豪斯博士学校，此后他开始在布雷斯劳的约翰·萨穆埃尔·柯尼希-威廉文理中学学习。他在文理中学一直读到1913年，正是在这所学校，他的数学天赋首次为老师们所清楚认识。然而，在其他科目上，他的成绩较差，很可能是因为他在体育运动上投入了过多时间——他尤其喜欢游泳和网球——而在学业科目上投入不足。他离开文理中学时，数学报告上写着：

他在这一领域表现出非凡的天赋，尤其是在代数方向。

1913年4月，霍普夫进入布雷斯劳的西里西亚弗里德里希·威廉大学攻读数学学位。在那里，他师从Kneser、艾尔哈德·施密特和Rudolf Sturm。他还听了在布雷斯劳理工学院任教的马克斯·登和施泰尼茨的讲座。然而，他的学业因1914年第一次世界大战爆发而中断。他立即入伍，并在整个战争期间作为中尉在西线作战。1917年，在服兵役期间的一次两周休假中，霍普夫去听了艾尔哈德·施密特在布雷斯劳大学开设的集合论课程。从那时起，他就知道自己想要从事数学研究。他在[13]中写到艾尔哈德·施密特的讲座对他的影响：

我为之着迷；这种着迷——对映射度方法的力量的着迷——从此从未离开过我，而且影响了我工作的大部分。当我寻找这种效果的原因时，我特别看到两点：第一，艾尔哈德·施密特在讲座中的生动和热情；第二，在多年兵役之后的两周休假期间，我自己接受能力的增强。

战后，霍普夫回到布雷斯劳继续学业，但大约一年后他离开那里，前往海德堡大学。此时艾尔哈德·施密特已经离开布雷斯劳，看来霍普夫想去海德堡与他的姐姐在一起，她前一年已在那里开始学习。在海德堡，霍普夫修了哲学和心理学课程，也听了奥斯卡·佩龙和Stäckel的课程。1920年，霍普夫前往柏林大学攻读博士学位，当时艾尔哈德·施密特正在那里任教。他在柏林听了伊赛·舒尔的几门课程，并于1925年获得博士学位，学位论文由艾尔哈德·施密特指导，研究流形的拓扑学。除其他结果外，他在这篇学位论文中分类了常曲率的单连通黎曼3-流形。这是一项令人印象深刻的工作，艾尔哈德·施密特在其报告中对它作了如下赞扬（例如见[11]）：

问题的胆识与解答的惊人结果同样值得钦佩。但学位论文中最美妙之处在于证明方法，这在该领域的工作中尤为罕见，它抽象且每一步都可理解，并且由于这种抽象性，同样清晰地展现了具体几何想象力的丰富。

路德维希·比贝尔巴赫和艾尔哈德·施密特考他数学，而马克斯·普朗克考他物理。

霍普夫于1925年前往哥廷根，在那里遇见了埃米·诺特。她的贡献对霍普夫发展自己的思想起了重要作用。也许更重要的是，帕维尔·亚历山德罗夫当时也在哥廷根逗留，而霍普夫在[13]中写道：-

我在哥廷根最重要的经历是遇见Pavel 帕维尔·亚历山德罗夫。这次相遇很快变成了友谊；讨论的不只是拓扑学，也不只是数学；那是一段幸运而也非常快乐的时光，不限于哥廷根，还在许多共同旅行中延续。

在哥廷根的这一年里，霍普夫研究他的教授资格论文（Habilitation）学位论文，该论文于1926年秋完成。论文包含了所罗门·莱夫谢茨刚刚证明的一个事实的另一种证明：对任何闭流形，一个一般向量场的指标之和是拓扑不变量，即莱昂哈德·欧拉示性类。帕维尔·亚历山德罗夫和霍普夫于1926年在法国南部与奥托·纽格包尔共度了一段时间。随后两人于1927-28学年在美国普林斯顿度过。这是拓扑学发展中的重要一年，帕维尔·亚历山德罗夫和霍普夫在普林斯顿，能够与所罗门·莱夫谢茨、奥斯瓦尔德·维布伦和詹姆斯·韦德尔·亚历山大合作。在普林斯顿的那一年里，帕维尔·亚历山德罗夫和霍普夫计划了一部关于Topology的多卷本联合著作，其第一卷直到1935年才出版。这是原计划三卷中唯一出版的一卷，因为第二次世界大战阻碍了剩余两卷的进一步合作。

霍普夫于1928年10月与Anja von Mickwitz结婚。1929年12月，普林斯顿向他提供助理教授职位，但他拒绝了这一职位。1930年，赫尔曼·外尔离开他在苏黎世ETH的讲席，去哥廷根担任讲席，1931年，有人与霍普夫接触，看他是否有兴趣接受这一讲席。这一提议部分是由于伊赛·舒尔寄给苏黎世的一封非常积极的推荐信促成的：-

霍普夫是一位优秀的讲师，一位气质强烈、影响力强大的数学家，是他学科中的领军典范……就他的举止、他的教养和他令人愉快的性格而言，我不能希望你有一位更好的同事。

霍普夫回复了苏黎世联邦理工学院的联系，表示他会接受正式聘任：-

去瑞士，去美丽的苏黎世城，确实会吸引我并给我荣誉，尤其是担任如此著名的讲席。因此我声明，我原则上愿意接受这样的聘任。

然而，在收到苏黎世的正式聘任之前，霍普夫收到了弗莱堡的讲席聘任，但他等待苏黎世的聘任并接受了它。他于1931年4月在苏黎世就职。接下来的几年对霍普夫来说并不容易。1933年纳粹在德国掌权后，霍普夫的父亲因为是犹太人而受到越来越大的压力。霍普夫一直去布雷斯劳探望父母，直到1939年。看到父亲面临的困难，霍普夫安排父母获得瑞士的移民文件。然而，他的父亲病倒了，无法旅行。

霍普夫能够在瑞士为那些在纳粹统治下不得不逃离德国的朋友提供避难所。特别是伊赛·舒尔来了一段时间，最后于1939年前往巴勒斯坦。然而，霍普夫自己的处境变得更加困难，因为他仍然是德国公民。所罗门·莱夫谢茨意识到霍普夫的困难，邀请他去普林斯顿，但霍普夫拒绝了。然后在1943年，他被告知要么搬回德国，要么将失去德国公民身份。面对这种情况，他别无选择，只能迅速申请瑞士公民身份，很快就获得了批准。

随着第二次世界大战的结束，II Hopf能够再次帮助他的德国朋友。然而，他所做的远不止这些，因为他投入了大量精力试图在德国重建数学界。他1946年8月访问奥伯沃尔法赫研究中心就是他努力的一部分。奥伯沃尔法赫访问后不久，霍普夫前往美国，在那里度过了六个月，并重续了许多旧日友谊。美国许多最负盛名的大学向他提供了教授职位，但经过仔细考虑，他决定继续忠于苏黎世。

在接下来的几年里，他享受受邀在领先的国际会议上演讲，并访问了许多地方，包括巴黎、布鲁塞尔、罗马和牛津。1955-56学年，他和妻子在美国度过。

霍普夫的大部分工作是在代数拓扑学中进行的，在那里可以认为他延续了勒伊岑·布劳威尔的工作。他研究了同伦类和向量场，得出了一个关于积分曲率的公式。

霍普夫在1928年进行的工作中扩展了所罗门·莱夫谢茨的不动点公式。正是在这篇1928年的论文中，他首次明确使用了同调群。他1927-28年在普林斯顿进行的关于流形同调的工作，导致他通过定义循环的相交乘积来定义相交环。这个想法后来被认为与上同调有关。

他在1931年定义了现在被称为“霍普夫不变量”的东西。这是在他研究不同维数球面之间的映射时所做的，这些映射无法在同调上区分，因此需要引入一个新的不变量。1939年，他研究了一个紧李群的同调。这是为了处理埃利·嘉当向他提出的问题。他在这次研究中引入的思想使他定义了今天所谓的霍普夫代数。

在20世纪40年代初，霍普夫发表了[11]：-

论文《基本群与第二贝蒂群》Ⓣ（基本群与恩里科·贝蒂的第二群）[这]被合理地视为同调代数的开端。它为定义群的同调与上同调开辟了道路。这一步在论文为人所知后不久，在不同地方独立完成……

霍普夫获得的荣誉几乎多得不胜枚举。他从1955年到1958年担任国际数学联盟主席。他获得了许多大学的荣誉博士学位，包括普林斯顿、弗莱堡、曼彻斯特、索邦、布鲁塞尔和洛桑。他获得了许多奖项，包括卡尔·弗里德里希·高斯-威廉·韦伯奖章和罗巴切夫斯基奖。他被选为世界各地许多学术团体的荣誉会员。

汉斯·弗洛伊登萨在[1]中这样描述霍普夫：-

霍普夫是个矮小、精力充沛的人，面容开朗愉快。他的声音抑扬顿挫，说话缓慢而咬字清晰。他的讲课风格清晰而引人入胜；在私下交谈中，他能传达出启发性的想法。

Frei和Stammbach在[11]中这样赞颂霍普夫：-

毫无疑问，霍普夫是二十世纪最杰出的数学家之一。他的工作与代数拓扑学的兴起紧密相连；正是由于他早期的著作，这一领域才确立为数学中一个新的重要分支。他的工作深刻影响了拓扑学乃至大部分数学的发展。但霍普夫不仅是一位有天赋的研究者：他还是一位优秀的教师，一位品格极为正直的人。同时，他洋溢着魅力与微妙的幽默。

Heinz Hopf's father was Wilhelm Hopf and his mother was Elizabeth Kirchner. Wilhelm Hopf was from a Jewish family. He joined Heinrich Kirchner at his brewery in Breslau in 1887. Wilhelm married Elizabeth, Heinrich Kirchner's eldest daughter, in 1892 and by that time he owned the brewery firm. They had two children, the eldest Hedwig was born in 1893 while Heinz was born in the following year. Elizabeth Hopf was a Protestant and, in 1895, Wilhelm converted to his wife's religion.

Heinz attended Dr Karl Mittelhaus's school from 1901 until 1904 and following this he began his studies at the König-Wilhelm Gymnasium in Breslau. He attended the Gymnasium until 1913 and it was at this school that his talent for mathematics first became clear to his teachers. In his other subjects, however, his results were less good and it is probable that he devoted too much time to sport, he was particularly fond of swimming and tennis, and not enough to his academic subjects. He left the Gymnasium with the mathematics report stating:-

He has shown an extraordinary gift in this topic, especially in the algebraic direction.

In April 1913 Hopf entered the Silesian Friedrich Wilhelms University in Breslau to read for a degree in mathematics. There he was taught by Kneser, Schmidt, and Rudolf Sturm. He also attended lectures by Dehn and Steinitz who taught at the polytechnic in Breslau. However, his studies were interrupted by the outbreak of World War I in 1914. He immediately enlisted and for the duration of the war he fought on the Western front as a lieutenant. During a fortnight's leave from military service in 1917 Hopf went to a class by Schmidt on set theory at the University of Breslau. From that time on he knew that he wanted to undertake research in mathematics. He wrote in [13] about the influence Schmidt's lectures had on him :-

I was fascinated; this fascination - of the power of the method of the mapping degree - has never left me since, but has influenced major parts of my work. And when I look for the cause of this effect, I see particularly two things: firstly, Schmidt's vividness and enthusiasm in his lecture, and secondly my own increased receptiveness during a fortnight off after many years of military service.

After the war Hopf returned to his studies in Breslau but after about a year he left and went to the University of Heidelberg. By this time Schmidt had left Breslau and it appears that Hopf wanted to go to Heidelberg to be with his sister who had begun her studies there in the previous year. At Heidelberg Hopf took courses in philosophy and psychology as well as attending courses by Perron and Stäckel. In 1920 Hopf went to study for his doctorate at the University of Berlin where Schmidt was now teaching. He attended several courses by Schur in Berlin and he received his doctorate in 1925 with a thesis, supervised by Schmidt, studying the topology of manifolds. Among other results, he classified simply connected Riemannian 3-manifolds of constant curvature in this thesis. It was an impressive piece of work which received the following praise from Schmidt in his report (see for example [11]):-

The boldness of the questions deserves as much admiration as the surprising results of the solutions. But the most beautiful thing in the thesis is the method of proving, which is, particularly rarely found in works in that area, abstract and comprehensible in every step, and which, due to the abstractness, shows equally clearly the richness of the concrete geometric imagination.

Bieberbach and Schmidt examined him in mathematics, while Planck examined him in physics.

Hopf went to Göttingen in 1925 where he met Emmy Noether. Her contributions would play an important part in Hopf's developing ideas. Perhaps even more significant was the fact that Aleksandrov was also spending time in Göttingen and Hopf wrote in [13]:-

My most important experience in Göttingen was to meet Pavel Aleksandrov. The meeting soon became friendship; not only topology, not only mathematics was discussed; it was a fortunate and also a very happy time, not restricted to Göttingen but continued on many joint journeys.

During this year in Göttingen Hopf worked on his habilitation thesis which was completed by the autumn of 1926. The thesis contains a different proof of the fact just shown by Lefschetz that for any closed manifold the sum of the indices of a generic vector field is a topological invariant, namely the Euler characteristic. Aleksandrov and Hopf spent some time in 1926 in the south of France with Neugebauer. Then the two spent the academic year 1927-28 at Princeton in the United States. This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander. During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes.

Hopf married Anja von Mickwitz in October 1928. He was offered an assistant professorship by Princeton in December 1929 but he rejected the offer. In 1930 Weyl left his chair in the ETH in Zürich to take up a chair at Göttingen and in 1931 Hopf was approached to see if he was interested in accepting this chair. In part the offer had been prompted by a very positive recommendation which Schur had sent to Zürich:-

Hopf is an excellent lecturer, a mathematician of strong temperament and strong influence, a leading example in his discipline ... I cannot wish you a better colleague in respect to his manners, his education and his sympathetic nature.

Hopf replied to the approach of the ETH in Zürich indicating that he would accept a formal offer:-

A call to Switzerland, to the beautiful city of Zürich, could indeed tempt and honour me, particularly to such a famous chair. I therefore declare that I am in principle willing to accept such an offer.

However, before receiving the formal offer from Zürich, Hopf received the offer of a chair at Freiburg but he waited for the Zürich offer and accepted it. He took up his duties in Zürich in April 1931. The next few years were not easy ones for Hopf. After the Nazis came to power in Germany in 1933, Hopf's father, being Jewish, came under increasing pressure. Hopf continued to visit his parents in Breslau up until 1939. Seeing the difficulties that his father faced Hopf arranged for his parents to receive immigration papers for Switzerland. However, his father fell ill and could not travel.

Hopf was able to provide refuge in Switzerland for friends who had to flee Germany under the Nazis. In particular Schur came for a while before finally going to Palestine in 1939. Hopf's own position became more difficult, however, for he was still a German citizen. Lefschetz, realising Hopf's difficulties, invited him to Princeton but Hopf refused. Then in 1943 he was told to move back to Germany or he would lose his German citizenship. Faced with this he had little choice but to quickly apply for Swiss citizenship, which was soon granted.

With the end of World War II Hopf was able to help his German friends again. He did much more than this, however, for he put much energy into trying to re-establish a mathematical community in Germany. His visit to the research centre in Oberwolfach in August 1946 was part of his efforts. Soon after the Oberwolfach visit, Hopf went to the United States where he spent six months and there he renewed many old friendships. He was offered professorships by many of the most prestigious of the American universities but, after careful consideration, he decided to remain loyal to Zürich.

Over the next few years he enjoyed invitations to lecture at leading international conferences, and he visited many places including Paris, Brussels, Rome and Oxford. He spent the academic year 1955-56 with his wife in the United States.

Most of Hopf's work was in algebraic topology where he can be thought of as continuing Brouwer's work. He studied homotopy classes and vector fields producing a formula about the integral curvature.

Hopf extended Lefschetz's fixed point formula in work which he undertook in 1928. It is in this 1928 paper that he first explicitly used homology groups. His work on the homology of manifolds, undertaken in Princeton in 1927-28, led to his definition of the intersection ring by defining a product on cycles by their intersection. This idea was later seen to be connected to cohomology.

He defined what is now known as the 'Hopf invariant' in 1931. This was done in his work on maps between spheres of different dimensions which cannot be distinguished homologically so required the introduction of a new invariant. In 1939 he examined the homology of a compact Lie group. This was to attack questions posed to him by Élie Cartan. The ideas which he introduced in this investigation led to him defining what is today called a Hopf algebra.

In the early 1940s Hopf published [11]:-

The paper Fundamentalgruppe und zweite Bettische Gruppe Ⓣ [which] is legitimately regarded to be the beginning of homological algebra. It opened the way for the definition for the homology and cohomology of a group. This step was made independently at different places shortly after the paper became known ...

The honours which Hopf received are almost too numerous to list. He was President of the International Mathematical Union from 1955 until 1958. He received honorary doctorates from many universities including Princeton, Freiburg, Manchester, Sorbonne, Brussels, and Lausanne. He was awarded many prizes including the Gauss-Weber medal and the Lobachevsky award. He was elected to honorary membership of many learned societies throughout the world.

Freudenthal gives this description of Hopf in [1]:-

Hopf was a short, vigorous man with cheerful, pleasant features. His voice was well modulated, and his speech slow and strongly articulated. His lecture style was clear and fascinating; in personal conversation he conveyed stimulating ideas.

Frei and Stammbach in [11] pay this tribute to Hopf:-

Without doubt Heinz Hopf was one of the most distinguished mathematicians of the twentieth century. His work is closely linked with the emergence of algebraic topology; it is most decisively thanks to his early works that this area established itself as a new and important branch of mathematics. his work has influenced profoundly the evolution not only of topology but of a large part of mathematics. But Heinz Hopf was not only a gifted researcher: he was also an excellent teacher and a personality of the highest integrity. at the same time, he effervesced with charm and subtle humour.