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We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute (or at least transform) a large number of contour integrals even without knowing any explicit antiderivative of . There are many forms and variants of Cauchy’s theorem. To give one such version, we need the basic topological notion of a homotopy:
Definition 1 (Homotopy) Let be an open subset of , and let , be two curves in .
- (i) If have the same initial point and final point , we say that and are homotopic with fixed endpoints in if there exists a continuous map such that and for all , and such that and for all .
- (ii) If are closed (but possibly with different initial points), we say that and are homotopic as closed curves in if there exists a continuous map such that and for all , and such that for all .
- (iii) If and are curves with the same initial point and same final point, we say that and are homotopic with fixed endpoints up to reparameterisation in if there is a reparameterisation of which is homotopic with fixed endpoints in to a reparameterisation of .
- (iv) If and are closed curves, we say that and are homotopic as closed curves up to reparameterisation in if there is a reparameterisation of which is homotopic as closed curves in to a reparameterisation of .
In the first two cases, the map will be referred to as a homotopy from to , and we will also say that can be continously deformed to (either with fixed endpoints, or as closed curves).
For a similar reason, in a convex open set , any two closed curves will be homotopic to each other as closed curves.
- (i) Prove that the property of being homotopic with fixed endpoints in is an equivalence relation.
- (ii) Prove that the property of being homotopic as closed curves in is an equivalence relation.
- (iii) If are closed curves with the same initial point, show that is homotopic to with fixed endpoints if and only if is homotopic to as closed curves.
- (iv) Define a point in to be a curve of the form for some and all . Let be a closed curve in . Show that is homotopic with fixed endpoints to a point in if and only if is homotopic as a closed curve to a point in . (In either case, we will call homotopic to a point, null-homotopic, or contractible to a point in .)
- (v) If are curves with the same initial point and the same terminal point, show that is homotopic to with fixed endpoints in if and only if is homotopic to a point in .
- (vi) If is connected, and are any two curves in , show that there exists a continuous map such that and for all . Thus the notion of homotopy becomes rather trivial if one does not fix the endpoints or require the curve to be closed.
- (vii) Show that if is a reparameterisation of , then and are homotopic with fixed endpoints in U.
- (viii) Prove that the property of being homotopic with fixed endpoints in up to reparameterisation is an equivalence relation.
- (ix) Prove that the property of being homotopic as closed curves in up to reparameterisation is an equivalence relation.
We can then phrase Cauchy’s theorem as an assertion that contour integration on holomorphic functions is a homotopy invariant. More precisely:
- (i) If and are rectifiable curves that are homotopic in with fixed endpoints up to reparameterisation, then
- (ii) If and are closed rectifiable curves that are homotopic in as closed curves up to reparameterisation, then
This version of Cauchy’s theorem is particularly useful for applications, as it explicitly brings into play the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate. This formulation of Cauchy’s theorem also highlights the close relationship between contour integrals and the algebraic topology of the complex plane (and open subsets thereof). Setting to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):
An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on . The conclusion of Cauchy’s theorem only involves the values of a function on the images of the two curves . However, in order for the hypotheses of Cauchy’s theorem to apply, the function must be holomorphic not only on the images on , but on an open set that is large enough (and sufficiently free of “holes”) to support a homotopy between the two curves. This point can be emphasised through the following fundamental near-counterexample to Cauchy’s theorem:
As a consequence of this and Cauchy’s theorem, we conclude that the contour is not contractible to a point in ; note that this does not contradict Example 2 because is not convex. Thus we see that the lack of holomorphicity (or singularity) of at the origin can be “blamed” for the non-vanishing of the integral of on the closed contour , even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of , not just the behaviour in the local neighbourhood of , has an impact on the contour integral.
One can of course rewrite this example to involve non-closed contours instead of closed ones. For instance, if we let denote the half-circle contours and , then are both contours in from to , but one has
In order for this to be consistent with Cauchy’s theorem, we conclude that and are not homotopic in (even after reparameterisation).
In the specific case of functions of the form , or more generally for some point and some that is holomorphic in some neighbourhood of , we can quantify the precise failure of Cauchy’s theorem through the Cauchy integral formula, and through the concept of a winding number. These turn out to be extremely powerful tools for understanding both the nature of holomorphic functions and the topology of open subsets of the complex plane, as we shall see in this and later notes.
Jordan’s theorem is a basic theorem in the theory of finite linear groups, and can be formulated as follows:
Informally, Jordan’s theorem asserts that finite linear groups over the complex numbers are almost abelian. The theorem can be extended to other fields of characteristic zero, and also to fields of positive characteristic so long as the characteristic does not divide the order of , but we will not consider these generalisations here. A proof of this theorem can be found for instance in these lecture notes of mine.
I recently learned (from this comment of Kevin Ventullo) that the finiteness hypothesis on the group in this theorem can be relaxed to the significantly weaker condition of periodicity. Recall that a group is periodic if all elements are of finite order. Jordan’s theorem with “finite” replaced by “periodic” is known as the Jordan-Schur theorem.
The Jordan-Schur theorem can be quickly deduced from Jordan’s theorem, and the following result of Schur:
Theorem 2 (Schur’s theorem) Every finitely generated periodic subgroup of a general linear group is finite. (Equivalently, every periodic linear group is locally finite.)
Remark 1 The question of whether all finitely generated periodic subgroups (not necessarily linear in nature) were finite was known as the Burnside problem; the answer was shown to be negative by Golod and Shafarevich in 1964.
Let us see how Jordan’s theorem and Schur’s theorem combine via a compactness argument to form the Jordan-Schur theorem. Let be a periodic subgroup of . Then for every finite subset of , the group generated by is finite by Theorem 2. Applying Jordan’s theorem, contains an abelian subgroup of index at most .
In particular, given any finite number of finite subsets of , we can find abelian subgroups of respectively such that each has index at most in . We claim that we may furthermore impose the compatibility condition whenever . To see this, we set , locate an abelian subgroup of of index at most , and then set . As is covered by at most cosets of , we see that is covered by at most cosets of , and the claim follows.
Note that for each , the set of possible is finite, and so the product space of all configurations , as ranges over finite subsets of , is compact by Tychonoff’s theorem. Using the finite intersection property, we may thus locate a subgroup of of index at most for all finite subsets of , obeying the compatibility condition whenever . If we then set , where ranges over all finite subsets of , we then easily verify that is abelian and has index at most in , as required.
Below I record a proof of Schur’s theorem, which I extracted from this book of Wehrfritz. This was primarily an exercise for my own benefit, but perhaps it may be of interest to some other readers.