The topology of Lie groups

Miguel Santos

February 8, 2018

1 Introduction

The following theorem shows that, in order to understand the topology of Liegroups, it suffices to understand the topology of compact Lie groups.

Theorem 1.1. Every Lie group G contains a maximal compact subgroup K,which is unique up to conjugacy. Moreover, G is diffeomorphic to K ×E, withE an Euclidean space.

Corollary 1.2. Every Lie group G has the homotopy type of a compact Liegroup.

A proof can be found in [3].In theory, the topology of Lie groups could be understood from Theorem

1.1, together with the following facts:

• Lie’s theorems;

• Lie algebras of compact Lie groups are the direct product of an abelianand a semisimple Lie algebra with negative definite Killing form;

• The classification of (complex) semisimple Lie algebras.

In practice, this is not so easy. Although some information about the topologyof the compact semisimple Lie groups has been known for decades (like theirBetti numbers), other information is yet to be fully understood, like their Hopfalgebra structure. Understanding the topology of the classical and exceptionalLie groups is an ongoing field of research (e.g. [4]).

General results on the topology of Lie groups, or compact Lie groups, arethus desirable. In this text we present the proof of two such results:

1. The cohomology ring H∗(G;K) with coefficients in a characteristic 0 fieldis isomorphic to an exterior algebra on odd degree generators;

2. The second homotopy group π2(G) is 0.

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For the first result we present the usual proof with Hopf algebras and theirclassification, and for the second result we follow Bott’s proof, as presented in[5].

This report is the result of a project developed for the Lie Groups and LieAlgebras course taught in the 1st semester of 2017/2018 by professor GustavoGranja, which culminated in a presentation given out in February 1.

2 The torsion-free homology and cohomology ofLie groups

For the remainder of this section, let K be a field of characteristic 0 and R bea commutative ring.

By the Kunneth formula, the cohomology ring

H∗

(m∏i=1

S2di+1;R

)∼=

m⊗i=1

H∗(S2di+1;R

) ∼= m⊗i=1

ΛR(xi) ∼= ΛR(x1, . . . , xm),

where xi has degree di for i = 1, . . . ,m.

Definition 2.1. Let X be a topological space and R be a commutative ring. Wesay that X has the R-cohomology of a product of odd-dimensional spheres if itscohomology ring H∗(X;R) is isomorphic to an exterior algebra on odd-degreegenerators with coefficients in R.

In this section the goal is to prove the following theorem.

Theorem 2.2. Let G be a connected Lie group. Then G has the K-cohomologyof a product of odd-dimensional spheres.

Remark 2.3. If G0 is the connected component of the identity, then

H∗(G;K) =⊕

g∈G/G0

H∗(G0;K)

so Theorem 2.2 readily gives a result about the cohomology of Lie groups.

The proof is done in the context of a connected H-space X, and follows twosteps:

1. Showing that H∗(X;K) is a connected Hopf algebra;

2. Classifying finite dimensional connected graded-commutative associativeHopf algebras over K.

We follow [2] closely. We also prove an analogue to Theorem 2.2 for thehomology H∗(G;K), which becomes a ring with the Pontryagin product.

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2.1 Graded algebras

We begin by setting some notation and nomenclature on graded algebras.

Definition 2.4. An algebra A over a commutative ring R is a graded algebraif it can be decomposed as

A =⊕n≥0

An,

in such a way that AnAm ⊂ An+m for all n ∈ Z≥0. Given a ∈ An we write|a| = n, and call |a| the degree of a.

We call the elements of any An generators.We say that an algebra is graded-commutative if ba = (−1)|a|·|b|ab for all

generators a, b.A graded-algebra homomorphism is a homomorphism of graded algebras f :

A→ B such that f(An) ⊂ Bn for all n ≥ 0.

Remark 2.5. If A has an identity 1 then 1 ∈ A0. We demand from a graded-algebra homomorphism between algebras with identity that f(1) = 1.

Remark 2.6. Sometimes graded algebras are called Z≥0-graded algebras, toallow for generalizations in the grading.

Example 2.7. Given a topological space X and a commutative ring R, thecohomology ring H∗(X;R) is an associative and graded-commutative R-algebrawith identity.

Example 2.8. Given a smooth manifold M the algebra of differential formsΩ∗(M) is an associative and graded-commutative R-algebra with identity.

DenoteA+ =

⊕n>0

An;

notice that A+ is an ideal of A. We can decompose

A = A0 ⊕A+, (1)

and denote by ρ0 : A → A0 and ρ+;A → A+ the canonical projections for thisdecomposition. Notice that ρ0 + ρ+ = idA, and, for a algebra homomorphismf : A→ B we must have f ρ0 = ρ0 f ρ0 and f ρ+ = ρ+ f ρ+.

Definition 2.9. Let A and B be graded algebras over a commutative ring R.The tensor algebra A ⊗ B is the R-module A ⊗ B together with the productdefined on generators by

(a⊗ b)(c⊗ d) = (−1)|b|·|c|(ac)⊗ (bd)

and grading

(A⊗B)n =

n⊕k+l=n

(Ak ⊗Bl

)for all n ≥ 0.

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Remark 2.10. Notice that the projections associated to the decomposition (1)for the tensor algebra are ρ0⊗ ρ0 and ρ+⊗ ρ0 + ρ0⊗ ρ+ + ρ+⊗ ρ+, respectively.

Remark 2.11. If A,B are associatived and graded-commutative R-algebraswith identity, then so is A⊗B, with identity 1⊗ 1.

2.2 The cohomology cross product

If X and Y are topological spaces such that each Hn(X) is a finite-dimensionalfree R-module, then the cohomology cross product gives a R-module isomor-phism

× :⊕k+l=n

(Hk(X;R)⊗H l(Y ;R)

)→ Hn(X × Y ;R). (2)

Recall the following properties of the cohomology cross product:

1. ×(1⊗ α) = α = ×(α⊗ 1), where 1 = [c1] ∈ H0(X;R);

2. (f × g)∗ × = × (f∗ ⊗ g∗), for maps f : Z → X, g : W → Y ;

3. × is a graded algebra homomorphism.

2.3 H-spaces and Hopf algebras

We begin by defining a H-space.

Definition 2.12 (H-space). A H-space is a topological space X together with acontinuous map µ : X ×X → X and a basepoint e ∈ X such that

µ (idX × ce) ' idX ' µ (ce × idX).

Remark 2.13. A topological group is a H-space. Although the H-space con-dition is much weaker than being a topological group, it is enough for showingtopological conditions on topological groups. For example, if X is a connectedH-space, then π1(X) is abelian.

Definition 2.14. A connected Hopf algebra is a graded algebra

A =⊕n≥0

An

over a comutative ring R, with identity 1, together with a graded algebra homo-morphism ∆ : A→ A⊗A satisfying the following two conditions:

1. Connectedness: A0 is isomorphic to R via r 7→ r1;

2. (ρ+ ⊗ ρ0 + ρ0 ⊗ ρ+)(∆(a)) = a⊗ 1 + 1⊗ a for all a ∈ A+.

The map ∆ is called a coproduct.

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Remark 2.15. Condition 2 in the definition of connected Hopf algebra is usu-ally written as

∆(a) = a⊗ 1 + 1⊗ a+

n∑i=1

bi ⊗ ci

for a generator a of positive degree and some |bi|, |ci| < |a|.Remark 2.16. Henceforth, all connected Hopf algebras are also graded-commutativeand associative.

The following proposition is the actual historical motivation for Definition2.14.

Proposition 2.17. Let R be a PID and X be a connected H space with productµ and basepoint e such that each Hn(X;R) a finitely generated free R-module.

Then the cohomology ring H∗(X;R) is a Hopf algebra with a coproduct ∆such that × ∆ = µ∗.

Proof. Firstly, the map ∆ exists: by the Kunneth formula in section 2 thecohomology cross product gives an R-algebra isomorphism. Thus we can define∆ = ×−1 µ∗, and since µ∗ is a R-algebra homomorphism, ∆ is a R-algebrahomomorphism.

The graded R-algebra H∗(X;R) is a commutative and associative algebrawith identity the class of the constant map [1] ∈ H0(X;R), and ∆(1) = 1.Moreover, since X is connected, the map r 7→ r[1] = [r] is an isomorphismR 7→ H0(X;R).

Recall that

H∗(X;R)⊗H0(X;R) ∼= H∗(X;R)⊗R ∼= H∗(X;R)

and α×1 = α so the restriction of × to H∗(X;R)⊗H0(X;R) is an isomorphism.Similarly, the restriction of × to H0(X;R)⊗H∗(X;R) is also an isomorphism.Thus, abbreviating idH∗(X;R) by id, condition 2 in Definition 2.14 can be rewrit-ten as the two conditions

× (ρ+ ⊗ ρ0) ∆ ρ+ = ρ+

and× (ρ0 ⊗ ρ+) ∆ ρ+ = ρ+.

We have µ (idX × ce) ' idX and c∗e = ρ0, so

id = id∗X = (idX×ce)∗µ∗ = (idX×ce)∗×∆ = ×(id∗X⊗c∗e)∆ = ×(id⊗ρ0)∆.

Composing with ρ+ on the right, we get

ρ+ = × (id⊗ ρ0) ∆ ρ+= × (id⊗ ρ0) (ρ0 ⊗ ρ+ + ρ+ ⊗ ρ0 + ρ+ ⊗ ρ+) ∆ ρ+= × (ρ+ ⊗ ρ0) ∆ ρ+,

which is the first condition.Similarly, from µ (ce × idX) ' idX we get the second condition, finishing

the proof.

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2.4 The classification of Hopf algebras over K and thecohomology ring H∗(G;K)

We now classify the finite dimensional of Hopf algebras over K and thus proveTheorem 2.2.

Definition 2.18. Let A be a Hopf algebra. A subalgebra B ⊂ A is said to be aHopf subalgebra if ∆(B) ⊂ B ⊗B.

Given S ⊂ A let 〈S〉 denote the subalgebra generated by S; i.e. the smallestsubalgebra of A containing S.

Theorem 2.19 (Hopf-Samelson). Let A be a finite-dimensional connected Hopfalgebra over K. Then A is isomorphic to an exterior algebra with odd-degreegenerators with coefficients in K.

Proof. Let B be a maximal Hopf subalgebra of A isomorphic to an exterioralgebra with odd-degree generators (whose existence is guaranteed because Ais finite-dimensional). If B 6= A, let x be an element of minimal degree inA \B. We claim that A′ := 〈B ∪ x〉 is isomorphic to an exterior algebra withodd-degree generators, contradicting the maximality of B.

Let I ⊂ A′ be the ideal generated by B+∪x2. Consider the map ϕ : A′ →A′ ⊗ A′/I given by ϕ = (idA′ ⊗ π) ∆, where π : A′ → A′/I is the canonicalprojection. The map ϕ is a K-algebra homomorphism.

Moreover, since x /∈ I, by minimality of |x|, the K-algebra A′/I has a basis1, π(x), so (A′)2 ∼= A′ ⊗A′/I as vector spaces via (a, b) 7→ a⊗ 1 + b⊗ π(x)

We have:

• ∆(r) = r ⊗ 1 for r ∈ R, so ϕ(r) = r ⊗ 1;

• ϕ(b) = b⊗ 1 for b ∈ B+, because ρ+(B) = B+ ⊂ I;

• ϕ(x) = x ⊗ 1 + 1 ⊗ π(x), since all lower degree elements on ∆(x) are inB+ ⊂ I.

In particular, ϕ(b) = b⊗ 1 for all b ∈ B.Suppose x has even degree. We claim that all xn are nonzero. For the sake

of contradiction, suppose that n ≥ 0 is minimal such that xn = 0. Then n > 1,and, by commutativity and Newton’s formula,

0 = ϕ(xn) = ϕ(x)n = (x⊗ 1 + 1⊗ π(x))n

=n∑k=0

(n

k

)xn−k ⊗ π(x)k = xn ⊗ 1 + nxn−1 ⊗ π(x),

since π(x)k = π(xk) = 0 for k ≥ 2. By the above identification (A′)2 ∼=A′ ⊗ A′/I, we conclude that nxn−1 = 0. Since K has characteristic 0, we getxn−1 = 0, a contradiction with the minimality of n.

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Thus all xn are nonzero and have distinct degrees, and so A′ is infinite-dimensional, a contradiction. Thus x has odd degree. We claim that the map

B ⊗ Λ(x)→ A′

b⊗ (r + sx) 7→ b(r + sx)

is a K-algebra isomorphism. It is easily seen to be well-defined, a K-algebraisomorphism, and surjective. To see that it is injective, notice again that, as aK-vector spaces, Λ(x) ∼= K2 and so B2 ∼= B ⊗ Λ(x) via (b, c) 7→ b ⊗ 1 + c ⊗ x.Suppose then that b⊗ 1 + c⊗ x is mapped to 0. Then b+ cx = 0, and

0 = ϕ(b+ cx) = ϕ(b) + ϕ(c)ϕ(x)

= b⊗ 1 + (c⊗ 1)(x⊗ 1 + 1⊗ π(x)) = b⊗ 1 + cx⊗ 1 + c⊗ π(x)

= (b+ cx)⊗ 1 + c⊗ π(x);

thus c = 0, and b = 0 as well.Finally, notice that ∆(x) = x⊗1+ 1⊗x+

∑ni=1 ai⊗ bi for some ai, bi ∈ B+,

so A′ is actually a Hopf subalgebra. This finishes the proof.

Proof of Theorem 2.2. By Corollary ??, it suffices to prove the result for com-pact G. The product on G gives it a H-space structure. Moreover, a compactmanifold has finite type homology in all dimensions, and its cohomology onlygoes up to its dimension, so H∗(G;K) is a finitely generated K-algebra. ByProposition 2.17, the cohomology ring H∗(G;K) is a finite-dimensional con-nected Hopf algebra over K. By Theorem 2.19, this ring is isomorphic to anexterior algebra in odd-degree generators with coefficients in K.

Remark 2.20. Since the classification in Theorem 2.19 makes no mention ofthe coproduct, what we call “classification of Hopf algebras over K” is actually a“classification of the algebra structure of finite-dimensional Hopf algebras overK”. Proposition 2.26 together with Theorem 2.19 provide an actual classifica-tion of the finite-dimensional Hopf algebras over K.

Remark 2.21. We can replace the “finite-dimensional” assumption with theassumption that An = 0 for large enough n, using Zorn’s Lemma to show thatthe maximal Hopf subalgebra B exists. This allows us not to use Theorem 1.1in the proof of Theorem 2.2.

We finish this section by stating the classification of finite-dimensional Hopfalgebras over perfect fields (namely Q and Z/pZ for prime p). A proof can befound in [6].

Theorem 2.22 (Hopf-Borel). Finite-dimensional Hopf algebras over a perfectfield F are finite tensor products of algebras of the following kinds:

• If the characteristic of F is 0, exterior algebras ΛF (x) with odd |x|;

• If the characteristic of F is p 6= 2, either ΛF (x) with odd |x| or F [x]/(xpn

)with n ≥ 0 and even |x|;

• If the characteristic of F is 2, truncated polynomial rings F [x]/(x2n

) withn ≥ 0.

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2.5 The Dual Hopf algebra, Lie algebra cohomology andthe torsion-free homology of Lie groups

In this section we drop the assumption that Hopf algebras are associative andcommutative. We follow the content of section 4.2. in [6] closely, although itsnomenclature or notation not so much.

Let X be a connected H-space with multiplication µ, and R be a commuta-tive ring. The map µ : X ×X → X allows us to define a product on H∗(X;R),called the Pontryagin product, defined by µ∗ ×, where we now denote by ×the homology cross product

× : H∗(X;R)⊗H∗(X;R)→ H∗(X;R).

Also, if each Hn(X;R) is a finite-dimensional free R-module, then H∗(X;R)and H∗(X;R) are dual (in the sense that each Hn(X;R) ∼= Hom(Hn(X;R);R)),and the Pontryagin product is dual to the coproduct ∆ on H∗(X;R). Moreover,the dual of the cup product on H∗(X;R) is a coproduct on H∗(X;R), makingit into a connected Hopf algebra.

This is a particular case of a dual Hopf algebra, which is an abstract con-struction.

Definition 2.23. Let A be a connected Hopf algebra with product π : A⊗A→ Aand coproduct ∆. The dual algebra to A is

A∗ =

∞⊕n=0

A∗n,

with product ∆∗ and coproduct π∗.

Proposition 2.24. The dual algebra of a connected Hopf algebra is also aconnected Hopf algebra.

Proof. Let A be a graded algebra, let 1 ∈ A and π : A⊗ A→ A, and ∆ : A→A⊕ A be graded R-module homomorphisms. For A to be a Hopf algebra withidentity 1, product π and coproduct ∆ we need to check:

1. π(a⊗ 1) = a = π(1⊗ a) for all a ∈ A;

2. Condition 2 in the definition of Hopf algebra;

3. That ∆ is an algebra homomorphism.

We can rewrite these as the following:

1. π (idA ⊗ 1) = idA = π (1⊗ idA);

2. (idA ⊗ ρ0) ∆ = idA = (ρ0 ⊗ idA) ∆;

3. ∆π = (π⊗π)τ (∆⊗∆), where τ : (A⊗A)⊗(A⊗A)→ (A⊗A)⊗(A⊗A)is defined on generators by τ((a⊗ b)⊗ (c⊗d)) = (−1)|b|·|c|(a⊗ c)⊗ (b⊗d).

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The dual of the map 1 : R → A given by r 7→ r1 is the projection of A∗ onto(A0)∗, and the dual of ρ0 is the map ρ∗0 : R → A∗ given by r 7→ r1∗. The mapτ∗ behaves the same as τ . Thus it is clear that the duals of conditions 1,2,3 forthe algebra A are the conditions 2,1,3, respectively, for the algebra A∗. Thisfinishes the proof.

Even if A is associative and/or graded-commutative, it does not mean thatA∗ is associative and/or graded-commutative. It is easily seen that:

• A∗ is associative if and only if (∆⊗ 1) ∆ = (1⊗∆) ∆;

• A∗ is graded-commutative if and only if T ∆ = ∆, where T : A ⊗ A →A⊗A is defined on generators by T (a⊗ b) = (−1)|a|·|b|(b⊗ a).

This inspires the following definition.

Definition 2.25. We say that A is coassociative if (∆⊗ 1) ∆ = (1⊗∆) ∆and cocommutative if T ∆ = ∆.

The Pontryagin product is not a priori associative or graded-commutative.For a Lie group, it is associative since the product is associative. We now showthat the Pontryagin product is graded-commutative for coefficients over a fieldof characteristic 0.

We need to set up some notation for exterior algebras. Consider an exterioralgebra ΛK(x1, . . . , xn). Given 1 ≤ i1 < · · · < ik ≤ n we write I = i1, . . . , ikand xI = xi1 · · ·xik . Given disjoint I, J ⊂ 1, . . . , n we then have xI∪J =σI,JxIxJ for some σI,J ∈ ±1. Moreover, given disjoint I, J,K ⊂ 1, . . . , nwe have

xI∪J∪K = σI,J∪KxIxJ∪K = σI,J∪KσJ,KxIxJxK

andxI∪J∪K = σI∪JxI∪JxK = σI∪JσI,JxIxJxK ,

whenceσI∪J,KσI,J = σI,J∪KσJ,K . (3)

Also, we abbreviate A ]B = S to mean that A ∪B = S and A ∩B = ∅.

Proposition 2.26. Let ΛK(x1, . . . , xn) be an exterior algebra on odd-degreegenerators with a coproduct ∆ making it a Hopf algebra. Suppose that ∆ isassociative. Then the xi can be chosen such that ∆(xi) = xi ⊗ 1 + 1 ⊗ xi, forall i = 1, . . . , n.

Proof. Without loss of generality assume that the xi are ordered by degree.Then clearly ∆(x1) = x1 ⊗ 1 + 1⊗ x1. We now proceed by induction: supposethat the xi are such that ∆(xi) = xi ⊗ 1 + 1⊗ xi for i = 1, . . . ,m− 1.

An easy computation shows that

∆(xI) =∑

J]K=I

σJ,K (xJ ⊗ xK) . (4)

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Moreover,

∆(xm) = xm ⊗ 1 + 1⊗ xm +∑I,J

aI,J (xI ⊗ xJ) (5)

for some aI,J ∈ K, where the I, J run over the pairs of disjoint subsets of1, . . . ,m− 1. We have

(∆⊗ id)(∆(xm)) = (∆⊗ id)

xm ⊗ 1 + 1⊗ xm +∑I,J

aI,J (xI ⊗ xJ)

= ∆(xm)⊗ 1 + ∆(1)⊗ xm +

∑I,J

aI,J (∆(xI)⊗ xJ)

= xm ⊗ 1⊗ 1 + 1⊗ xm ⊗ 1 +∑I,J

aI,J (xI ⊗ xJ ⊗ 1)

+ 1⊗ 1⊗ xm +∑I,J,K

aI∪J,K · σI,J (xI ⊗ xJ ⊗ xK)

and similarly

(id⊗∆)(∆(xm)) = 1⊗ xm ⊗ 1 + 1⊗ 1⊗ xm +∑I,J

aI,J (1⊗ xI ⊗ xJ)

+ xm ⊗ 1⊗ 1 +∑I,J,K

aI,J∪K · σJ,K (xI ⊗ xJ ⊗ xK) .

Comparing the two expressions, we get

σI,J · aI∪J,K = σJ,K · aI,J∪K

for all disjoint I, J,K ⊂ 1, . . . ,m − 1 with I,K 6= ∅. By equation (3), thiscan be rewritten as

σI∪J,K · aI∪J,K = σI,J∪K · aI,J∪K .

In particular, if A ] B = A′ ] B′ ⊂ 1, . . . ,m − 1 are non-empty, then,setting I = A, J = A′ \A and K = B′ we get

σA,B · aa,B = σA′,B′ · aI′,J′ .

Thus, for each I ⊂ 1, . . . ,m− 1 there is aI ∈ K such that aJ,K = aIσJ,K fornon-empty J ]K = I. Equation (5) can now be written as

∆(xm) = xm⊗1+1⊗xm+∑

I⊂1,...,m−1

aI

(−xI ⊗ 1− 1⊗ xI +

∑J]K=I

σJ,KxJ ⊗ xK

),

and so, setting

x′m = xm −∑

I⊂1,...,m−1

aIxI

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we have∆(x′m) = x′m ⊗ 1 + 1⊗ x′m.

This finishes the step of induction and hence the proof.

Corollary 2.27. The homology ring H∗(G;K) with the Pontryagin product isan associative and graded-commutative Hopf algebra.

Proof. By Proposition 2.24 H∗(G;K) is a Hopf algebra. As we have mentioned,it is also associative. Finally, Proposition 2.26 shows that the coproduct ∆ iscommutative.

Theorem 2.28. Let G be a connected Lie group. There are generators xi ∈H∗(G;K) of odd degree such that

H∗(G;K) = ΛK (x1, . . . , xn)

andH∗(G;K) = ΛK (x∗1, . . . , x

∗n) .

In particular, the homology ring H∗(G;K) with the Pontryagin product is iso-morphic to an exterior algebra on odd-degree generators with coefficients in K;i.e. G has the K-homology of a product of odd-dimensional spheres.

Proof. By Theorem 2.2, there are x1, . . . , xn ∈ H∗(G;K) of odd-degree suchthat H∗(G;K) = ΛK(x1, . . . , xn). By Proposition 2.26, these can be chosenin such a way that ∆(xi) = xi ⊗ 1 + 1 ⊗ xi. Denoting by (x∗I) the basis ofH∗(G;K) dual to H∗(G;K), using formula (4) it is easily seen that ∆∗(x∗I , x

∗J)

is 0 if I ∩ J 6= ∅ and is σI,Jx∗Ix∗J otherwise. Thus H∗(G;K) = ΛK(x∗1, . . . , x

∗n),

finishing the proof.

The following examples display two different ways in which the Pontryaginproduct can fail to be graded-commutative.

Example 2.29. Let G be a discrete group. The homology ring H∗(G;R) isisomorphic to the group ring R[G] (concentrated on degree 0), and hence it is(graded-)commutative if and only if G is abelian.

Example 2.30. Theorem A in [4] in particular implies that the cohomologyring H∗(E7;Z/3Z) has generators ei ∈ Hi(E7;Z/3Z) for i = 3, . . . , 35 such that∆(e11) = e11 ⊗ 1 + 1 ⊗ e11 + e8 ⊗ e3, and so the Pontryagin product is notgraded-commutative.

2.6 Some straightforward applications

By the universal coefficient theorem, H∗(G;K) ∼= (H∗(G)/Tor(H∗(G))) ⊗ Kand H∗(G;K) ∼= (H∗(G)/Tor(H∗(G))) ⊗ K, and the isomorphisms carry thering structure. Since:

• Each Hn(G)/Tor(Hn(G)) and Hn(G)/Tor(Hn(G)) is free-abelian;

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• Zm ⊗K ∼= Km for each m ≥ 0;

• ΛK(S) = ΛZ(S)⊗K for any set of odd-degree generators S;

we get the following corollary.

Corollary 2.31. Let G be a connected Lie group. Then H∗(G)/Tor(H∗(G))and H∗(G)/Tor(H∗(G)) are isomorphic to exterior rings on odd-degree genera-tors.

The following corollary is weaker than Theorem 3.1, but is much easier toprove.

Corollary 2.32. Let G be a connected Lie group. Then π2(G) is torsion.

Proof. Let G be the universal cover of G. Then π1(G) = 0 and, by the Hurewicztheorem, H1(G) = 0. Also, by the lifting lemma and the Hurewicz theorem,π2(G) ∼= π2(G) ∼= H2(G).

By corollary 2.31, Tor(H∗(G)) is isomorphic to an exterior ring on odd-degree generators, with no elements of degree 1. Thus all its generators havedegree at least 3, so Tor(H∗(G)) has no elements of degree 2. This means thatH2(G)/Tor(H2(G)) = 0; that is, H2(G) is torsion.

3 The second homotopy group of a Lie group

In this section the goal is to prove the following theorem.

Theorem 3.1. Let G be a connected Lie group. Then π2(G) = 0.

This is a stronger result than Corollary 2.32. We follow the idea of the proofdue to Bott, as detailed in [5]. We will not need the same kind of homotopymachinery as Milnor uses to get the results in section 3.8, as any representativeof a class in π2(G) lives inside a finite energy subspace, as we will argue.

The idea of the proof is to use Morse theory on the energy function on thepath space of a Lie group G. This needs a Riemannian metric, which we actuallyneed to be bi-invariant - this is where the compactness assumption comes in.We begin by quickly reviewing some concepts and notation in Morse theoryand Riemannian geometry. We then describe the path space of a Riemannianmanifold and its energy map, the finite-dimensional approximation to this pathspace, compute the index of the critical points of the energy map, and deriveTheorem 3.1.

3.1 A brief review of Morse theory

The following definition is standard for symmetric bilinear forms.

Definition 3.2. Let B be a symmetric bilinear form on a real vector space V .Let B : V → V ∗ be the map B(v) : u 7→ B(v, u) for v, u ∈ V . The nullity ofB is dim ker B. The index of B is the maximal subspace U ⊂ V on which B isnegative-definite.

12

Let M be a smooth manifold. The Hessian of a map f ∈ C∞(M) at a criticalpoint p ∈M can be defined in terms of coordinates, but it is often more usefulto think of it as in [5]. Given vector fields X,Y ∈ X(M), define

(d2f)p(X,Y ) = X · (Y · f).

This expression depends only on the value of X at p and on Y . Moreover,

X · (Y · f)− Y · (X · f) = [X,Y ]p · f = 0,

so X · (Y · f) = Y · (X · f) depends only on the values of X and Y at p. Thusit is a symmetric 2-tensor.

Notice that

(d2f)p(X,Y ) =∂2f(φsX(φtY (p)))

∂s ∂t

∣∣∣∣s=t=0

,

and so for a map c :]− ε, ε[2→M with c(0, 0) = p we have

(d2f)p

(∂c

∂s(0, 0),

∂c

∂t(0, 0)

)=∂2f(c(s, t))

∂s ∂t

∣∣∣∣s=t=0

. (6)

Definition 3.3. Let f ∈ C∞(M) and p ∈M a critical point of f . The nullityand index at p are the nullity and index, respectively, of (d2f)p as a symmetricbilinear form. The point p is said to be non-degenerate if its nullity is 0.

The following theorem is Theorem 3.5 in [5].

Theorem 3.4. If f is a Morse function on a compact manifold with boundaryM , then M has the homotopy type of a CW-complex with one cell of dimensionλ.

3.2 A brief review of Riemannian manifolds

In this subsection we review some concepts we need about Riemannian mani-folds. A more detailed description can be seen in [1] and [5].

Let M be a Riemannian manifold. We denote its metric by 〈·, ·〉p for p ∈M .We omit the point whenever it is clear from context or otherwise irrelevant. Thenorm

√〈v, v〉 of a vector v ∈ TM is denoted ||v||.

The Levi-Civita connection on M is denoted by ∇, and satisfies the followingproperties:

• Symmetry: ∇XY −∇YX = [X,Y ];

• Compatibility with the metric: X · 〈Y,Z〉 = 〈∇XY, Z〉+ 〈Y,∇XZ〉.

The curvature tensor is R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z.Let U ⊂ Rn be an open subset and f : U → M be a smooth map. Given

X ∈ X(M), we defineDX

∂xi= ∇∂f/∂xiX,

13

where ∂f/∂xi denotes f∗(∂/∂xi).

If n = 1 we simply denote DX/∂x by DX/dx, and call it the “covariantderivative”. In terms of the covariant derivative, compatibility with the metriccan be seen as the following equation for a curve t 7→ c(t) and vector fields X,Yalong c:

∂

∂t〈X(t), Y (t)〉 =

⟨DX

dt, Y (t)

⟩+

⟨X(t),

DY

dt

⟩(7)

The symmetry is rewritten as the following identity:

DX

∂x

(f∗

(∂

∂y

))=

DX

∂x

(f∗

(∂

∂y

)). (8)

As for the curvature tensor, we have the following identity:

D

∂x

D

∂yX − D

∂y

D

∂x= R

(∂f

∂x,∂f

∂y

)X (9)

We denote the geodesic exponential by expp : U → M for all p ∈ M andsome open subset U ⊂ TpM containing 0. The Riemannian manifold M is saidto be geodesically complete if we can always take U = TpM .

Definition 3.5. Let M be a Riemannian manifold and α : [a, b] → M a path.The length of α is

L(α) =

∫ b

a

||α(t)|| dt.

A Riemannian manifold M can be made into a metric space with distanced defined by

d(p, q) = infαL(α),

where p, q ∈M and α runs all the paths in M from p to q. Since the length of apath is invariant under parametrization changes, we can consider without lossof generality only α ∈ Ω(M ; p, q).

By the Hopf-Rinow theorem (in particular, the formulation in [1]), geodesi-cally complete Riemannian manifolds are complete as metric spaces and vice-versa, and we simply call such manifolds complete. In particular, closed andbounded subsets in complete Riemannian manifolds are compact.

We need the two following statements for a Riemannian manifold M , whichare standard in Riemannian Geometry, and are proven in [5].

Lemma 3.6. Given p ∈ M there is an open neighborhood U of p in M and anumber ε > 0 such that:

1. Any two points of U are joined by a unique geodesic γ : [0, 1] → M oflength less than ε;

2. The pair (γ(0), γ(0)) depends smoothly on the pair (γ(0), γ(1));

3. For each q ∈ U the map expq maps the open ε-ball in TqM diffeomorphi-cally onto an open subset of U .

14

Lemma 3.7. Given a compact set K ⊂M there exists a number δ > 0 so thatany two points of K with distance less than δ are joined by a unique geodesic oflength less than δ. Furthermore this geodesic is minimal; and depends differen-tiably on its endpoints.

3.3 The path space of a Riemannian manifold

We begin by defining our path space.

Definition 3.8. Let M be a smooth manifold and p, q ∈M . Denote by Ω(M ; p, q)the set of piecewise-smooth maps α : [0, 1]→M with α(0) = p and α(1) = q.

Remark 3.9. Let X be a path-connected topological space. Given p, q ∈ X, letΩqpX denote the space of paths [0, 1] → X from p to q, with the compact opentopology.

Given p′, q′ ∈ X, there are paths α, β in X from p to p′ and from q to q′,respectively. The maps γ 7→ α ? γ ? β and δ 7→ α ? δ ? β are homotopy inverses

of each other, and so define an homotopy equivalence between ΩqpX and Ωq′

p′X.

In particular, πk(ΩqpX) ∼= πk(Ωq′

p′X).The usual loop space ΩX is simply ΩppX for some p ∈ X, and it is well-known

(e.g. [2]) that πk(ΩX) ∼= πk+1(X) for k ≥ 0.

Remark 3.10. For a smooth manifold M , the homotopy groups πk(ΩqpM) do

not depend on the Ck-topology chosen. Moreover, from the compactness of theinterval [0, 1] these topologies are induced by the Ck-norm.

When the context is clear, we abbreviate Ω = Ω(M ; p, q).

Definition 3.11. Let M be a Riemannian manifold and p, q ∈M .We define an energy function E : Ω→ R by

E(α) =

∫ 1

0

||α(t)||2 dt

for all α ∈ Ω.Given a > 0 we abbreviate Ωa = E−1([0, a]) and int Ωa = E−1([0, a[).Given a partition 0 = t0 < t1 < · · · < tk−1 < tk = 1 of [0, 1], we define

Ω(t0, . . . , tk) ⊂ Ω as the set of broken geodesics which break at most at timesti; i.e. paths γ ∈ Ω such that γ|[ti−1,ti] is a geodesic for i = 1, . . . , k.

DenoteΩ(t0, . . . , tk)a = Ω(t0, . . . , tk) ∩ E−1([0, a]),

int Ω(t0, . . . , tk)a = Ω(t0, . . . , tk) ∩ E−1([0, a[).

Notice that for any α ∈ Ω we have L(α)2 ≤ E(α), by the Cauchy-Schwarzinequality. Also, if γ is a geodesic of minimal length, then E(γ) is minimal,since

E(α) ≥ L(α)2 ≥ L(γ)2 = E(γ)

for any α ∈ Ω.

15

Definition 3.12. Given α ∈ Ω, we define TαΩ to be the space of piecewisesmooth vector fields along α which vanish at p and q.

Let 0 = t0 < · · · < tk = 1 be a partition of [0, 1]. Given a smooth manifoldN , a map f : N → Ω is said to be a piecewise smooth map which possiblybreaks at (t0, . . . , tk) in Ω if each restriction

N × [ti−1, ti]→ Ω

(p, t) 7→ f(p)(t)

is smooth for i = 1, . . . , k.In particular, a variation is such a map with N =]− ε, ε[ for some ε > 0.The tangent vector field to a piecewise smooth curve at c at c(0) is X ∈

Tc(0)Ω given by

X(t) =∂c(s)(t)

∂s

∣∣∣∣s=0

for all t ∈ [0, 1].

Remark 3.13. We will often abuse notation, writing f both for the map N ×[0, 1]→M and for the map N → Ω.

Remark 3.14. All piecewise vector fields X ∈ TαΩ which vanish at p and q aretangent to some curve which breaks at the same points: for example, considerc(s, t) = expα(t)(sX(t)).

Remark 3.15. Just as we define the differential of a scalar map with a curve,we define the differential of E: given α ∈ Ω and X ∈ TαΩ, let c be a curve withc(0) = α and tangent vector X at α,

(dE)α(c(0)) =dE(c(s))

ds

∣∣∣∣s=0

.

Similarly, we define the Hessian of E by equation (6). That equation showsthe symmetry of the Hessian, because of the identity ∂2/∂t ∂s = ∂2/∂s ∂t, but itdoes not show that it is well-defined. This is however a consequence of Proposi-tion 3.17.

Proposition 3.16. Suppose that X ∈ TαΩ is a piecewise vector field whichbreaks at t1, . . . , tk−1. Then

1

2(dE)α(X) = −

k−1∑i=1

〈X(ti), α(t+i )− α(t−i )〉 −∫ 1

0

⟨X(t),

Dα

dt

⟩dt.

Proof. It is clear from the equation that it is enough to show, for a smooth mapc :]− ε, ε[×[0, 1]→M with c(0) = α, that

1

2

dE(c(s))

ds

∣∣∣∣s=0

= 〈X(1), α(1)〉 − 〈X(0), α(0)〉 −∫ 1

0

⟨X(t),

DX

dt

⟩dt,

16

where X(t) = ∂c∂s (0, t).

Indeed, using identities (7) and (8), we get

1

2

d

ds

(∫ 1

0

∣∣∣∣∣∣∣∣∂c∂t∣∣∣∣∣∣∣∣2 dt

)∣∣∣∣s=0

=1

2

∫ 1

0

d

ds

⟨∂c

∂t,∂c

∂t

⟩dt

∣∣∣∣s=0

=

∫ 1

0

⟨∂c

∂t,

D

ds

∂c

∂t

⟩dt

∣∣∣∣s=0

=

∫ 1

0

⟨∂c

∂t,

D

dt

∂c

∂s

⟩dt

∣∣∣∣s=0

=

∫ 1

0

(d

dt

⟨∂c

∂t,∂c

∂s

⟩−⟨

D

dt

∂c

∂t,∂c

∂s

⟩)dt

∣∣∣∣s=0

=

⟨∂c(s, 1)

∂t,∂c(s, 1)

∂s

⟩−⟨∂c(s, 0)

∂t,∂c(s, 0)

∂s

⟩−∫ 1

0

⟨∂c(s, t)

∂s,

D

dt

∂c(s, t)

∂t

⟩dt

∣∣∣∣s=0

= 〈X(1), α(1)〉 − 〈X(0), α(0)〉 −∫ 1

0

⟨X(t),

Dα

dt

⟩dt,

since ∂c(s, t)/∂t→ α(t) as s→ 0.

Proposition 3.17. Let α ∈ Ω be a geodesic. Suppose that X,Y ∈ TαΩ arepiecewise vector fields and X breaks at t1, . . . , tk−1. The Hessian of E, calculatedas in equation (6), satisfies

1

2(d2E)α(X,Y ) = −

k−1∑i=1

⟨Y (ti),

DX

dt

(t+i)− DX

dt

(t−i)⟩

−∫ 1

0

⟨Y (t),

D2X

dt2−R(α(t), X(t))α(t)

⟩dt.

In particular, the Hessian is a well-defined, bilinear and symmetric map onTαΩ.

Proof. Let c :] − ε, ε[2×[0, 1] → M be a piecewise smooth map which possiblybreaks at t1, . . . , tk−1, with α(t) = c(0, 0, t), and X(t) = ∂c

∂u (0, 0, t), and Y (t) =

17

∂c∂s (0, 0, t). Using Proposition 3.16 and identity 7, we get

1

2

∂2E(c(u, s))

∂u ∂s

∣∣∣∣s=0

=∂

∂u

(1

2

∂E(c(u, s))

∂s

∣∣∣∣s=0

)=

∂

∂u

(k−1∑i=1

⟨∂c

∂s(u, 0, ti),

∂c

∂t

(u, 0, t+i

)− ∂c

∂t

(u, 0, t−i

)⟩−∫ 1

0

⟨∂c

∂s(u, 0, t),

D

∂t

∂c

∂t(u, 0, t)

⟩dt

)=

k−1∑i=1

(⟨D

∂u

∂c

∂s(u, 0, ti),

∂c

∂t

(u, 0, t+i

)− ∂c

∂t

(u, 0, t−i

)⟩+

⟨∂c

∂s(u, 0, ti),

D

∂u

∂c

∂t

(u, 0, t+i

)− D

∂u

∂c

∂t

(u, 0, t−i

)⟩)−∫ 1

0

(⟨D

∂u

∂c

∂s(u, 0, t),

D

∂t

∂c

∂t(u, 0, t)

⟩+

⟨∂c

∂s(u, 0, t),

D

∂u

D

∂t

∂c

∂t(u, 0, t)

⟩)dt.

Since α is a geodesic, it is smooth, so

∂c

∂t

(u, t+i , 1

)− ∂c

∂t

(u, t−i , 1

) ∣∣∣∣u=0

= α(t+i )− α(t−i ) = 0.

Moreover, Dα∂t = 0, so⟨

D

∂u

∂c

∂s(u, 0, t),

D

∂t

∂c

∂t(u, 0, t)

⟩ ∣∣∣∣u=0

= 0.

This, together with identity (8), yields

1

2

∂2E(c(u, s))

∂u ∂s

∣∣∣∣u=s=0

=

k−1∑i=1

⟨Y (ti),

DX

∂t

(t+i)− DX

∂t

(t−i)⟩−∫ 1

0

⟨Y (t),

D

∂u

D

∂tα(t)

⟩dt.

Now, using the identities (8) and (9), we get

D

∂u

D

∂t

∂c

∂t= −R

(∂c

∂t,∂c

∂u

)∂c

∂t+

D

∂t

D

∂u

∂c

∂t= −R

(∂c

∂t,∂c

∂u

)∂c

∂t+

D

∂t

D

∂t

∂c

∂u,

and evaluating at u = s = 0 we get

D

∂u

D

∂tα(t) = −R (α(t), X(t)) α(t) +

D2X

dt2,

finishing the proof.

Definition 3.18. A vector field J along a geodesic γ is called a Jacobi field ifit satisfies the Jacobi equation:

D2J

dt2= R(γ(t), J(t))γ(t).

18

Remark 3.19. Choosing orthonormal vector fields X1, . . . , Xn which are par-allel along γ and writing J(t) =

∑ni=1 J

i(t)Xi(t) for some J i ∈ C∞([0, 1]), theJacobi equation becomes a system of n equations

d2J i

dt2−

n∑j=1

aij(n)Jj(t) = 0, i = 1, . . . , n,

where aij(t) = 〈R(γ(t), Xj(t))γ(t), Xi(t)〉 for i, j = 1, . . . , n. This is a linearODE of second order, hence a Jacobi field can always be defined for [0, 1], anddepends only on the initial conditions J(0) and DJ

dt (0).

The importance of Jacobi fields becomes clear from the following proposition.

Proposition 3.20. Jacobi fields along a geodesic γ are exactly the tangentvector fields to variations of γ by geodesics.

Proof. Let γ : [0, 1]→M be a geodesic and c :]− ε, ε[×[0, 1]→M be a smoothmap such that each c(s) is a geodesic and c(0) = γ. Furthermore, let J = ∂c/∂s.Then

0 =D

dt

∂c

∂t.

Using identities (8) and (9), we get

0 =D

ds

D

dt

∂c

∂t=

D

dt

D

ds

∂c

∂t−R

(∂c

∂t,∂c

∂s

)∂c

∂t=

D

dt

D

dt

∂c

∂s−R

(∂c

∂t,∂c

∂s

)∂c

∂t

evaluating at s = 0 we get

0 =D2J

dt2−R(γ(t), J(t))γ(t),

for all t ∈ [0, 1], so J is a Jacobi field.

Remark 3.21. Let p ∈M and an open neighborhood U of p in M , and ε > 0 inthe conditions of Lemma 3.6. Consider a geodesic γ : [0, 1]→ U , and a variationα :]−ε, ε[×[0, 1]→ U of γ by geodesics (of length less than ε). Then each α(s, ·)depends only and smoothly on α(s, 0) and α(s, 1), and so the Jacobi field J

tangent to α at γ depends only on ∂α(s,0)∂s |s=0 = J(0) and ∂α(s,1)

∂s |s=0 = J(1).Thus the map J 7→ (J(0), J(1)) ∈ Tγ(0)M×Tγ(1)M is a surjective linear map.

Since Jacobi fields along γ are defined by the values of J(0) and DJdt (0), they form

a vector space of dimension 2 dimM . Moreover, the vector space Tγ(0)M ×Tγ(1)M also has dimension 2 dimM , so it is in fact a linear isomorphism.

In conclusion, for close enough endpoints of a geodesic, a Jacobi field alongit depends only on its values at the endpoints.

Definition 3.22. Let a, b ∈ [0, 1]. The multiplicity of a and b as conjugateparameter values with respect to γ is the dimension of the vector space consistingof all Jacobi fields along γ which vanish at a and b.

We say that a and b are conjugate with respect to γ if their multiplicity asconjugate points is positive.

19

Remark 3.23. It is usually said that γ(a) and γ(b) are conjugate points, whichmight be ambiguous. For our purposes the distinction between parameter valuesand points on a curve is of vital importance.

Proposition 3.24. Let γ ∈ Ω be a geodesic. The kernel of ˜(d2E)γ is the spaceof Jacobi fields along γ which vanish at 0 and 1. In particular, the nullity of(d2E)γ is the multiplicity of 0 and 1 as conjugate parameter values with respectto γ.

Proof. Let J be a Jacobi field along γ which vanishes at 0 and 1. By Proposition

3.17, it is clear that J is in the kernel of ˜(d2E)γ .

Conversely, suppose that X ∈ TγΩ is in the kernel of ˜(d2E)γ . Of course, Xvanishes at 0 and 1. Define Y ∈ TγΩ

Y (t) = f(t)

(D2X

dt2−R(α(t), X(t))α(t)

)for all t ∈ [0, 1], for some smooth f : [0, 1]→ R such that f(t) ≥ 0 and f(ti) = 0for i = 1, . . . , k − 1. By Proposition 3.17, we have (d2E)γ(X,Y ) < 0 unless

D2X

dt2−R(α(t), X(t))α(t) = 0

away from the ti. Thus X|[ti−1,ti] is a Jacobi field for i = 1, . . . , k.Now choose some Y ∈ TγΩ such that

Y (ti) =DX

dt

(t+i)− DX

dt

(t−i).

Unless X is an unbroken vector field, we have (d2E)γ(X,Y ) < 0. Thus X is anunbroken Jacobi field along γ which vanishes at 0 and 1, finishing the proof.

Lemma 3.25. If γ is a minimal geodesic, then the index of (d2E)γ is 0.

Proof. Has as already been discussed, in this case γ minimizes E on Ω, and sothere can be no non-zero vector subspace of TγΩ on which (d2E)γ is negative-definite.

3.4 The finite-dimensional approximation and the indextheorem

For this section we suppose that M is a complete Riemannian manifold anda > 0.

Let S = x ∈M : d(x, p) ≤√a. The image of any path α ∈ Ωa lies in S.

Since M is complete, S is compact. By Lemma 3.7 there is ε > 0 so thatwhenever x, y ∈ S and d(x, y) < ε there is a unique geodesic from x to y oflength smaller than ε.

We fix a partition 0 = t0 < t1 < · · · < tk−1 < tk = 1 of [0, 1] such that allti − ti−1 < ε2/a for the remainder of this section.

20

Proposition 3.26. The map int Ω(t0, . . . , tk)a →Mk−1 given by

α 7→ (α(t1), . . . , α(tk−1)) (10)

is an embedding.

Proof. For any i = 1, . . . , k and α ∈ Ω(t0, . . . , tk)a we have

d (α (ti−1) , α (ti)) ≤∫ ti

ti−1

||α(t)||dt ≤

√∫ ti

ti−1

1 dt

√∫ ti

ti−1

||α(t)||2 dt

≤√ti − ti−1

√∫ ti

ti−1

||α(t)||2 dt ≤√ti − ti−1

√E(α) < ε,

by the Cauchy-Schwarz inequality. Hence α is uniquely defined by (α(t1), . . . , α(tk−1)).Also, by Lemma 3.6, map (10) is a homeomorphism onto its image.

Notice that

E(α) =

k∑i=1

d (α (ti−1) , α (ti))2

ti − ti−1

is a smooth function of (α(t1), . . . , α(tk−1)). Thus the image of int Ω(t0, . . . , tk)a →Mk−1 under the map (10) is an open subset of Mk−1, and hence a manifold.We can thus endow Ω(t0, . . . , tk)a with a smooth structure, on which E is alsosmooth.

Now let α ∈ int Ω(t0, . . . , tk)a. A tangent curve inMk−1 to (α(t1), . . . , α(tk−1))corresponds to a variation of α by broken geodesics, and so a tangent vector to(α(t1), . . . , α(tk−1)) corresponds to a broken Jacobi field along α. By Remark3.21, this is a bijective correspondence, and so we write TαΩ(t0, . . . , tk) for thespace of broken Jacobi fields along α.

The formulas in Propositions 3.16 and 3.17 hold for the function E restrictedto this space.

Proposition 3.27. The critical points of the restriction of E to Ω(t0, . . . , tk)are the geodesics.

Proof. By Proposition 3.16, geodesics are critical points of E.Conversely, let α ∈ Ω(t0, . . . , tk) be a critical point of E. Then Dα

dt = 0 butfor a finite number of points. As a consequence of Remark 3.21, there is a brokenJacobi field J along α such that J(ti) = α(t+i )−α(t−i ) for i = 1, . . . , k−1. Then,by Proposition 3.16, we have

0 =1

2(dE)α = −

k∑i=1

∣∣∣∣α (t+i )− α (t−i )∣∣∣∣2 ,so each α(t+i ) = α(t−i ). Thus α is a geodesic.

The next proposition, together with Proposition 3.26, shows that Ω(t0, . . . , tk)a

is the finite-dimensional approximation we need.

21

Proposition 3.28. The subspace Ω(t0, . . . , tk)a is a deformation retract of Ωa.

Proof. The inequalities in the proof of Proposition 3.26 still hold for α ∈ Ωa.Define then r(α) ∈ Ω(t0, . . . , tk)a as the unique broken geodesic such that eachrestriction r(α)|[ti−1,ti] is the unique geodesic of length smaller than ε fromω(ti−1) to ω(ti).

The map r : Ωa → Ω(t0, . . . , tk)a thus obtained is continuous, since it is thecomposition of α 7→ (α(t1), . . . , α(tk−1)) with the inverse of the map (10).

Now define R : Ωa × [0, 1] → Ωa as follows: if ti−1 ≤ s ≤ ti let R(α, s) bethe path such that

R(α, s)|[0,ti−1] = r(α)|[0,ti−1],

R(α, s)|[ti−1,s] is the minimal geodesic from α(ti−1) to α(s),

R(α, s)|[s,1] = α|[0,ti−1].

To finish the proof we only need to see that R is continuous. This is oncemore seen as an easy consequence of Lemma 3.6.

3.5 The index theorem

We begin by showing that the index and nulities of the finite-dimensional ap-proximation and the whole path-space are the same.

Proposition 3.29. Let γ ∈ Ωa be a geodesic. The index and nulity of (d2E)γon int Ωa and int Ω(t0, . . . , tk)a are the same.

Proof. We decompose

TγΩ = TγΩ(t0, . . . , tk)⊕ V,

whereV = X ∈ TγΩ : X(ti) = 0, i = 1 . . . , k − 1.

Notice that TγΩ(t0, . . . , tk) is orthogonal to V with the bilinear form definedby (d2E)γ : if X ∈ TγΩ(t0, . . . , tk) and Y ∈ V , then (d2E)γ(X,Y ) = 0 byProposition 3.17. It is then enough to show that (d2E)γ is positive definite onV .

Let X ∈ V . Then there is a piecewise smooth map c :] − ε, ε[×[0, 1] → Msuch that c(s, ti) = γ(ti) for i = 0, . . . , k and s ∈]− ε, ε[ and c(0, t) = γ(t) for allt ∈ [0, 1]. Then each c(s)|[ti−1,ti] is a path between γ(ti−1) and γ(ti), and sincea geodesic has minimal energy,

E(c(s)) ≥ E(γ)

for all s ∈]−ε, ε[. By equation (6), we get that (d2E)γ(X,X) ≥ 0 for all X ∈ V .Now suppose that (d2E)γ(X,X) = 0. Then (d2E)γ(X,Y ) = 0 for all Y ∈

V , since (d2E)γ is positive semi-definite on V . Moreover, by orthogonality,

(d2E)γ(X,Y ) = 0 for all Y ∈ TγΩ(t0, . . . , tk). Thus X is in the kernel of ˜(d2E)γ .

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By Proposition 3.24, the field X is a Jacobi field along γ which vanishes at 0and 1. By Remark 3.21, such a field with X(ti) = 0 for each i = 1, . . . , k − 1must be 0. This finishes the proof.

Theorem 3.30. The index of (d2E)γ is the total number of conjugate parametervalues t ∈]0, 1[ to 0 with respect to γ, counted with multiplicities.

Proof. Given τ ∈]0, 1], let γτ denote γ|[0,τ ] and Eτ (α) =∫ τ0||α(t)||dt for any

piecewise smooth α : [0, τ ] → M . Then γτ is a geodesic from γ(0) to γ(τ).Let λ(τ) and ν(τ) denote the index and nullity of (d2Eτ )γτ , respectively. Thetheorem follows from the following three statements, together with Proposition3.24:

1. λ(0+) = 0;

2. λ is constant on intervals [τ ′, τ ], for some τ ′ close enough to τ .

3. λ(τ+)− λ(τ) = ν(τ) for τ ∈]0, 1[.

For statement 1, notice that for small enough τ > 0 that γτ is a minimalgeodesic, and by Lemma 3.25 we thus have λ(τ) = 0 for such τ .

For statement 2, let τ ∈]0, 1[. Choose a partition 0 = t0 < · · · < tk = 1 suchthat Proposition 3.26 holds. Furthermore, do it in such a way that ti < τ < ti+1

for some i ∈ 0, . . . , k − 1. By Proposition 3.29, the index λ(τ) is the index ofa quadratic form Hτ on the vector space of broken Jacobi fields along γτ whichvanish at 0 and τ . By Remark 3.21, this vector space is isomorphic to

V =

i⊕j=1

Tγ(ti)M.

This does not depend on τ ∈]ti, ti+1[, and the quadratic form Hτ dependscontinuously on τ . Now, if V − ⊂ V is the maximal linear subspace on whichHτ is negative-definite, then Hτ ′ is also negative-definite on V − for τ ′ closeenough to τ . Thus λ(τ ′) ≥ λ(τ) for τ ′ sufficiently close to τ .

Now let τ ′ ≤ τ . Let U ′ be the maximal linear subspace of the space ofpiecewise smooth vector fields along γτ ′ which vanish on 0 and τ ′, on which(d2Eτ ′)γτ′ is negative-definite. Given any X ′ ∈ U ′, we can extend it to a vectorfield X along γτ by X|[τ ′,τ ] = 0, and, of course,(

d2Eτ)γτ

(X,X) =(d2Eτ ′

)γτ′

(X ′, X ′).

Thus (d2Eτ )γτ is negative-definite on U = X : X ′ ∈W ′, and so λ(τ) ≥ λ(τ ′).This finishes the proof of statement 2.

Finally, for statement 3, we first prove that λ(τ ′) ≤ λ(τ) + ν(τ) for τ ′ > τsufficiently close to τ . Using the same notation as in the first part of the proofof statement 2, let V + ⊂ V be the maximal subset on which Hτ is positivedefinite. Then dimV + = dimV − λ(τ) − ν(τ). By continuity, H ′τ is positivedefinite on V +, hence λ(τ ′) ≤ dimV − dimV + = λ(τ) + ν(τ).

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We now prove that λ(τ ′) ≥ λ(τ) + ν(τ). Let X1, . . . , Xλ(τ) be generatorsof V − as in the first part of the proof of statement 2, and let J1, . . . , Jν(τ) be

generators of the kernel of ˜(d2Eτ )γτ ). By Remark 3.19, the vectors

DJjdt

(τ) ∈ Tγ(τ)M

are linearly independent. We can thus chose Y1, . . . , Yν(τ) vector fields along γτ ′

which vanish at 0 and τ ′, such that⟨DJjdt

(τ), Yk(τ)

⟩= δjk

for all j, k = 1, . . . , ν(τ). Once more, for all j, k = 1, . . . , ν(τ) and l = 1, . . . , λ(τ)extend the vector fields Xl, Jj and Yk to vector fields X ′l , J

′j and Y ′k by Xj |[τ,τ ′] =

Jk|[τ,τ ′] = Yl|[τ,τ ′] = 0. By Proposition 3.17, and because J ′j is a broken Jacobifield which breaks at τ,

(d2Eτ ′

)γτ′

(J ′j , J

′k

)= −2

⟨Jk(τ),−DJj

dt(τ)

⟩= 0

(d2Eτ ′

)γτ′

(J ′j , X

′l

)= −2

⟨Xl(τ),−DJj

dt(τ)

⟩= 0

(d2Eτ ′

)γτ′

(J ′j , Y

′k

)= −2

⟨Yk(τ),−DJj

dt(τ)

⟩= 2δjk

for j, k = 1, . . . , ν(τ) and l = 1, . . . , λ(τ).Now, given ε > 0 consider the λ(τ) + ν(τ) vector fields(

X ′1, . . . , X′λ(τ), ε

−1J ′1 − εY ′1 , . . . , ε−1J ′ν(τ) − εY′ν(τ)

).

We claim that these vector fields span a vector space of dimension λ(τ) + ν(τ)on which (d2Eτ ′)γτ′ is negative definite, which will finish the proof. Indeed, thematrix which represents (d2Eτ ′)γτ′ in this candidate to a basis is

[(d2Eτ

)γτ

(Xj , Xk)]λ(τ)j,k=1

−ε[(d2Eτ ′

)γτ′

(X ′j , Y

′k

)]−ε[(d2Eτ ′

)γτ′

(X ′j , Y

′k

)]T−4I + ε2

[(d2Eτ ′

)γτ′

(Y ′j , Y

′k

)].

For ε = 0 this matrix is clearly negative definite, so for small enough ε > 0it is still negative definite. From negative definiteness it also follows that ourcandidate to a basis is indeed a basis.

3.6 Existence of non-conjugate points

In this short interlude we show that “almost all” pairs of points in a Riemannianmanifold M are not conjugate along geodesics.

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Proposition 3.31. Let M be a complete Riemannian manifold. Let p ∈ Mand v ∈ TpM . Then expp(v) is conjugate to p along the geodesic t 7→ expp(tv)if and only if v is critical point of expp.

Proof. Let γv : [0, 1]→M be the geodesic γv(t) = expp(tv).Let c :] − ε, ε[→ TpM be a smooth curve with c(0) = v. Then, defining

f :] − ε, ε[×[0, 1] → M by f(s, t) = expp(tc(s)). This is a variation of γv bygeodesics with starting point p, so the vector field

J(t) =∂

∂sexp(tc(s))

∣∣∣∣s=0

is a Jacobi field which vanishes at 0. Also,

J(1) =∂

∂sexp(c(s))

∣∣∣∣s=0

=(d expp

)v

(dc

dt(0)

),

so J(1) = 0 if and only if dcdt (0) ∈ ker(d expp)v.

Notice that, using the identity (8),

DJ

dt(0) =

D

∂t

∂

∂s(expp(tc(s)))

∣∣∣∣s=t=0

=D

∂s

∂

∂t(expp(tc(s)))

∣∣∣∣s=t=0

=D

∂sc(s)

∣∣∣∣s=0

=dc

dt(0).

The conclusion immediately follows.

Corollary 3.32. Let M be a complete Riemannian manifold and p ∈M . Then,for almost all q ∈M , the point p is not conjugate to q along any geodesic.

Proof. This is an immediate consequence of Proposition 3.31 together withSard’s theorem.

3.7 Compact Lie groups as Riemannian manifolds and themain theorem

Let G be a compact Lie group. Then there is an inner product ge on TeG whichis Ad invariant. This gives a Riemannian metric on G given by gh = L∗h−1gefor all h ∈ G, which is clearly left-invariant. It is also right-invariant, since gis invariant under Adh−1 = (Rh)∗ (Lh−1)∗ and (Lh−1)∗ , for all h ∈ G. We nowreturn to denoting g by 〈·, ·〉. We also denote by 〈·, ·〉 the inner product inducedon g.

The following proposition is standard in the theory of bi-invariant metricson Lie groups.

Proposition 3.33. In a compact Lie group with a bi-invariant Riemannianmetric and X,Y, Z ∈ g the following identities hold:

1. 〈[X,Y ], Z〉 = −〈Y, [X,Z]〉;

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2. R(X,Y )Z = 14 [Z, [X,Y ]].

Proof of Theorem 3.1. By Corollary ??, the Lie group G has the homotopy typeof a compact Lie group. It is thus enough to prove the case where G is compactand connected.

By Proposition 3.32, there exists g ∈ G which is not conjugate to e along anygeodesic. By Remarks 3.9 and 3.10, it is enough to show that π1(Ω(G; e, g)) = 0.Let α : [0, 1] → Ω be a continuous map. Since E is continuous and [0, 1] iscompact, there is a such that E(α(s)) < a for all s ∈ [0, 1]. Thus α can be seenas a continuous path in int Ωa. By Proposition 3.28, it is enough to show thatπ1(int Ω(t0, . . . , tk)a) = 0 for a sufficiently fine partition 0 = t0 < t1 < · · · <tk−1 < tk = 1 of [0, 1].

By Proposition 3.30 and Theorem 3.4, it suffices to show that given anygeodesic γ ∈ Ω the number of parameters in ]0, 1[ which are conjugate to 0 withrespect to γ, counted with multiplicities, is even. For this purpose, we writeR(V,W )V in a nicer way in order to explicitly solve Jacobi’s equation.

Let V ∈ g be the left-invariant vector field tangent to γ, as guaranteed byProposition ??. By Proposition 3.33, the map adX : g → g is skew-symmetric,so it is complex-diagonalisable with pure imaginary values λ1i, . . . , λni. Thus,it has an orthonormal basis

(V1, U1, . . . , Vn, Un,W1, . . . ,Wm)

such that

adX(Vi) = λiUi,

adX(Ui) = −λiVi,adX(Wj) = 0,

for i = 1, . . . , n and j = 1, . . . ,m. On the basis, the map Y 7→ R(X,Y )X, whichby Proposition 3.33 can be written as 4 adX adX , is then represented by thematrix

diag(−λ21,−λ21,−λ22,−λ22, . . . ,−λ2n,−λ2n, 0, . . . , 0

).

LetN = 2n+m and write (Y1, . . . , YN ) for the above basis and−4µ21, . . . ,−4µ2

N

for its eigenvalues.Now let J be a Jacobi field along γ which vanishes at 0. We can write

J =∑Ni=1 JiYi for some Ji ∈ C∞([0, 1]), and since the Yi are orthonormal and

eigenvalues of Y 7→ R(X,Y )X, the Jacobi equation becomes the system of Nequations

d2Jidt2

+ µ2i = 0, i = 1, . . . , N.

Thus, for each i = 1, . . . , 2n, we have Ji(t) = ai cos(µit) + bi sin(µit) for someai, bi ∈ R, and for each i = 2n + 1, . . . , N we have Ji(t) = ait + bi. SinceJ(0) = 0, we get bi = 0 and so each Ji(t) = ai cos(µit) for i = 1, . . . , 2n andeach Ji(t) = ait for i = 2n + 1, . . . , N . Thus the space of Jacobi fields along γ

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which vanish at 0 is generated by the basis of vector fields J ′i(t) = cos(µit)Yi(t)for i = 1, . . . , 2n and J ′i(t) = tYi(t) for i = 2n+ 1, . . . , N .

For i > 2n, the Jacobi field J ′i vanishes only at 0. For i = 1, . . . , 2n, each J ′ivanishes on a number of parameters which depends only on µi. Since µ2j−1 =µ2j for j = 1, . . . ,m, we conclude that the numbers of parameters in ]0, 1[ whichare conjugate to 0 with respect to γ, counted with multiplicites, is even. Thisfinishes the proof.

Remark 3.34. Of course, if G is connected then π2(G, x) = 0 for any base-pointx ∈ G.

3.8 Stronger results on Ω

In the proof of Theorem 3.1 we actually proved that, for any a > 0 the criticalpoints in int Ω(t0, . . . , tk)a have even indices. As such, as a consequence ofTheorem 3.4 and Propositions 3.28 and 3.26, we conclude that int Ωa has thehomotopy type of a CW-complex with no odd dimensional cells, and only finitelymany k-cells for each even k.

The following theorem, due to Bott, yields a stronger result, whose proofcan be found in [5].

Theorem 3.35 (Bott). Let G be a compact, simply connected Lie group. Thenthe loop space Ω(G) has the homotopy type of a CW-complex with no odd di-mensional cells, and with only finitely many k-cells for each even k.

Thus Hk(Ω(G)) = 0 for odd k and Hk(Ω(G)) is a free abelian group of finiterank for even k.

The proof of Theorem 3.35 is the same as the one done here, up to technicalresults on homotopy. In [5], two different ways to prove this are given:

• Proving explicitely that the space of piecewise smooth paths is homotopyequivalent to the space of continuous paths, which is a CW-complex;

• Or showing in general that the “homotopy direct limit” of CW-complexesis a CW-complex.

This description gives us all the information on the homology of the pathspace, provided we can compute the indices. For the homotopy groups (whichwould give the homotopy groups of G, by Remark 3.9) this is not so easy: weneed information on the gluing maps of the cells.

References

[1] L. Godinho, J. Natario, An Introduction to Riemannian Geometry WithApplications to Mechanics and Relativity, Springer International Publishing,2014.

[2] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

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[3] J. Hilgert, K.-H. Neeb, Structure and Geometry of Lie Groups, SpringerScience+Business Media, LLC, 2012

[4] A. Kono, M. Mimura, Cohomology operations and the Hopf algebra struc-tures of the compact, exceptional Lie groups E7 and E8. Proc. London Math.Soc. (3) 35 (1977), no. 2, 345–358.

[5] J. Milnor, Morse Theory, Princeton University Press, 1963.

[6] M. Mimure, H. Toda, Topology of Lie Groups, I and II, American Mathe-matical Society, 1991.

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