Eilenberg–Mazur Swindle
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Eilenberg–Mazur swindle, named after
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
and
Barry Mazur Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...
, is a method of proof that involves paradoxical properties of infinite sums. In
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
it was introduced by and is often called the Mazur swindle. In algebra it was introduced by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
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 In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of
knots A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Kn ...
or
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Example : A typical application of the Mazur swindle in geometric topology is the proof that the sum of two
non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group, topological space). The noun triviality usual ...
knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...
s ''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 In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus S^1\times D^2 into the 3-sphere. A knot is tame if and only if it can be represented as a fin ...
, not a tame knot. See 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 some dimensions, such as 7, there are examples of
exotic sphere In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of ...
s ''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 (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
. Example: A typical application of the Eilenberg swindle in algebra is the proof that if ''A'' is a
projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizati ...
over a ring ''R'' then there is a
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
''F'' with .Lam (1999), Corollary 2.7, p. 22; Eklof & Mekler (2002), Lemma 2.3, p. 9 To see this, choose a module ''B'' such that 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: 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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
such that ''X'' ≅ ''X'' ⊕ ''X'' (for example the direct sum of an infinite number of copies of any nonzero 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: 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 the restricted direct product of infinitely many copies of ''A'' ⨯ ''B''.


Other examples

The proof of the Cantor–Bernstein–Schroeder theorem might be seen as antecedent of the Eilenberg–Mazur swindle. In fact, the ideas are quite similar. If there are injections of sets from ''X'' to ''Y'' and from ''Y'' to ''X'', this means that formally we have and for some sets ''A'' and ''B'', where + means disjoint union and = means there is a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
between two sets. Expanding the former with the latter, : ''X'' = ''X'' + ''A'' + ''B''. In this bijection, let ''Z'' consist of those elements of the left hand side that correspond to an element of ''X'' on the right hand side. This bijection then expands to the bijection : ''X'' = ''A'' + ''B'' + ''A'' + ''B'' + ⋯ + ''Z''. Substituting the right hand side for ''X'' in ''Y'' = ''B'' + ''X'' gives the bijection : ''Y'' = ''B'' + ''A'' + ''B'' + ''A'' + ⋯ + ''Z''. Switching every adjacent pair ''B'' + ''A'' yields : ''Y'' = ''A'' + ''B'' + ''A'' + ''B'' + ⋯ + ''Z''. Composing the bijection for ''X'' with the inverse of the bijection for ''Y'' then yields : ''X'' = ''Y''. This argument depended on the bijections and as well as the well-definedness of infinite disjoint union.


Notes


References

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External links


Exposition by Terence Tao
on Mazur's swindle in topology {{DEFAULTSORT:Eilenberg-Mazur Swindle Knot theory Module theory