Holomorphic Logarithms and Roots



1. Motivation: Square Root of a Complex Number















So, what exactly is going on?

2. Square Roots of Holomorphic Functions










3. Covering Projections


More precisely, recall that there is a covering projection






Ques 1




since {f} has no poles.


Theorem 2 (Existence of Holomorphic {n}th Roots)



for every contour {\gamma} in {U}.

4. Complex Logarithms

The multivalued nature of the complex logarithm comes from the fact that




Ques 3



Theorem 4 (Existence of Logarithms)



for every contour {\gamma} in {U}.

5. Some Special Cases

The most common special case is

Corollary 5 (Nonvanishing Functions from Simply Connected Domains)



Theorem 6 (Branch Cut Functions)

There exist holomorphic functions


satisfying the obvious properties.



Example 7


Corollary 8


Facts about Lie Groups and Algebras


1. Lie groups


Definition 1





Proposition 2 (Reduction to connected Lie groups)



Proposition 3 (Reduction to simply connected Lie groups)


Here are some examples of Lie groups.

Example 4 (Examples of Lie groups)



Proposition 5


2. Haar measure

Recall the following result and its proof from representation theory:

Claim 6





Theorem 7 (Haar measure)




Example 8 (Examples of Haar measures)









Note that we have:

Corollary 9



Corollary 10 (Compact Lie groups are semisimple)


Indeed, we can now consider


as we described at the beginning.

3. The tangent space at the identity



Example 11 (Lie algebras corresponding to Lie groups)














By repeating this example in greater generality, we discover


4. The exponential map


Proposition 14 (Differential equations for Lie theorists)


\displaystyle \gamma_x : K \rightarrow G

which is a morphism of Lie groups, such that



This lets us define:

Definition 15




Example 16 (Exponential Maps of Lie algebras)

  • If {G = \mathbb R}, then {\mathfrak g = \mathbb R} too. Then {\gamma_x(t) = tx \in \mathbb R} (where {t \in \mathbb R}) is a morphism of Lie groups {\gamma_x : \mathbb R \rightarrow G}. Hence

    \displaystyle  \exp : \mathbb R \rightarrow \underbrace{\mathbb R}_{=G} \qquad \exp(x) = \gamma_x(1) = x \in \mathbb R = G.

    In other words, the exponential map is the identity.

  • Ditto for {\mathbb C}.


  • For {\mathop{\mathrm{GL}}(n, K)}, the map {\gamma_X : K \rightarrow \mathop{\mathrm{GL}}(n, K)} given by {t \mapsto e^{tX}} works nicely (now {X} is a matrix). (Note that we have to check {e^{tX}} is actually invertible for this map to be well-defined.) Hence the exponential map is given by

    \displaystyle  \exp : \mathfrak{gl}(n,K) \rightarrow \mathop{\mathrm{GL}}(n,K) \qquad \exp(X) = \gamma_X(1) = e^X \in \mathop{\mathrm{GL}}(n, K).

  • Similarly,

    \displaystyle  \exp : \mathfrak{sl}(n,K) \rightarrow \mathop{\mathrm{SL}}(n,K) \qquad \exp(X) = \gamma_X(1) = e^X \in \mathop{\mathrm{SL}}(n, K).

    Here we had to check that if {X \in \mathfrak{sl}(n,K)}, meaning {\mathop{\mathrm{Tr}} X = 0}, then {\det(e^X) = 1}. This can be seen by writing {X} in an upper triangular basis.





Here are a few more properties of {\exp}:


5. The commutator







Definition 17





Theorem 18

The bracket {[,]} satisfies the Jacobi identity







Example 19 (Examples of commutators attached to Lie groups)





Definition 20


A morphism of Lie algebras and preserves the bracket.


Example 21 (Associative algebra {\rightarrow} Lie algebra)


6. The fundamental theorems



we have described earlier, which is a functor

  • from the category of Lie groups
  • into the category of finite-dimensional Lie algebras.

The first theorem requires the following definition:

Definition 22



Theorem 23 (Lie I)




Theorem 24 (The Lie functor is an equivalence of categories)

Restrict {\mathscr{L}} to a functor





Algebraic Topology Functors

This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about {\mathsf{hPairTop}} at all.

In algebraic topology you (for example) associate every topological space {X} with a group, like {\pi_1(X, x_0)} or {H_5(X)}. All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.

The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup” work in algebraic topology.

1. Homology, {H_n : \mathsf{hTop} \rightarrow \mathsf{Grp}}

Note that {H_5} is a functor

\displaystyle H_5 : \mathsf{Top} \rightarrow \mathsf{Grp}

i.e. to every space {X} we can associate a group {H_5(X)}. (Of course, replace {5} by integer of your choice.) Recall that:

Definition 1

Two maps {f, g : X \rightarrow Y} are homotopy equivalent if there exists a homotopy between them.

Thus for a map we can take its homotopy class {[f]} (the equivalence class under this relationship). This has the nice property that {[f \circ g] = [f] \circ [g]} and so on.

Definition 2

Two spaces {X} and {Y} are homotopic if there exists a pair of maps {f : X \rightarrow Y} and {g : Y \rightarrow X} such that {[f \circ g] = [\mathrm{id}_X]} and {[g \circ f] = [\mathrm{id}_Y]}.

In light of this, we can define

Definition 3

The category {\mathsf{hTop}} is defined as follows:

  • The objects are topological spaces {X}.
  • The morphisms {X \rightarrow Y} are homotopy classes of continuous maps {X \rightarrow Y}.

Remark 4

Composition is well-defined since {[f \circ g] = [f] \circ [g]}. Two spaces are isomorphic in {\mathsf{hTop}} if they are homotopic.

Remark 5

As you might guess this “quotient” construction is called a quotient category.

Then the big result is that:

Theorem 6

The induced map {f_\sharp = H_n(f)} of a map {f: X \rightarrow Y} depends only on the homotopy class of {f}. Thus {H_n} is a functor

\displaystyle H_n : \mathsf{hTop} \rightarrow \mathsf{Grp}.

The proof of this is geometric, using the so-called prism operators. In any case, as with all functors we deduce

Corollary 7

{H_n(X) \cong H_n(Y)} if {X} and {Y} are homotopic.

In particular, the contractible spaces are those spaces {X} which are homotopy equivalent to a point. In which case, {H_n(X) = 0} for all {n \ge 1}.

2. Relative Homology, {H_n : \mathsf{hPairTop} \rightarrow \mathsf{Grp}}

In fact, we also defined homology groups

\displaystyle H_n(X,A)

for {A \subseteq X}. We will now show this is functorial too.

Definition 8

Let {\varnothing \neq A \subset X} and {\varnothing \neq B \subset X} be subspaces, and consider a map {f : X \rightarrow Y}. If {f(A) \subseteq B} we write

\displaystyle f : (X,A) \rightarrow (Y,B).

We say {f} is a map of pairs, between the pairs {(X,A)} and {(Y,B)}.

Definition 9

We say that {f,g : (X,A) \rightarrow (Y,B)} are pair-homotopic if they are “homotopic through maps of pairs”.

More formally, a pair-homotopy {f, g : (X,A) \rightarrow (Y,B)} is a map {F : [0,1] \times X \rightarrow Y}, which we’ll write as {F_t(X)}, such that {F} is a homotopy of the maps {f,g : X \rightarrow Y} and each {F_t} is itself a map of pairs.

Thus, we naturally arrive at two categories:

  • {\mathsf{PairTop}}, the category of pairs of toplogical spaces, and
  • {\mathsf{hPairTop}}, the same category except with maps only equivalent up to homotopy.

Definition 10

As before, we say pairs {(X,A)} and {(Y,B)} are pair-homotopy equivalent if they are isomorphic in {\mathsf{hPairTop}}. An isomorphism of {\mathsf{hPairTop}} is a pair-homotopy equivalence.

Then, the prism operators now let us derive

Theorem 11

We have a functor

\displaystyle H_n : \mathsf{hPairTop} \rightarrow \mathsf{Grp}.

The usual corollaries apply.

Now, we want an analog of contractible spaces for our pairs: i.e. pairs of spaces {(X,A)} such that {H_n(X,A) = 0} for {n \ge 1}. The correct definition is:

Definition 12

Let {A \subset X}. We say that {A} is a deformation retract of {X} if there is a map of pairs {r : (X, A) \rightarrow (A, A)} which is a pair homotopy equivalence.

Example 13 (Examples of Deformation Retracts)

  1. If a single point {p} is a deformation retract of a space {X}, then {X} is contractible, since the retraction {r : X \rightarrow \{\ast\}} (when viewed as a map {X \rightarrow X}) is homotopic to the identity map {\mathrm{id}_X : X \rightarrow X}.
  2. The punctured disk {D^2 \setminus \{0\}} deformation retracts onto its boundary {S^1}.
  3. More generally, {D^{n} \setminus \{0\}} deformation retracts onto its boundary {S^{n-1}}.
  4. Similarly, {\mathbb R^n \setminus \{0\}} deformation retracts onto a sphere {S^{n-1}}.

Of course in this situation we have that

\displaystyle H_n(X,A) \cong H_n(A,A) = 0.

3. Homotopy, {\pi_1 : \mathsf{hTop}_\ast \rightarrow \mathsf{Grp}}

As a special case of the above, we define

Definition 14

The category {\mathsf{Top}_\ast} is defined as follows:

  • The objects are pairs {(X, x_0)} of spaces {X} with a distinguished basepoint {x_0}. We call these pointed spaces.
  • The morphisms are maps {f : (X, x_0) \rightarrow (Y, y_0)}, meaning {f} is continuous and {f(x_0) = y_0}.

Now again we mod out:

Definition 15

Two maps {f , g : (X, x_0) \rightarrow (Y, y_0)} of pointed spaces are homotopic if there is a homotopy between them which also fixes the basepoints. We can then, in the same way as before, define the quotient category {\mathsf{hTop}_\ast}.

And lo and behold:

Theorem 16

We have a functor

\displaystyle \pi_1 : \mathsf{hTop}_\ast \rightarrow \mathsf{Grp}.

Same corollaries as before.

Uniqueness of Solutions for DiffEq’s

Let {V} be a normed finite-dimensional real vector space and let {U \subseteq V} be an open set. A vector field on {U} is a function {\xi : U \rightarrow V}. (In the words of Gaitsgory: “you should imagine a vector field as a domain, and at every point there is a little vector growing out of it.”)

The idea of a differential equation is as follows. Imagine your vector field specifies a velocity at each point. So you initially place a particle somewhere in {U}, and then let it move freely, guided by the arrows in the vector field. (There are plenty of good pictures online.) Intuitively, for nice {\xi} it should be the case that the trajectory resulting is unique. This is the main take-away; the proof itself is just for completeness.

This is a so-called differential equation:

Definition 1

Let {\gamma : (-\varepsilon, \varepsilon) \rightarrow U} be a continuous path. We say {\gamma} is a solution to the differential equation defined by {\xi} if for each {t \in (-\varepsilon, \varepsilon)} we have

\displaystyle  \gamma'(t) = \xi(\gamma(t)).

Example 2 (Examples of DE’s)

Let {U = V = \mathbb R}.

  1. Consider the vector field {\xi(x) = 1}. Then the solutions {\gamma} are just {\gamma(t) = t+c}.
  2. Consider the vector field {\xi(x) = x}. Then {\gamma} is a solution exactly when {\gamma'(t) = \gamma(t)}. It’s well-known that {\gamma(t) = c\exp(t)}.

Of course, you may be used to seeing differential equations which are time-dependent: i.e. something like {\gamma'(t) = t}, for example. In fact, you can hack this to fit in the current model using the idea that time is itself just a dimension. Suppose we want to model {\gamma'(t) = F(\gamma(t), t)}. Then we instead consider

\displaystyle  \xi : V \times \mathbb R \rightarrow V \times \mathbb R \qquad\text{by}\qquad \xi(v, t) = (F(v,t), 1)

and solve the resulting differential equation over {V \times \mathbb R}. This does exactly what we want. Geometrically, this means making time into another dimension and imagining that our particle moves at a “constant speed through time”.

The task is then mainly about finding which conditions guarantee that our differential equation behaves nicely. The answer turns out to be:

Definition 3

The vector field {\xi : U \rightarrow V} satisfies the Lipschitz condition if

\displaystyle  \left\lVert \xi(x')-\xi(x'') \right\rVert \le \Lambda \left\lVert x'-x'' \right\rVert

holds identically for some fixed constant {\Lambda}.

Note that continuously differentiable implies Lipschitz.

Theorem 4 (Picard-Lindelöf)

Let {V} be a finite-dimensional real vector space, and let {\xi} be a vector field on a domain {U \subseteq V} which satisfies the Lipschitz condition.

Then for every {x_0 \in U} there exists {(-\varepsilon,\varepsilon)} and {\gamma : (-\varepsilon,\varepsilon) \rightarrow U} such that {\gamma'(t) = \xi(\gamma(t))} and {\gamma(0) = x_0}. Moreover, if {\gamma_1} and {\gamma_2} are two solutions and {\gamma_1(t) = \gamma_2(t)} for some {t}, then {\gamma_1 = \gamma_2}.

In fact, Peano’s existence theorem says that if we replace Lipschitz continuity with just continuity, then {\gamma} exists but need not be unique. For example:

Example 5 (Counterexample if {\xi} is not differentiable)

Let {U = V = \mathbb R} and consider {\xi(x) = x^{\frac23}}, with {x_0 = 0}. Then {\gamma(t) = 0} and {\gamma(t) = \left( t/3 \right)^3} are both solutions to the differential equation

\displaystyle  \gamma'(t) = \gamma(t)^{\frac 23}.

Now, for the proof of the main theorem. The main idea is the following result (sometimes called the contraction principle).

Lemma 6 (Banach Fixed-Point Theorem)

Let {(X,d)} be a complete metric space. Let {f : X \rightarrow X} be a map such that {d(f(x_1), f(x_2)) < \frac{1}{2} d(x_1, x_2)} for any {x_1, x_2 \in X}. Then {f} has a unique fixed point.

For the proof of the main theorem, we are given {x_0 \in V}. Let {X} be the metric space of continuous functions from {(-\varepsilon, \varepsilon)} to the complete metric space {\overline{B}(x_0, r)} which is the closed ball of radius {r} centered at {x_0}. (Here {r > 0} can be arbitrary, so long as it stays in {U}.) It turns out that {X} is itself a complete metric space when equipped with the sup norm

\displaystyle  d(f, g) = \sup_{t \in (-\varepsilon, \varepsilon)} \left\lVert f(t)-g(t) \right\rVert.

This is well-defined since {\overline{B}(x_0, r)} is compact.

We wish to use the Banach theorem on {X}, so we’ll rig a function {\Phi : X \rightarrow X} with the property that its fixed points are solutions to the differential equation. Define it by, for every {\gamma \in X},

\displaystyle  \Phi(\gamma) : t \mapsto x_0 + \int_0^t \xi(\gamma(s)) \; ds.

This function is contrived so that {(\Phi\gamma)(0) = x_0} and {\Phi\gamma} is both continuous and differentiable. By the Fundamental Theorem of Calculus, the derivative is exhibited by

\displaystyle  (\Phi\gamma)'(t) = \left( \int_0^t \xi(\gamma(s)) \; ds \right)' = \xi(\gamma(t)).

In particular, fixed points correspond exactly to solutions to our differential equation.

A priori this output has signature {\Phi\gamma : (-\varepsilon,\varepsilon) \rightarrow V}, so we need to check that {\Phi\gamma(t) \in \overline{B}(x_0, r)}. We can check that

\displaystyle  \begin{aligned} \left\lVert (\Phi\gamma)(t) - x_0 \right\rVert &=\left\lVert \int_0^t \xi(\gamma(s)) \; ds \right\rVert \\ &\le \int_0^t \left\lVert \xi(\gamma(s)) \; ds \right\rVert \\ &\le t \max_{s \in [0,t]} \left\lVert \xi\gamma(s) \right\rVert \\ &< \varepsilon \cdot A \end{aligned}

where {A = \max_{x \in \overline{B}(x_0,r)} \left\lVert \xi(x) \right\rVert}; we have {A < \infty} since {\overline{B}(x_0,r)} is compact. Hence by selecting {\varepsilon < r/A}, the above is bounded by {r}, so {\Phi\gamma} indeed maps into {\overline{B}(x_0, r)}. (Note that at this point we have not used the Lipschitz condition, only that {\xi} is continuous.)

It remains to show that {\Phi} is contracting. Write

\displaystyle  \begin{aligned} \left\lVert (\Phi\gamma_1)(t) - (\Phi\gamma_2)(t) \right\rVert &= \left\lVert \int_{s \in [0,t]} \left( \xi(\gamma_1(s))-\xi(\gamma_2(s)) \right) \right\rVert \\ &= \int_{s \in [0,t]} \left\lVert \xi(\gamma_1(s))-\xi(\gamma_2(s)) \right\rVert \\ &\le t\Lambda \sup_{s \in [0,t]} \left\lVert \gamma_1(s)-\gamma_2(s) \right\rVert \\ &< \varepsilon\Lambda \sup_{s \in [0,t]} \left\lVert \gamma_1(s)-\gamma_2(s) \right\rVert \\ &= \varepsilon\Lambda d(\gamma_1, \gamma_2) . \end{aligned}

Hence once again for {\varepsilon} sufficiently small we get {\varepsilon\Lambda \le \frac{1}{2}}. Since the above holds identically for {t}, this implies

\displaystyle  d(\Phi\gamma_1, \Phi\gamma_2) \le \frac{1}{2} d(\gamma_1, \gamma_2)

as needed.

This is a cleaned-up version of a portion of a lecture from Math 55b in Spring 2015, instructed by Dennis Gaitsgory.

Constructing the Tangent and Cotangent Space







Definition 1




Definition 2




This suggests a third equivalent definition: suppose we define








  • Does not rely on any kind of parametrization, and



Theorem 3




Thus, it follows that there is an isomorphism

