Morse Theory Flashcards
Discuss brief history of Morse theory
Started in 1930’s by Morse. In 1980’s entirely new way of thinking of field emerged –> Floer theory
What is Morse theory about?
P = Reasonable space
f: P –> R a reasonably smooth function
Goal: Relate critical points of f to the topology of P (homology, homotopy etc)
can work both ways topology –> critical points and critical points –> topology
In particular prove Morse Inequalities
|Crit_k(f)| >= dim H_k(P)
Inductive move from min to max. See how topology of sublevels change as we go through critical points
pg 9, 19
Discuss setting of Morse theory for f.d. manifolds
P = M = Closed manifold.
f at least C^2 normally just C^inf
Crit(f) = {p | df(p) = 0}
Main question: Produce meaningful lower bound for Crit(f)
pg 10
What is very weak lower bound on number of critical points of closed manifold?
2 - max min
pg 10
What happens if f:M –> R has exactly 2 critical points? M closed manifold
M homeo to S^2, not nec. diffeo
Use Morse theory
pg 10, pg 63
Discuss critical points of height function for closed surface. Lower bound? Draw examples
p in Crit(f) <=> T_pP horizontal <– prove
Almost true that >= 2 + 2g
It is possible to construct a function on surface of g >0 with exactly 3 critical points
But is true for broad class of functions –> Morse functions
pg 11-12
Discuss Lie derivatives
See Lee book
Define: Hessian and discuss/prove properties
quadratic form/bilinear pairing
14-16
Can we define Hessian when p is not critical point?
Example?
No - not well-defined
Can find coordinates so f(x) = x_1 <– Hessian = 0
or more generally f(x) = x_1 + any quadratic <– Hessian = whatever you want
pg 15
Consider P^n < R^n+1 and f projection onto some coordinate - x_n+1. Discuss how P is locally graph(f)
pg 17
Define: p non-degenerate
Morse index
Morse function
Crit_k(f)
pg 18
What happens to {f<c} sublevel sets as we move through an interval where c is regular?
Proof?
Nothing happens – diffeomorphism type does not change
Pf. Use anti-gradient flow, key tool of Morse theory. Fix Riemannian metric on P. Now have well-defined gradient that looks the same as classical gradient as long as we choose nice coordinates
Scale gradient to have speed -1 everywhere. Call this X.
phi^t = flow of X
By construction, f decreases along the flow lines of X with unit speed => f(phi^t(x)) = f(x) - t
phi^(b-a) gives diffeo
pg 20 -21
What is Morse Lemma? Importance? Examples?
Answers question of what happens at critical points
Prop. p in Crit(f) is non-degen => exists a coordinate system s.t. f(x) = f(p) + 1/2 (-x1^2 - x2^2 - … - xk^2 + xk+1^2 + … + xn^2) = f(p) + 1/2 d_p^2f
k = ind_p (f)
By local change of coordinates at p, eliminate higher order terms in Taylor expansion
going from d^2_pf to (-x1^2 - x2^2 - … - xk^2 + xk+1^2 + … + xn^2) just linear algebra - diagonalize
Index is only local invariant of non-degen critical point. Two critical points with same index are locally diffeo
Notice in examples critical points f = 0 lie on a variety with singularities – not smooth. All other level sets give smooth manifolds
pg 22-23
What is Hadamard Lemma? Why is result non-obvious? Proof?
Corollary?
Lemma. f: R^n –> R, f(0) = 0. Then there exist g_i: R^n –> R s.t. g_i(0) = df/dx_i(0) and f(x) = sum g_i(x) x_i near 0.
Can replace R^n by any star-shaped neighborhood.
Pf. FTC used cleverly
Corollary. Assume f(x) = f(0) + h_ij(x)x_ix_j
Note: In general, Hadamard lemma takes over role of converging Taylor expansion
pg 24-27
Prove Morse Lemma
- The question is local - work in R^n
- f(x) = Q(x) + R(x). Want a diffeo phi so f o phi = Q
- Use Moser Homotopy Method
- Set f_t = Q + tR <– homotopy between Q and f. Look for family of diffeos phi_t s.t. f_t o phi_t = Q
- Differentiate and look for a generating time dependent vector field v_t
- Get linear system and solve
pg 28 - 32
Discuss normal forms and singularities of functions
Consider examples
1. Matrices - Jordan normal form
- Symmetric matrices - diagonal with +1, -1, 0 (or eigenvalues if we only allow orthogonal transformations)
- Functions near 0 (germs) - ?
Now (3) in more detail:
(a) df_p != 0, then can write f(x) = x1 + c
(b) df_p = 0. Morse Lemma provides normal form as long as Hessian non-degenerate - only invariant is index
(c) Can go down hierarchy to more and more degenerate functions - classification of singularities very popular in 1970s - difficult
pg 33-35
Discuss attaching cells to topological space. Define CW complex. Discuss cell structures on surfaces and projective spaces over R and C
Just form disjoint union and quotient by relation. Build CW inductively, attaching cells in increasing dimension
pg 36-38
Discuss 2 criteria for homotopy equivalence from Hatcher
See chapter 0 of Hatcher
Prove: Attaching a cell to a CW complex arbitrarily gives a space homotopic to a CW complex
If we are attaching a k+1 cell, we need to push the attaching map, which might hit cells of dim > k, down into k-skeleton.
Idea: If S^k hits interior of ball, we can homotope S^k to boundary of ball by radially projection from point of ball that is not in image of S^k. Then continue pushing inductively until we have pushed into k-skeleton
Problem: Space filling curves.
Solution: Approximate attaching map by map which is smooth on part mapped into ball (Weierstrass approximation theorem). Then Sard’s thm shows curve not space filling.
pg 39-41
Discuss cellular homology intuitively - chain complex, differential
C_k = free module over Z generated by the k-cells
differential = sum degree of attaching maps * e_k-1
pg 42-43
Discuss cellular homology with differential coming from l.e.s. of pair. Prove we get a chain complex…
See Chris/Hatcher
Attempt to do from purely geometric description as well…he failed so don’t spend too much time
pg 44-46
Show cellular homology = singular homology whenever both are defined
So cellular homology is a homotopy invariant.
See Chris/Hatcher
pg 47-49
Use CW structure to calculate homology groups of surfaces and projective spaces over R and C
See Chris/Hatcher
pg 49
What can we say about topology of a sublevel containing a critical point? Corollaries? Proof of corollaries? Don’t prove main.
Thm. If only critical value in [a,b] is c, then Pb obtained from Pa by attaching a k-cell for every critical point x on the critical level f = c, where k = index(x).
Cor. P has homotopy type of CW complex with one cell for each crit pt with dim = index
In essense exists a complex (C_k, d) with C_k generated by Crit_k(f) over F
Cor. Morse Inequalities. |Crit_k(f)| >= dim H_k(P, F) = b_k or rk H_k((P, Z).
pg 50-51
Discuss refining Morse inequalities
Proof?
Consider 2 polynomials:
h(t) = sum dimH_k(P) t^k
m(t) = sum |Crit_k(f)|t^k
the Poincare polynomial and the Morse polynomial. Call coeffic of h b_k - betti and of m C_k
Prop. (1) There exists a poly r(t) “remainder” with coef >= 0 s.t.
m(t) = (1+t)r(t) + h(t) <– in particular coeff of m(t) >= h(t)
(2) Ck- Ck-1+Ck-2-… -+ C0 >= bk-bk-1+…
Cor. Euler Char of complex = Euler Char of manifold. Plug in -1
pg 51-52
Prove passing through a critical point adds a k-cell.
Idea: Use unit speed flow along gradient as we did for case without singularities. Use cut-off functions to ensure small neighborhood of singularity remains fixed.
This unretracted neighborhood can be homotoped to a disc…
pg 53-55
Discuss handles - examples of handles - and how Morse theory gives handle body decomp. Examples of handle body decomp for surfaces?
A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory.
The cellular decomp we obtain in Morse theory is actually a handle body decomp - attach a k-handle for each critical pt of index k
Cor. A closed smooth manifold has a “handlebody decomposition”
pg 55-57
Why doesn’t Morse lemma matter?
Already know that near a non-degen critical point the smooth and top picture is completely dominated by quadratic term - local dynamics
pg 57
Define perfect Morse function. Discuss motivation, steps to make perfect, possible? Examples? h-cobordism?
Start with some Morse function f. Want the simplest possible handle-body decomp - smallest number of singularities.
f is PERFECT if d=0 in the associated complex: |crit_k(f)| = b_k
Example surfaces. In reality perfect functions are rare - normally not possible to modify an in imperfect f to a perfect f.
2 steps always work to simplify:
1. Kill extra max’s and min’s
2.Can move lower index pts below higher index pts
In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
M –> W and N –> W
are homotopy equivalences. (Recall: Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher)
The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.
The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture.
pg 58-59
Discuss classification of closed surfaces via Morse theory
Hirsch
pg 60
Discuss the Poincare conjecture and Morse theory
Milner
pg 60
Discuss Heegard Splitting and Morse theory. Proof?
For every closed orientable 3-mfld P, exists a diffeo phi: S_g –> S_g s.t. P = union of 2 solid bodies with g handles glued via diffeo phi.
Proof. Mores theory…nice
pg 60-62
What is Morse complex? Goal? Idea? Do examples of torus and weird sphere
Form a chain complex with chains generated by critical points.
Goal: describe differential explicitly without using algebraic topology or CW complex structure
Idea: m(x,y) = number of trajectories from x to y. <– this depends on choice of metric
If index differs by 1, there are generically just a finite number of anti-gradient trajectories from x to y.
pg 65-66
Define: mu(x,y). Key theorem? Proof?
Stable, unstable manifolds?
mu(x,y) = { z | phi_t(z) goes to x as t –> -inf and y as t –> inf}
where phi_t is the anti-gradient flow of f and x, y critical points.
Call mu(x,y) moduli space
Thm. For a generic metric, mu(x,y) is a smooth manifold of dimension mu(x) - mu(y) <–difference in index.
stable = incoming trajectories
unstable = outgoing trajectories
Each of these diffeo to open balls - by Morse Lemma locally just quadratic … think of parabolas on different axises
mu(x,y) = W^u(x) int W^s(y) and with a generic metric, we can ensure these sub-manifolds intersect transversely
=> m(x,y) is a manifold and dim mu(x,y) = n - dim W^u(x) - dim W^s(y) = … = mu(x) - mu(y).
pg 68-70
Discuss trajectories of height function on torus. Stable/unstable manifolds
Generically, we shouldn’t have trajectories between critical points on saddle.
Tilt slightly.
pg 70
Discuss how to order critical points s.t. lower points have lower index.
Sliding handles. We start with an arbitrary Morse function f and modify it by moving critical points past each other
pg 71-72
Discuss the modern perspective on mu(x,y)
think of as set of solutions of antigradient equation instead of thinking of as submanifold of P.
mu(x,y) = space of parameterized trajectories
Note: we have a free R-action on mu(x,y). Taking quotient get space of unparameterized trajectories mu hat(x,y) = mu(x,y)/R
This is a smooth manifold of dim mu(x)-mu(y) - 1.
In particular, when m(x) - mu(y) = 1, this is a 0-dim manifold = discrete collection of pts
pg 73
Discuss compactification of mu hat (x,y). Importance?
This is a key result which requires the modern view - m(x,y) not being subset.
Thm. For a generic metric, mu hat has a compactification formed by broken trajectories – form manifold with corners.
In the case where index differs by 1, there are no broken paths. mu hat already compact –> finite set. Only finitely many trajectories from x to y.
pg 74
Define Morse differential over Z
Do examples of torus and weird sphere
Need to account for orientations. Fix orientations of stable and unstable manifolds by orienting tangent spaces. This gives orientation of mu(x,y)
Trajectories also oriented by flow so we have 2 orientations on each trajectory. If they agree +1 if they disagree -1.
pg 75-76
Show Morse boundary^2 = 0, i.e. the Morse complex is actually a complex
Explicitly computing we see that we need to sum over broken trajectories with one brake between critical points whose index differs by 2. mu hat (x,y) is a 1-dim manifold in this case. Compactification is either S^1 or closed interval. If S^1, no broken trajectories - already comapct. Otherwise, we have a pair of broken trajectories - two end points of interval…
pg 77
Show Morse homology and cellular homology are the same
Claims differential is the same <– try to check. Note: we proved cellular = singular = topological invariant. So we see Morse homology is a topological invariant of manifold — actually homotopy invariant.
pg 78
Do examples of computations with Morse homology – surfaces, projective spaces Real and Complex
pg 79-81
Discuss Kunneth formula. Proof?
Kunneth: P0, P1 closed manifolds. Then H_(P0 x P1) = H_(P0) (x) H_*(P1).
Proof. Choose Morse functions f0, f1 on P0, P1 and generic metrics. Product metric on P0 x P1.
pg 83
Discuss Poincare duality. Proof?
P smooth closed manifold - homology in dim k matches cohomology in codim k. Palindromic
b_k = b_n-k
Pf. Idea: Choose a Morse function f and look at f and -f. Gradient flows in opposite direction – critical points flip index.
pg 84
What can be said about the existence of Morse functions? (Just statements, no proof)
Thm. If P is a closed manifold, then Morse functions form an open dense subset in C^k(P).
Corollary. Every manifold admits a Morse function. Moreover, every function can be C^k-approximated by Morse functions.
Recast the definition of critical points, non-degenerate, and Morse function in terms of graphs and transversality
Identify the 0 section with P
x is critical <=> x in graph(df) int P
x is nondegenerate <=> graph(df) int P transverse at x
f is Morse <=> graph(f) transverse to P
pg 86-87
Discuss the idea of proof of Morse function using transversality. Difficulty?
- Start with f0 and approximate by Morse function
- Transversality thms => df0 approximated by 1-forms whose graphs intersect P transversely.
Difficulty: Need 1-forms exact i.e. = df
pg 87
Define: F transverse to Z in Y
dF and TpZ together span TpY
pg 88
Let L0 and L1 in R^3 be two embedded loops. For almost all v in R^3 (L0 - v) int L1 = empty.
Proof?
pg 88-89
Discuss transversality theorems - increasing constraints on F (4 settings)
X a closed manifold.
Thm 1. F transverse to Z for open dense set in C^k(X,Y)
Y = X x K > Z, F: X–>K
Thm 2. Graph(f) transverse to Z for an open dense set in C^k(X,K)
K –> Y –> X Fiber bundle, Z < Y, s a section.
Thm 3. s transverse to Z for an open dense set of C^k sections
First jet bundle J^1X = T*X x R –> X
First jet of function j^1f = (df, f)
Thm 4. j^1f transverse to Z for an open and dense set of f in C^k(X) k>= 2
pg 90 - 93
Discuss proof of existence, density of Morse functions via Jet transversallity
As a result of Jet transversality, we see that df transverse Z in T*X for an open and dense set f in C^k(X) k>= 2.
Now just take Z= P = zero section and we’re done.
pg 93-94
Discuss proof of ordinary transversality thm
Guillemin-Pollock
pg 95-96
What is C^k topology?
Look up on internet
Discuss how –
Thm: Morse Functions form an open and dense set in C^k(P)
– follows from thm about height functions
Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse.
h_v is projection onto line spanned by v. This can be interpreted as a height function
pg 97-98
Prove:
Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse.
In the case P a closed hypersurface
Prop. H-v Morse <=> +-v are regular values of Gauss map G
This implies thm, sense regular values form open dense subset of full measure by Sard’s thm.
Notice Crit(h_v) = points where tangent plane is horizontal = points where unit outer normal is vertical –> v or -v = G^-1(v) U G^-1(-v)
pg 99-102
Discuss Sard’s Thm, Normal bundle, unit normal bundle, tubular neighborhoods, Gauss map
Lee Ch 6
pg 103
Discuss the proof of the general case
Thm. For an open dense (full measure) subset of S^m-1, h_v is Morse.
With the generalized version of Gauss map, have: h_v Morse <=> +- v regular values of G
as before.
PG 104 - 106
Define/discuss cup product and intersection product
One way to think of cup product is wedge product of differential forms.
If closed manifold, have Poincare duality - can use to define product in homology
Want to think of intersection product as intersection of 2 chains yielding another chain –> think of as everything as submanifolds
Also read in Hatcher
pg 107
Discuss 2 interpretations of intersection product in terms of Morse homology
This requires modern perspective in terms of Morse complex, not just handlebody stuff.
- Think of Morse homology groups as module over geometric homology groups. - product obtained by looking at trajectories from x to z that pass through B…
- Use 3 Morse functions to define product…
pg 107 - 111
What is the setting for Lusternik-Schrirelmann theory? Compare to Morse theory.
Again we have f:P–>R and want to understand lower bound on |Crit(f)| but here we do not require f non-degenerate.
Underlying idea LS and Morse the same: change in topology from sublevel Pa to Pb => critical pts between a and b.
LS better adapted to inf dimensional settings – think variational problems
Morse deeper connection with topology in finite dims
pg 112
Compare lower bounds on critical pts for surfaces, spheres, projective, tori for Morse and LS
In general, LS very far from sharp but any P^n admits f with |Crit(f)| <= n+1. By this, the examples given are sharp.
pg 113
Define: LS category, examples? Pros/cons
minimum size of an open cover by contractible sets (IN AMBIENT SPACE - S^1 can be contractible) , Contractible sets need not be connected - may have nontrivial topology
cat(S^n) = 2
cat(surface) = 3
cat(CW) <= n+1
Difficult to compute - upper bounds by explicit constructions. Lower bounds??
Advantage: easy definition
pg 114 - 115
Define: cup length, examples?
advantages, disadvantages
max product of cohomology classes of positive degree before getting 0.
Clearly between 0 and dim X
depends on ground field
Easier to calculate. Harder to define - need cohomology
pg 116 -117
Discuss with proof the relationship between LS category and cup length
cat(X) >= cl(X) + 1
claim: A < X contractible to a point => H(X,A) –» H(X) onto.
Note: It need not be 1-to-1. Consider collapsing a circle to a point on S^2, creates a wedge of two S^2
pg 117-119
Discuss cat_X(A). Advantages over cat(A)?
covering A, but contractible in ambient space X.
With this notion, monotone, continuous, homotopy invariant, subadditive…
pg 119
Discuss with proof the relationship of |Crit(f)| and cat(P). Corollaries?
LS approach vs min/max approach
Topological approach (LS approach) to proof. P closed manifold.
Thm. Assume that Crit(f) are isolated. Then |crit values of f| >= cat(P).
Cor. |Crit(f)| > = cat(P) >= cl(P) + 1
Inequalities established for some of our favorite examples are sharp.
Pf. Idea: what is responsible for presence of critical points is change in topology.
Proceed by induction. Since critical points isolated, can cover every critical point on a critical level by little geodesic ball which is contractible. Using anti-gradient flow we see the rest is diffeomorphic to sublevel from previous induction step.
pg 120 - 122
Min/Max Approach
By “LS Inequality” since critical pts isolated, intersecting with non-unit homology class strictly reduces the value of the critical value selector. Writing a maximal intersection = cup length, we get k+1 distinct critical values.
pg 138
How few critical points can a function on P have? Pf idea?
n+1 critical pts.
pg 123-125
What is the min/max principle? Examples
Says that inf sup f|A is a critical value
min f = given by choosing F = collection of all pts of F
max f = given by choosing F = {P}
pg 126-127
Prove: Min/Max principle
pg 128-129
3 definitions of CRITICAL VALUE SELECTOR?
Example? Equivalence? Properties?
Let alpha in H_k(P) and define F_alpha = { A : [A] = alpha} = {cycles representing alpha}
Def 1. C_alpha(f) = inf_F_alpha max_A f
Def 2. Assume f is Morse. Then A = linear combo of critical points (in Morse homology) <– if P compact as we assume throughout, there are finitely many critical points so inf becomes min and we only have to calculate on finitely many chains. (just max/min over critical points having non-zero coeff in chain)
If f not Morse, approximate by Morse and take limit
Def 3. c_alpha(f) = inf {a <– regular value : alpha in image(H_(P_a) –> H_(P) }
Properties:
1. Criticality
2. monotonicity
3. C^0 continuity
4. subadditivity
pg 130 - 136
Discuss/prove LS inequaility
Assume critical points are isolated and beta != fundamental class, then:
C_alpha int beta (f) < c_alpha(f) KEY: inequality strict.
IDEA: intersecting A with B we get a submanifold of positive codim in A. No reason for it to pass through the isolated max point in A. measure 0 chance
pg 137
Discuss/prove Courant-Fischer Thm
Thm, Let Q(x) be a quadratic form, f = Q|S^n, and l1 <= l2 <= … <= ln the eigenvalues of Q. Then lk = inf max f|Lk
Pf. Crit(f) = unit eigenvectors
cv(f) = eigenvalues
This is an immediate consequence of Lagrange multipliers…
pg 141-144
Discuss the number of distinct closed geodesics on S^2
Original motivation for LS theory. Any metric on S^2 has >= 3 simple closed geodesics.
Pf. Difficult. Cleaned up by Ballman – required tool of curvature forms to develop.
Outline:
1. Look at space Lambda of smooth embedded loops in S^2
2. Lambda ~ RP^2 ~ Space of Great circles
3. Critical points of the length functional L:RP^2 –> R = simple closed geodesics
4. LS theory => |Crit(L)| >= cl(Lambda) +1 >= 2+1 = 3
pg 145-146
What is Polais-Smale condition? Importance?
P = f.d. or Hilbert/Banach manifold
f: P –> R satisfies PS if every sequence {xi} < P s.t. |f(xi)| < C < inf and df(xi) –> 0 contains a convergent subsequence.
IMPORTANT: This condition is very fragile. Easily breaks down.
Without PS, change of topology from Pa to Pb does not imply existence of critical points. Loose connection between topology and critical points.
pg 147 - 148
Discuss assumptions accompanying PS which guaranty min value obtained (critical point exists)
- f bounded from below
- f satisfies PS
- The flow for anti-gradient-like vector field is defined for all t >= 0.
=> f has a critical pt, actually min is attained
Pf. Start with x in P, Create sequence using anti-gradient-like flow. PS + bounded from below ==> converges to a critical point y.
pg 149
Discuss min/max principle in PS setting. Proof?
Works out very similarly to min/max in compact case and example of PS
pg 150-152
Discuss 4 different definitions of geodesic
Let M be a closed manifold with fixed Riemannian metric.
Def 1. A geodesic is a curve locally minimizing length
Def 2. A geodesic is a critical point of the length functional on space of all curves between p and q.
Def 3. A geodesic is a critical point of the energy functional.
Def 4. 0 acceleration curves
3 => 1,2
pg 153-155, 156, 158
How can you think of critical points in inf dim spaces?
The change in function over any variation = 0.
pg 155-156
How does a critical point of energy functional come with a natural parameterization?
pg 157
Define: injectivity radius
an r > 0 s.t. any pts in a r-ball are connected by a unique geodesic.
radius where exponential map is injective
pg 158
What is Cartan’s Thm on closed geodesics? Pf?
Discuss pf via finite approximation and infinite dim approach
Thm. For any non-trivial free homotopy class [alpha] in [S^1, M] there is a closed geodesic representative of [alpha].
Idea: take shortest loop in [alpha] or minimize energy over all geodesics
Pf.
- Introduce idea of geodesic polygons with edge lengths < injectivity radius
- Every loop in [alpha] can be approximated by geodesic polygons in [alpha]. Every connected component of loop space contains geodesic polygon ==> P_k int Connected component of alpha = union of components of P_k = smooth manifold with boundary
- Replace E by cursive E (write CE). See both have same critical points. (broken geodesic actually a geodesic if critical point of CE)
- Find minimum
pg 159-168
175-179
What is Lusternik-Fet thm? Proof?
Discuss pf via finite approximation and infinite dim approach
Thm. If pi_1 = 1, there exists a non-constant closed geodesic.
Pf.
1. Topology of loop space: e: Gamma –> M. Use LES of fibration…
2. State exactly how P_k approximates Lambda –> can get isomorphisms of homotopy groups for all pi_n with n < N <– arbitrary, by taking k big enough.
pg 169 - 179
Discuss and prove Bott periodicity of U(n)
pg 202
