Celebratio Mathematica — Kister

James Milton Kister

by Rob Kirby

James M. Kister was born on 29 June 1930 in Cleveland, Ohio to James Leonard Kister and Katherine Sherrick Kister. Jim’s paternal grandfather came to Ohio from Pennsylvania in 1840, found land with a stream; built a dam, waterwheel and mill; and then ground wheat and corn and made cider.

Jim’s father, JLK, did not finish high school, but had a talent with machinery. In 1908 his father bought a car in Chicago, which was then disassembled, boxed, and shipped by train to Millbrook, Ohio (named for the Kister mill). JLK and his father took a buggy to the train station where JLK reassembled the car and drove it home! (That seems an odd way to get a car from Chicago to Ohio, even in 1908.) JLK went on to work as an inspector of cars and trucks for White Motor Company (later bought by Mack) in Cleveland.

Jim lived in Cleveland through high school, winning prizes in math and physics. He went to Wooster College (Ohio) from 1948–52, winning the Rensselaer Prize in math. Melcher Fobes, a Hassler Whitney student at the time, was the main math teacher at Wooster, and encouraged Jim to go to Harvard for graduate work.

Jim started well at Harvard in 1952, but fell ill with a lung infection and had to leave Harvard before Thanksgiving and go to live in Tucson for its drier air. After a few months Los Alamos called with a job offer (George Birkhoff helped). Jim worked there for almost two years, with a three month break when the lower lobe of his left lung was removed.

On the advice of Stanislaw Ulam and C. J. Everett, and for the better air, Jim began graduate school at the University of Wisconsin in Madison in fall, 1955. That semester Jim took a point set topology course from Bob Williams (a Gordon Whyburn Ph.D. at the University of Virginia) and this got Jim interested in topology.

Jim met Sue Spence in summer 1955. They were married in 1956 and daughter Karen was born in December 1957. A Wisconsin fellowship only paid \$1500 so the young couple went to Los Alamos in summer 1956 to make extra money.

Ulam wanted Jim to join his team to work on a chess playing program. This seemed too formidable so the chess board was shrunk to $ 6 \times 6 $ and the rooks were eliminated. They wrote a program and got a good chess player, Martin Kruskal, to test the program. Kruskal spotted the program a queen, and then did not have an easy time beating it. Jim gave a talk on the chess playing program before a large audience at a meeting of the Association for Computing Machinery in Los Angeles, and published a paper [1]. A couple of courses from R H Bing in 1956–57 led to Bing becoming Jim’s Ph.D. advisor. Bing was away at the Institute for Advanced Study in 1957–58 and when he returned he suggested that Jim work on the Side Approximation Theorem (later proved by Bing [e4]). Instead Jim, in early 1959, saw that the space of bounded homeomorphisms (with the compact-open topology) of $ \mathbb{R}^n $ is contractible, for any $ n $. (Bounded means that there is some constant $ K $ such that the homeomorphism does not move any point more than distance $ K $.) The proof took one page, and can be seen as a canonical version of the Alexander isotopy [e1] which shows that a homeomorphism of the $ n $-ball, $ B^n $, which is equal to the identity on the bounding $ (n-1) $-sphere, is isotopic to the identity. Jim saw that if one squeezes the homeomorphism of $ \mathbb{R}^n $ into a homeomorphism of $ B^n $, then it will be equal to the identity on the boundary. In his Math Review, S. S. Cairns called the proof “remarkably efficient”, which it certainly is.

Bing reportedly asked the authorities whether the University would accept a one page thesis for a Ph.D. The answer is unknown and irrelevant for Jim also at the same time proved that the space of homeomorphisms of a 3-manifold is locally connected [2]. Jim put his bounded homeomorphisms theorem into [2], which had 3-manifolds in the title, but no mention of higher dimensions. That led to my overlooking the result for many decades, and led me to cite instead a paper of Ed Connell [e5] in my paper [e7] using the torus trick to prove local contractibility for all homeomorphisms (not just bounded) of $ \mathbb{R}^n $. Jim’s result is key to my theorem, and it’s embarrassing that my acquaintances and I didn’t know of its existence!

In 1963 Jim proved a very important theorem, that a topological manifold has a tangent bundle [6]. Milnor had defined the notion of a microbundle and showed that microbundles had many of the useful properties of actual bundles [e6]. Jim showed that every microbundle (over a locally finite, finite dimensional complex) contains a bundle (with fiber $ \mathbb{R}^n $ and group equal to the space of homeomorphisms of $ \mathbb{R}^n $ with the compact open topology). The key to the proof is the theorem stating that the space of homeomorphisms of $ \mathbb{R}^n $ is a weak deformation retract of the space of embeddings of $ \mathbb{R}^n $ into $ \mathbb{R}^n $.

Kervaire called the proof, “elementary but delicate”, in his Math Review, which is accurate but misses the fact that many geometric proofs from that era looked elementary, but only after the fact.

If $ T $ is a homeomorphism of period $ r $ on Euclidean space $ E^n $, then P. A. Smith [e2] proved that $ T $ has a fixed point if $ r $ is prime or a prime power. Then Conner and Floyd [e3] gave examples when $ r $ is not a prime power of contractible spaces with periodic maps with no fixed points.

Jim wrote two papers [3] [5] giving counterexamples for all $ r $ which are not prime powers. The second paper constructs a map on a contractible infinite complex $ K $ which is then embedded equivariantly in $ E^n $ for $ n > 7 $, and then a regular neighborhood of $ K $ is the desired Euclidean space. The map can be made differentiable, and these examples are nearly best possible, the only case left open was $ n=7 $. The case $ n=7 $ has been settled: R. Oliver announced fixed point free maps in [Notices of the AMS, 26 (1979), no. 2, A251], but details were never published. In [e8] the four authors give the relevant details of the $ n=7 $ proof.

Jim and Russ McMillan constructed [4] a contractible open manifold $ M $ which cannot be embedded in $ E^3 $. All previous such examples were embeddable in $ E^3 $. However, $ M \times E^1 $ and $ E^4 $ are combinatorially equivalent.

While on sabbatical at Oxford University in 1977, Jim met Jane Bridge, a mathematical logician, tenured at Somerville College of Oxford University. She was a student of Robin Gandy and thus a granddaughter of Alan Turing. Jane and Jim were married in 1978 and returned to Ann Arbor where Jane became an editor of Math Reviews. In 1998 she became the Executive Director of MR.

I met Jim in 1966–67 when he spent a year at UCLA. He gave a course on topological bundles and his notes from that course and my notes from a course two years later on triangulations of topological manifolds led us to plan a book together. Finishing this book kept getting put off — even though it is now 90% done. It should have been finished before Jim died, and will be soon.

Jim was a very good friend, a good athlete (despite being plagued by bronchiectasis all his life) and in particular was a cagey tennis player.

References

[e1] J. W. Alexander: “On the deformation of an $ n $ cell,” Proc. Natl. Acad. Sci. USA 9 : 12 (1923), pp. 406–7. article

[e2] P. Smith: “Fixed-point theorems for periodic transformations,” Amer. J. Math. 63 (1941), pp. 1–8. MR 3199 article

[e3] P. E. Conner and E. E. Floyd: “On the construction of periodic maps without fixed points,” Proc. Amer. Math. Soc. 10 (1959), pp. 354–360. MR 105115 article

[e4]R. H. Bing: “Approximating surfaces from the side,” Ann. Math. (2) 77 : 1 (January 1963), pp. 145–192. MR 0150744 Zbl 0115.40603 article

[e5] E. H. Connell: “Approximating stable homeomorphisms by piecewise linear ones,” Ann. of Math. (2) 78 (1963), pp. 326–338. MR 154289 article

[e6]J. Milnor: “Microbundles, I,” Topology 3 : suppl. 1 (1964), pp. 53–80. MR 0161346 article

[e7]R. C. Kirby: “Stable homeomorphisms and the annulus conjecture,” Ann. of Math. (2) 89 (1969), pp. 575–582. MR 0242165 Zbl 0176.22004

[e8] R. Haynes, S. Kwasik, J. Mast, and R. Schultz: “Periodic maps on $ \mathbb{R}^7 $ without fixed points,” Math. Proc. Cambridge Philos. Soc. 132 : 1 (2002), pp. 131–136. MR 1866329 article

Works

[1] J. Kister, P. Stein, S. Ulam, W. Walden, and M. Wells: “Experiments in chess,” Jour. Assoc. Computing Machinery 4 (1957), pp. 174–177. article

Stanisław Marcin Ulam	Related
Mark Brimhall Wells	Related
Paul Stein	Related

[2] J. Kister: “Small isotopies in Euclidean spaces and 3-manifolds,” Bull. Am. Math. Soc. 65 : 6 (1959), pp. 371–373. MR 107232 Zbl 0089.39502 article

[3] J. M. Kister: “Examples of periodic maps on Euclidean spaces without fixed points,” Bull. Am. Math. Soc. 67 : 5 (1961), pp. 471–474. MR 130929 Zbl 0101.15602 article

Let $ T $ be a map of period $ r $ on a Euclidean space $ E^n $. Smith seems to have been the first to consider fixed points of $ T $. He showed that $ T $ has a fixed point if $ r $ is a prime in [Smith 1934], extended this result to $ r $ a power of a prime, and raised the question concerning the existence of a fixed point for $ r $ not a prime power in [Smith 1941]; also cf. Problem 33 in [Eilenberg 1949]. Conner and Floyd gave an example of a contractible manifold $ M_r $ for every $ r $ not a prime power, and a map $ T $ of period $ r $ on $ M_r $ without fixed points [Conner and Floyd 1959]. They conjectured that $ M_r $ was a Euclidean space. This note shows that a slight modification of their example is Euclidean, hence:

If $ r $ is an integer which is not a power of a prime, then there exists a triangulation $ \tau $ of $ E^{9r} $, a map $ T $ of period $ r $ on $ E^{9r} $ without fixed points, and $ T $ is simplicial relative to $ \tau $.

[4] J. M. Kister and D. R. McMillan, Jr.: “Locally euclidean factors of $ E^4 $ which cannot be imbedded in $ E^3 $,” Ann. Math. (2) 76 : 3 (November 1962), pp. 541–546. MR 144322 Zbl 0115.40703 article

[5] J. M. Kister: “Differentiable periodic actions on $ E^8 $ without fixed points,” Am. J. Math. 85 : 2 (April 1963), pp. 316–319. MR 154278 Zbl 0119.18801 article

[6]J. M. Kister: “Microbundles are fibre bundles,” Ann. of Math. (2) 80 (1964), pp. 190–199. MR 0180986 Zbl 0131.20602 article

[7] J. Cheeger and J. M. Kister: “Counting topological manifolds,” Topology 9 : 2 (May 1970), pp. 149–151. MR 256399 Zbl 0199.58403 article

We consider the class $ \mathscr{C} $ of all compact topological manifolds, boundaries permitted. It is known that there are only a countable number of homotopy types in $ \mathscr{C} $, [Mather 1965] and [Kister 1968]. The subclass $ \mathscr{C}_{PL} $, of piecewise linear manifolds has only a countable number of topological distinct elements, since each could be regarded as a finite simplicial complex and a simple argument shows there are only a countable number of those, up to isomorphism. Recently, however, Kirby and Siebenmann [1969] have discovered some examples of topological manifolds admitting no $ PL $ structure, so that route for counting homeomorphism types in $ \mathscr{C} $ is rather unpromising (whether manifolds can be triangulated without a $ PL $ structure is still open).

We take a direct approach and with the aid of a very useful result of Edwards and Kirby [1971] examine overlapping coordinate neighborhoods to show that $ \mathscr{C} $ has only countably many elements up to homeomorphism. Our argument is similar to one in a differential setting in [Kirby and Siebenmann 1969].