Eilenberg–Mazur swindle

In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums. In geometric topology it was introduced by Mazur (1959, 1961) and is often called the Mazur swindle. In algebra it was introduced by Samuel Eilenberg and is known as the Eilenberg swindle or Eilenberg telescope (see telescoping sum).

The Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0:

1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 1) + ... = 0

This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge, but the analogous argument can be used in some contexts where there is some sort of "addition" defined on some objects for which infinite sums do make sense, to show that if A + B = 0 then A = B = 0.

Mazur swindle

In geometric topology the addition used in the swindle is usually the connected sum of knots or manifolds.

Example (Rolfsen 1990, chapter 4B): A typical application of the Mazur swindle in geometric topology is the proof that the sum of two non-trivial knots A and B is non-trivial. For knots it is possible to take infinite sums by making the knots smaller and smaller, so if A + B is trivial then

$A=A+(B+A)+(B+A)+\cdots = (A+B)+(A+B)+\cdots=0\,$

so A is trivial (and B by a similar argument). The infinite sum of knots is usually a wild knot, not a tame knot. See (Poénaru 2007) for more geometric examples.

Example: The oriented n-manifolds have an addition operation given by connected sum, with 0 the n-sphere. If A + B is the n-sphere, then A + B + A + B + ... is Euclidean space so the Mazur swindle shows that the connected sum of A and Euclidean space is Euclidean space, which shows that A is the 1-point compactification of Euclidean space and therefore A is homeomorphic to the n-sphere. (This does not show in the case of smooth manifolds that A is diffeomorphic to the n-sphere, and in most dimensions at least 7 there are examples of exotic spheres A with inverses that are not diffeomorphic to the standard n-sphere.)

Eilenberg swindle

In algebra the addition used in the swindle is usually the direct sum of modules over a ring.

Example: A typical application of the Eilenberg swindle in algebra is the proof that if A is a projective module over a ring R then there is a free module with A + F = F.^[1] To see this, choose a module B such that A + B is free, which can be done as A is projective, and put

F = B + A + B + A + B + ....

so that

A + F = A + (B + A) + (B + A) + ... = (A + B) + (A + B) + ... = F.

Example: (Eisenbud 1995, p.121) Finitely generated free modules over commutative rings R have a well-defined natural number as their dimension which is additive under direct sums, and are isomorphic if and only if they have the same dimension. This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows. Let X be an abelian group such that X = X + X (for example the direct sum of an infinite number of copies of an abelian group), and let R be the ring of endomorphisms of X. Then the left R-module R is isomorphic to the left R-module R + R.

Example: (Lam 2003, Exercise 8.16) If A and B are any groups then the Eilenberg swindle can be used to construct a ring R such that the group rings R[A] and R[B] are isomorphic rings: take R to be the group ring of A + B + A + B + ...

Other examples

The proof of the Cantor–Bernstein–Schroeder theorem uses a similar idea. If there are injections of sets from X to Y and from Y to X, this means that formally we have X=Y+A and Y=X+B for some sets A and B, where + means disjoint union and = means there is a bijection between two sets. Then

X=A+B+A+B+...+Z

and

Y=B+A+B+A+...+Z

for some set Z, which shows that X=Y.

Notes

1. ^ Lam (1999), Corollary 2.7, p. 22; Eklof & Mekler (2002), Lemma 2.3, p. 9.

References

• Bass, Hyman (1963), "Big projective modules are free", Illinois J. Math. 7: 24–31, MR0143789, http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1255637479 
• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, New York: Springer-Verlag, pp. xvi+785, ISBN 0-387-94268-8, MR1322960 
• Eklof, Paul C.; Mekler, Alan H. (2002), Almost free modules: set-theoretic models, Elsevier, ISBN 0444504923 
• Lam, Tsit-Yuen (2003), Exercises in Classical Ring Theory, New York, NY: Springer, ISBN 978-0387005003 
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Springer, ISBN 0387984283 
• Mazur, Barry (1959), "On the structure of certain semi-groups of spherical knot classes", Publications Mathématiques de l'IHÉS 3: 19–27, MR0116347, http://www.numdam.org/item?id=PMIHES_1959__3__19_0 
• Mazur, B. C. (1961), "On embeddings of spheres", Acta Math. 105: 1–17, doi:10.1007/BF02559532, MR0125570 
• Poénaru, Valentin (2007), "What is ... an infinite swindle?", Notices Amer. Math. Soc. 54 (5): 619–622, MR2311984, http://www.ams.org/notices/200705/what-is-poenaru-web.pdf 
• Rolfsen, Dale (1990), Knots and links. Corrected reprint of the 1976 original., Mathematics Lecture Series, 7, Houston, TX,: Publish or Perish, Inc., pp. xiv+439, ISBN 0-914098-16-0, MR1277811 

External links

• Talk by John Stallings on Mazur's swindle in topology
• Exposition by Terence Tao on Mazur's swindle in topology