Heinz Hopf

How to use this course

Heinz Hopf worked at the moment when topology changed from counting numbers to studying groups and maps. He took Brouwer’s degree of a map, Poincaré’s index of a vector field and Riemann’s curvature, and turned each into a global theorem. His name now sits on a fibration, an invariant, an algebraic structure, a formula in group theory and a theorem in Riemannian geometry. He also trained, directly or through his seminar, much of the next generation of topologists.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his work roughly in order: geometry and vector fields (1925–1927), degree and fixed points (1926–1933), maps between spheres (1931–1935), Lie groups (1941), group homology (1942) and differential geometry throughout.

Timeline

Breslau and the war, 1894–1919

Heinz Hopf was born on 19 November 1894 in Gräbschen, a village just outside Breslau, then in the Prussian province of Silesia and now part of Wrocław in Poland. His father, Wilhelm Hopf, came from a Jewish family and had joined the brewery of Heinrich Kirchner; by the time he married Kirchner’s eldest daughter Elizabeth, a Protestant, he owned the firm. Wilhelm converted to Protestantism in 1895, a year after Heinz was born. Heinz had an older sister, Hedwig.

He attended Karl Mittelhaus’s private school from 1901 to 1904 and then the König-Wilhelm-Gymnasium in Breslau. His teachers noticed a gift for mathematics, especially algebra. In April 1913 he entered the University of Breslau, where his teachers included Adolf Kneser, Erhard Schmidt, Ernst Steinitz, Max Dehn and Rudolf Sturm.

The front

When war broke out in 1914 he volunteered at once. He spent the whole war on the Western Front, rising to the rank of lieutenant. He was wounded twice and in 1918 received the Iron Cross, first class. During a fortnight’s leave in 1917 he went back to Breslau and sat in on Schmidt’s course on set theory. The course convinced him that he wanted to do research in mathematics. Set theory and the young subject of topology, then being transformed by L. E. J. Brouwer, were where he would begin.

Heidelberg, Berlin and the thesis, 1919–1925

After demobilization Hopf spent about a year at Heidelberg, attending lectures by Oskar Perron and Paul Stäckel and studying philosophy and psychology. In 1920 he followed Schmidt to Berlin, where he also heard the algebra lectures of Issai Schur. Schur’s influence was lasting; years later Schur would recommend him for the most important appointment of his career.

The thesis

Hopf’s dissertation, Über Zusammenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten (On connections between the topology and the metric of manifolds), was accepted in Berlin in 1925. Schmidt and Ludwig Bieberbach examined him in mathematics and Max Planck in physics. The thesis attacked a question of Wilhelm Killing and Felix Klein: which closed spaces can carry a geometry of constant curvature? Hopf showed that every simply connected complete three-manifold of constant curvature is Euclidean space, the round sphere or hyperbolic space, so the problem reduces to classifying groups of isometries (Lesson 04). The title already states his lifelong programme: to find what the local geometry of a space forces on its global shape.

A second early result, on the total curvature of closed hypersurfaces, led him to vector fields and their singularities (Lesson 05).

Göttingen, Princeton and Alexandrov, 1925–1931

In 1925 Hopf went to Göttingen. There he met two people who changed his mathematics. Emmy Noether was urging topologists to treat Betti numbers and torsion coefficients as invariants of groups; Hopf adopted her view and acknowledged it in print. Pavel Alexandrov, a young Moscow topologist, became his closest friend for the rest of his life. In 1926 Hopf completed his habilitation, with the general theorem on vector fields, and returned to Berlin to teach.

For the academic year 1927–28 Rockefeller fellowships took Hopf and Alexandrov to Princeton, then the leading centre of combinatorial topology. They worked with Solomon Lefschetz, Oswald Veblen and James Alexander. Hopf learned Lefschetz’s fixed-point theory and generalized it (Lesson 06). He also found, while there, the map from the three-sphere to the two-sphere that carries his name, and began to suspect it could not be deformed to a constant. The two friends planned a three-volume treatise on topology. Only the first volume, Topologie I, appeared, in 1935; it was the standard reference for a generation.

In October 1928 Hopf married Anja von Mickwitz. Princeton offered him an assistant professorship in 1929; he declined. In 1930 Hermann Weyl left ETH Zürich for Göttingen, and the following year ETH chose Hopf to succeed him. He took up the chair in April 1931.

Space forms: the Clifford–Klein problem

For the general reader. A flat sheet of paper can be rolled into a cylinder or, with stretching avoided, a torus can be built from a square by gluing opposite edges. Each looks flat to an ant living on it. Hopf answered a general version of the question: which closed spaces look, locally, exactly like flat space, like a sphere, or like the hyperbolic plane?

Let \(M^n_K\) denote the simply connected model space of constant sectional curvature \(K\): the round sphere \(S^n\) of radius \(1/\sqrt K\) if \(K>0\), Euclidean space \(\mathbb R^n\) if \(K=0\), and hyperbolic space \(\mathbb H^n\) scaled to curvature \(K\) if \(K<0\).

Proof sketch

The universal cover \(\tilde M\) is complete and simply connected with curvature \(K\). Fix \(p\in\tilde M\) and \(o\in M^n_K\), and a linear isometry \(T_o M^n_K\to T_p\tilde M\). Compose with the exponential maps to get \(F=\exp_p\circ\, \iota\circ\exp_o^{-1}\), defined on a normal ball. Because the curvature tensors agree, Jacobi fields agree, and \(F\) is a local isometry (the Cartan–Ambrose–Hicks argument). Completeness lets one continue \(F\) along every path; simple connectivity makes the continuation single valued. A local isometry from a complete manifold onto a simply connected one is a global isometry. The deck group of \(\tilde M\to M\) then acts by isometries on \(M^n_K\), freely and discretely.

The classification of space forms is therefore the classification of discrete, fixed-point-free isometry groups. For \(K>0\) these are finite subgroups of \(\mathrm O(n+1)\) acting freely on \(S^n\). In dimension 3 they include the lens spaces and the quotient of \(S^3\) by the binary icosahedral group, which is Poincaré’s homology sphere. The complete list of spherical space forms was finished by Joseph Wolf in the 1960s.

Vector fields and the Euler characteristic

For the general reader. You cannot comb a hairy ball flat: there is always a crown. On a doughnut you can. Henri Poincaré explained this for surfaces by counting the singular points of a flow with signs. Hopf proved the count works in every dimension and always gives the same number, the Euler characteristic.

Index

Let \(v\) be a continuous vector field on an open set of \(\mathbb R^n\) with an isolated zero at \(p\). On a small sphere \(S_\varepsilon(p)\) the map \(x\mapsto v(x)/|v(x)|\) goes to \(S^{n-1}\); its degree is the index \(\operatorname{ind}_p(v)\). On a manifold, use a chart; the index does not depend on the chart. At a non-degenerate zero, \(\operatorname{ind}_p(v)=\operatorname{sign}\det Dv(p)\).

Proof sketch

The proof has two halves. First, the index sum is the same for all such fields: given two of them, join them by a homotopy and observe that indices are degrees, which are locally constant. Hopf’s proof of this step identified the sum with the degree of a map on the boundary of a tubular neighbourhood, using his own theorems on degree (Lesson 06). Second, exhibit one field whose index sum is visibly \(\chi(M)\). Triangulate \(M\) and let the flow run from the barycentre of each simplex toward the barycentres of its faces. Each barycentre of a \(k\)-simplex is a zero which is repelling in \(k\) directions and attracting in \(n-k\), with index \((-1)^k\). The sum is \(\sum_k(-1)^k\,\#\{k\text{-simplices}\}=\chi(M)\).

The Gauss map

Hopf’s earlier result is a special case with geometric content. If \(M^n\subset\mathbb R^{n+1}\) is a closed hypersurface with outward unit normal \(\nu:M\to S^n\), then for \(n\) even

where \(K\) is the Gauss–Kronecker curvature. For \(n=2\) this is the Gauss–Bonnet theorem \(\int_MK\,dA=2\pi\chi(M)\). Hopf proved it by projecting a fixed vector \(e\) onto the tangent spaces: the zeros of the projected field are the points where \(\nu=\pm e\).

Degree, homotopy and fixed points

For the general reader. Wrap a rubber sheet around a ball. The number of times it covers the ball, counted with orientation, is the degree. Brouwer showed the degree does not change under continuous deformation. Hopf proved the converse: the degree is the only obstruction, so two maps of a sphere to itself can be deformed into each other exactly when their degrees agree.

Hopf’s method was to make a map simplicial, then cancel preimages of opposite sign by pushing them together along paths. In 1933 he extended the result to any finite \(n\)-dimensional polyhedron \(K\):

This is the first case of the modern principle that cohomology classifies maps into Eilenberg–MacLane spaces, since \(S^n\) agrees with \(K(\mathbb Z,n)\) through dimension \(n\).

The trace formula and fixed points

Let \(X\) be a finite polyhedron and \(f:X\to X\) continuous. The Lefschetz number is \(L(f)=\sum_k(-1)^k\operatorname{tr}\big(f_*:H_k(X;\mathbb Q)\to H_k(X;\mathbb Q)\big)\). Solomon Lefschetz had proved a fixed-point formula for manifolds in 1926. During the Princeton year Hopf gave a new proof and extended it to all finite polyhedra, using a lemma now called the Hopf trace formula.

Proof. With cycles \(Z_k\) and boundaries \(B_k\), the exact sequences \(0\to Z_k\to C_k\xrightarrow{\partial}B_{k-1}\to0\) and \(0\to B_k\to Z_k\to H_k\to0\) are preserved by \(\varphi\). Trace is additive on such sequences, so \(\operatorname{tr}_{C_k}=\operatorname{tr}_{Z_k}+\operatorname{tr}_{B_{k-1}}=\operatorname{tr}_{B_k}+\operatorname{tr}_{H_k}+\operatorname{tr}_{B_{k-1}}\). In the alternating sum the boundary terms cancel in pairs.

Sketch: if \(f\) has no fixed point, subdivide finely and approximate \(f\) by a simplicial-type chain map that moves every simplex off itself. All diagonal entries of the chain map vanish, so the left side of the trace formula is \(0\), hence \(L(f)=0\). Applied to \(f=\mathrm{id}\), the formula gives \(L(\mathrm{id})=\chi(X)\), recovering Lesson 05.

The Hopf fibration and the Hopf invariant

For the general reader. Before 1931 it seemed natural that a map from a higher-dimensional sphere onto a lower one could always be shrunk to a point. Hopf found a map from the three-sphere to the ordinary sphere where every point pulls back to a circle, and any two of those circles are linked like the rings of a chain. The linking cannot be undone, so the map cannot be shrunk. Higher homotopy theory begins here.

The map

View \(S^3=\{(z_1,z_2)\in\mathbb C^2:|z_1|^2+|z_2|^2=1\}\) and \(S^2\subset\mathbb C\times\mathbb R\). The Hopf map is

equivalently \((z_1,z_2)\mapsto[z_1:z_2]\in\mathbb{CP}^1\cong S^2\). Its fibres are the circles \(\{(\lambda z_1,\lambda z_2):|\lambda|=1\}\), so \(\eta\) is a principal circle bundle.

The invariant

Let \(f:S^3\to S^2\) be smooth and \(p\ne q\) regular values. Then \(f^{-1}(p)\) and \(f^{-1}(q)\) are disjoint oriented closed curves in \(S^3\). Hopf defined

Later work by Hans Freudenthal and Lev Pontryagin showed that \(H:\pi_3(S^2)\to\mathbb Z\) is an isomorphism. The Hopf fibration also gives the long exact sequence \(\pi_k(S^1)\to\pi_k(S^3)\to\pi_k(S^2)\to\pi_{k-1}(S^1)\), so \(\pi_k(S^3)\cong\pi_k(S^2)\) for \(k\ge3\).

The cohomological definition and 1935

For \(f:S^{2n-1}\to S^n\) with \(n\ge2\), attach a \(2n\)-cell to \(S^n\) along \(f\): \(C_f=S^n\cup_fe^{2n}\). Let \(a\in H^n(C_f)\) and \(b\in H^{2n}(C_f)\) be generators. Then

In 1935 Hopf built the analogous maps \(S^7\to S^4\) from the quaternions and \(S^{15}\to S^8\) from the octonions, each of Hopf invariant \(1\). He showed \(H(f)=0\) when \(n\) is odd and that for even \(n\) there are maps with \(H(f)=2\). Whether Hopf invariant one occurs in other dimensions became a central problem. Frank Adams settled it in 1960: maps \(S^{2n-1}\to S^n\) of Hopf invariant one exist only for \(n=2,4,8\). A consequence is that \(\mathbb R^n\) carries a bilinear division algebra structure only for \(n=1,2,4,8\).

Group manifolds and Hopf algebras

For the general reader. A Lie group is a space that is also a group, such as the rotations of space. Élie Cartan and others had computed the Betti numbers of many such groups and found a strange pattern: they always look like those of a product of odd-dimensional spheres. In 1941 Hopf explained why. The explanation uses only the fact that points can be multiplied continuously, and the algebraic structure he extracted is now called a Hopf algebra.

A space \(X\) with a continuous map \(\mu:X\times X\to X\) and a point \(e\) such that \(\mu(e,x)=x=\mu(x,e)\) (up to homotopy) is an H-space, after Hopf. Every topological group is one. With field coefficients \(k\) of characteristic \(0\) and \(H^*(X)\) finite-dimensional, the Künneth formula gives an algebra homomorphism

with \(0<\deg x'_i,\deg x''_i<\deg x\). The cup product together with this coproduct makes \(H^*(X)\) a graded, connected, commutative and cocommutative-up-to-sign bialgebra. This is the structure later named a Hopf algebra.

Proof sketch

Choose generators inductively by degree so that each \(x_i\) is primitive modulo decomposables. Suppose a generator \(x\) has even degree. Then for every \(k\), expanding \(\mu^*(x^k)=(x\otimes1+1\otimes x+\cdots)^k\) shows that the component \(\binom{k}{1}x^{k-1}\otimes x\) is non-zero whenever \(x^{k-1}\ne0\), since binomial coefficients are invertible in characteristic \(0\). By induction no power of \(x\) vanishes, contradicting finite dimensionality. So all generators have odd degree, and graded commutativity gives \(x_i^2=0\). A counting argument with \(\mu^*\) shows that the monomials in the \(x_i\) are independent, so the algebra is exterior. For a compact Lie group the number \(r\) of generators equals the rank.

Hopf’s student Hans Samelson showed in 1941 that the generators can be taken primitive. Armand Borel extended the theory to finite fields in his 1953 thesis, and John Milnor and John Moore set out the general algebraic theory of Hopf algebras in 1965. Hopf algebras now appear far from topology: in combinatorics, in renormalization and as quantum groups.

Hopf’s formula and homological algebra

For the general reader. The shape of a space constrains its fundamental group, but Hopf noticed the converse: the fundamental group alone forces part of the second homology. In 1942 he wrote that part down as a formula in terms of generators and relations. The formula showed that groups have homology of their own, and the subject of homological algebra grew from it.

In modern terms this is the integral homology of an Eilenberg–MacLane space \(K(G,1)\), and the theorem gives the exact sequence \(\pi_2(X)\to H_2(X)\to H_2(G)\to0\). Its proof builds a \(K(G,1)\) from \(X\) by attaching cells of dimension \(\ge3\). The 3-cells, attached along generators of \(\pi_2(X)\), kill exactly the spherical classes in \(H_2\); cells of dimension \(\ge4\) do not affect \(H_2\). The expression in \(F\) and \(R\) is computed from the two-complex of the presentation. Within a few years Beno Eckmann (Hopf’s student), Samuel Eilenberg and Saunders Mac Lane, and Hans Freudenthal had defined homology and cohomology of groups in all degrees. Henri Cartan and Eilenberg’s Homological Algebra (1956) made the theory general.

In 1944 Hopf also showed that an infinite finitely generated group has 1, 2 or infinitely many ends. Freudenthal had studied ends of spaces, and the theory of ends of groups later became a pillar of geometric group theory through John Stallings’ theorem.

Global differential geometry

For the general reader. Hopf never stopped being a geometer. He proved that the tangent of a simple closed curve turns exactly once, gave the definitive meaning of a “complete” surface with Willi Rinow, and showed that a soap bubble shaped like a sphere must be round. These short theorems are now first chapters of the subject.

Hopf’s proof. Parametrize by arc length, starting at a lowest point. On the triangle \(\{0\le s\le t\le L\}\) define the secant direction \(\sigma(s,t)=\frac{\gamma(t)-\gamma(s)}{|\gamma(t)-\gamma(s)|}\), with \(\sigma(s,s)=\gamma'(s)\) on the diagonal and \(\sigma(0,L)=-\gamma'(0)\). Simplicity makes \(\sigma\) continuous. Its rotation along the diagonal equals its rotation along the other two sides. Along each of those sides the secant turns through \(\pi\), in the same sense, because the curve lies above its lowest point. Total: \(2\pi\).

Hopf and Rinow proved it for surfaces; the proof works verbatim in all dimensions. The space forms of Lesson 04 needed exactly this notion of completeness.

Constant mean curvature

In 1951 Hopf proved that an immersed sphere in \(\mathbb R^3\) with constant mean curvature is a round sphere. The key is a holomorphic quadratic differential on the surface, now called the Hopf differential, which must vanish on a sphere because the sphere carries no non-zero holomorphic quadratic differentials. Hopf asked whether closed CMC surfaces of higher genus exist; Henry Wente constructed an immersed torus in 1986. Two questions associated with Hopf remain open: whether \(S^2\times S^2\) admits a metric of positive sectional curvature, and whether a closed even-dimensional manifold of positive curvature has positive Euler characteristic.

Zürich, 1931–1971

Hopf arrived at ETH Zürich in April 1931 and stayed for the rest of his life. He was a clear, unhurried lecturer, who enjoyed teaching elementary number theory as much as topology, and his seminar became one of the main meeting places of the subject. Zürich hosted the International Congress of Mathematicians in 1932, where he gave an invited address. His students from the Zürich years include Eduard Stiefel, whose 1935 thesis introduced what became Stiefel–Whitney classes, James Stoker, Alice Roth, Werner Gysin, Hans Samelson, Beno Eckmann, Ernst Specker, Alfred Frölicher and Michel Kervaire. Earlier, in Berlin, Hans Freudenthal and Willi Rinow had written their theses under him. Friedrich Hirzebruch, formally a Münster student, worked with him in Zürich in 1949–50.

The war

Hopf remained a German citizen after moving to Switzerland. Under the Nazi racial laws his father’s Jewish origin made his status precarious. During the Second World War German authorities confiscated his property and pressed him to return; he applied for Swiss citizenship and became a Swiss citizen. He and Anja helped refugees and persecuted colleagues throughout these years. He declined an offer from Princeton in 1940 and, after the war, several further American offers.

In August 1946 he took part in a meeting at Oberwolfach, part of an effort to reconnect German mathematicians with the international community, and then spent six months in the United States. He made a further American visit in 1955–56, at Princeton, New York University and Stanford.

The IMU, honours and legacy

Hopf gave a plenary address at the 1950 International Congress in Cambridge, Massachusetts. From 1955 to 1958 he was president of the International Mathematical Union, the body that awards the Fields Medals. He received honorary doctorates from Princeton (1947), Freiburg, Manchester, Paris, the Free University of Brussels and Lausanne, and was elected to the US National Academy of Sciences (1957), the American Academy of Arts and Sciences (1961) and the American Philosophical Society (1963), and was a member of the Accademia dei Lincei. He became an honorary member of the London Mathematical Society in 1956 and received the Lobachevsky Prize in 1969.

He retired from ETH in 1965. Anja Hopf died in 1967. Heinz Hopf died on 3 June 1971 in Zollikon, near Zürich, aged 76. ETH now awards the Heinz Hopf Prize for outstanding work in pure mathematics.

Legacy

• Topology: the degree theorem and the classification of maps into spheres, the Lefschetz–Hopf theorem, the Hopf invariant and the Hopf fibration, which opened the study of homotopy groups of spheres.
• Algebra: Hopf algebras, H-spaces, and the first theorem of group homology, from which homological algebra developed.
• Geometry: the Killing–Hopf theorem, Poincaré–Hopf, Hopf–Rinow, the Umlaufsatz, the Hopf differential and the Hopf conjectures on positive curvature.
• Physics: the Hopf fibration appears as the Dirac monopole bundle, in the Bloch sphere of a qubit, and in knotted solutions of field equations.
• School: a line of students including Stiefel, Eckmann, Samelson, Kervaire and Hirzebruch, who in turn shaped topology, numerical analysis and geometry after the war.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Heinz Hopf, MacTutor History of Mathematics.
2. Heinz Hopf, Wikipedia.
3. B. Eckmann, Heinz Hopf, Dictionary of Scientific Biography, online at Encyclopedia.com.
4. Heinz Hopf, Mathematics Genealogy Project.
5. B. Eckmann, Heinz Hopf, memorial text, MacTutor Extras.
6. H. Hopf, Selecta Heinz Hopf, Springer, 1964; Collected Papers, ed. B. Eckmann, Springer, 2001.

Mathematics

1. H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95 (1926).
2. H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann. 96 (1927).
3. H. Hopf, Abbildungsklassen n-dimensionaler Mannigfaltigkeiten, Math. Ann. 96 (1927).
4. H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931).
5. H. Hopf and W. Rinow, Über den Begriff der vollständigen differentialgeometrischen Fläche, Comment. Math. Helv. 3 (1931).
6. H. Hopf, Die Klassen der Abbildungen der n-dimensionalen Polyeder auf die n-dimensionale Sphäre, Comment. Math. Helv. 5 (1933).
7. H. Hopf, Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25 (1935).
8. H. Hopf, Über die Drehung der Tangenten und Sehnen ebener Kurven, Compositio Math. 2 (1935).
9. P. Alexandroff and H. Hopf, Topologie I, Springer, 1935.
10. H. Hopf, Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, Ann. of Math. 42 (1941).
11. H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942).
12. H. Hopf, Differential Geometry in the Large, Lecture Notes in Mathematics 1000, Springer, 1983.
13. J. W. Milnor, Topology from the Differentiable Viewpoint, Princeton, 1965 (degree, Poincaré–Hopf, Hopf’s theorem).
14. J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965).
15. D. W. Lyons, An elementary introduction to the Hopf fibration, Math. Mag. 76 (2003).
16. H. Hopf, Enden offener Räume und unendliche diskontinuierliche Gruppen, Comment. Math. Helv. 16 (1944).