Algebraic Topology Flashcards
Define: deformation retraction
A deformation retraction of a space X onto a subspace A is a family of maps f_t:X –> X for t in [0,1] s.t. f0 = I_X f1(X) is in A and f_t restricted to A is identity in A for all t and f_t jointly continuous.
1st semester pg 74
Examples of deformation retracts?
R^n to 0, R^n - {0} to S^n-1, Mobius band onto circle, Disk D^2 with 2 open disks removed pgs 75-76
Define: Homotopy, homotopic maps
a family of maps F_t: X –> Y st F:X x I –> Y is continuous. Say f0 and f1 are homotopic maps pg 77
Define: Homotopy relative to A
Describe def retraction in this language
If A a subspace of X, a homotopy relative to A is a homotopy F_t : X –> Y st f_t|A does not depend on t - i.e. points of A don’t move
A deformation retraction of X onto A is a homotopy rel A from identity on X to a map r: X –> A
Define: homotopy equivalence
A map f:X –> Y is a homotopy equivalence if exists a map g: Y –> X st gof = I_X and fog = I_Y
Generalizes homeomorphism
Describe how deformation retraction gives homotopy equivelence
S1 pg 78
Define: homotopy type
Compare to homeomorphism
If spaces X and Y are homotopy equivalent, we say they have the same homotopy type
looser than homeomorphism - R^2 and R not homeo but both def retract to a point so homotopy equiv. Measures in some sense the connectedness properties of spaces
Define: contractible
X is contractible if X has the same homotopy type as a one-point space
Prove: X contractible iff I_X nullhomotopic
nullhomotopic = homotopic to a constant map. pg 80
Define: CW Complex, attaching maps, characteristic maps
pg 81 weak topology Hatcher 5,7
What is a CW complex graph?
Just a 1-dim CW complex
Discuss cell structures for S^n
pg 84,
- S^n = D^n / boundary(D^n) collapse boundary to a point. Two cells: one zero cell, one n-cell. phi maps boundary to 0-cell
- Two hemispheres attached along S^n-1. Inductive. Two cells of each dimension
- Iterated suspension of 0 sphere pg 92
Note 2 and 3 same
Discuss constructions of RP^n
- Space of all lines through the origin in R^n+1 - modulii space of 1-dim subspaces - generalized by Grassmanian
= R^n+1 - {0} / x ~ lamda*x for all lamda in R - {0} - S^n/ x ~ -x
- D^n / x ~ -x for all x on boundary of D
Viewing RP^n as D^n/~ suggests a simple cell structure. The boundary of D^n is a copy of RP^n-1 so we RP^n has a cell in each dimension:
85 - 88
What is Closed Map Lemma? Proof?
Mapping from compact space to Hausdorff space –> always closed map
Define: subcomplex of CW complex, CW pair
Examples?
A closed subset this is a union of open cells (Hatcher just calls these open cells e^n_alpha cells) - notice a subcomplex is a CW complex in its own right
If X = CW and A = subcomplex, (X,A) is a CW pair
n-skeleton best example
Topological properties of CW complexes?
- Finite CW complex is compact
- Every compact subset lies in a finite subcomplex
- A set is open [closed] if its intersection with the closure of each cell is open [closed] - i.e. weak/coherent topology
- CW complexes are normal
- Locally path-connected ( so connected iff path-connected)
pg 89-90
List basic topological operations on spaces.
- Suspension: SX = X x [0,1]/ ~ where X x {0} is collapsed to a point and X x {1} is collapsed to a point
- Cone: CX = X x [0,1]/~ where X x {0} collapsed to a point
- Join: X * Y = X x Y x [0,1]/~ where we collapse X x Y x {0} to X and collapse X x Y x {1} to Y. X on one end of space, Y on other end. Cone, suspension are special cases
- Wedge sum: Glue two spaces at a point X v Y
Discuss criteria for homotopy equivalence
Examples?
- Collapsing Subspaces - If (X,A) a CW pair st A contractible, the quotient X –> X/A is a homotopy equivalence
- Graphs - maximal tree
- S^2 v S^1 = S^2/S^0 - Attaching Spaces - If (X1,A) a CW pair and f:A –> X0, g:A –>X0 are two attaching maps, then if f homotopy equiv to g, the spaces attained by attaching X1 to X0 are homotopy equivalent
- Dunce Cap
- S^2 v S^1
What is the homotopy extension property?
Equivalent property?
A pair of top spaces (X,A) satisfies the hom extension property if given any map f0: X –> Y and a homotopy f_t : A –> Y, there always exists an extension to a homotopy f_t: X –> Y
A pair (X,A) has the h.e.p. <=> X x {0} U A x I is a retract of X x I.
Prove: Every CW pair (X,A) has the Homotopy Extension Property
pg 99
Prove: If (X,A) has the H.E.P. and A is contractible, then q: X –> X/A is a homotopy equivalence.
pg 101
Prove: If (X,A) is a CW pair and attaching maps f,g: A –> X0 are homotopic, then the two spaces obtained by gluing are homotopy equivalent [rel X0]
pg 103. Do an example with simple spaces
Discuss the classification of surfaces, CW structure, proof idea
pg 105-111. Key points: Compact no boundary. Orientable vs. Nonorientable.
Rado - Every surface has a triangulation
Part 1 - Show every compact surface without boundary is homeo to one of the following S^2, M1, M2, M3, … N1, N2, N3, ….
-Triangulate and manipulate into nice form - polygon by glueing - cut and paste all justified since compact Hausdorff
Part 2 - These surfaces are all in fact topologically distinct (different homology groups)
Basic approach to showing two spaces are homeomorphic? not homeomorphic?
Homeomorphic: Exhibit explicit homeomorphism between the spaces - cts bijection with cts inverse
Not Homeomorphic: Harder - need topological invariants
Show B(0,1) is homeo to R^2
pg 1
Discuss how to visualize R, R^2, R^3 not homeomorphic
lasso, disconnected, bubble pgs 2- 3
What is the main aim of algebraic topology
Convert topology to algebra. Generate algebraic invariants of a space. Functors from Category of top spaces and cts maps to some algebraic category. Here we will primarily be going to groups and homs
What are the basic types of homology? pros/cons
- Simplicial - combinatorial
- space made of simplices
- simplest defs - Singular - maps of simplex onto space. Most abstract - easy invariance
- Cellular - ‘polygons glued together’ easy examples
All agree where defined
Define: standard n-simplex, barycentric coordinates, vertices, dimension, face, boundary, open simplex
Discuss properties
pg 6 : let ai = (0, … , 0 , 1 , 0 , … 0) where 1 is in the ith spot. The standard n-simplex is the set of all points x = sum tiai s.t. ti sum to 1 and each ti >= 0
convex hull of {a0, …. , an}
barycentric coordinates are the ti
ai are the vertices, they span the simplex. A face of an n-simplex is a simplex spanned by a proper nonempty subset of {a0,…an}
The boundary of a simplex is the union of all proper faces
An open simplex or interior is just simplex - its boundary. All points of sigma st t_i(x) >0 for all i
n-simplex is compact, n dimensional, homeomorpic to closed n-ball
Define: simplicial complex, abstract simplicial complex K, dimension of simplex, dim(K), n-skeleton, vertices
Example?
Motivation?
A simplicial complex K in R^N is a collection of simoplicies in R^N st
(1) Every face of a simplex of K is in K
(2) The intersection of any two simplexes of K is a face of each of them
An abstract simplicial complex K consists of. collection S of finite nonempty sets called simplices st every nonempty subset of a simplex is also a simplex
dim(simplex) = cardinality - 1
dim(K) = sup dim(simplices in K) - could be infinite
dim(empty set) = -1
n-skeleton K^(n) = all p-simplices for p <= n
vertices are just the single elements
pg 8-9
Geometric approach gets very messy - all the essential data is in the combinatorial structure of the vertices
Define: simplicial map
example?
A simplicial map f: K1 –> K2 is a function defined on vertices V(K1) –> V(K2) st if x is a simplex, then f(x) is a simplex
ie a map defined on vertices sending simplices to simplices
Define: geometric realization of an abstract simplicial complex
Example?
pg 11 - 12
First, the geometric realization of an n-simplex is a standard geometric n-simplex . View as a set of functions from vertices –> R.
The geometric realization of an abstract simplicial complex K with vertex set V is denoted |K| inside R^V and defined |K| = {all functions a:V –>R} s.t.
- For all a in |K|, {v in V : a(v) != 0} is a simplex of K
- For all a in |K|, sum over V of a is 1
- For all a, v, a(v) is in [0,1]
How can the geometric realization of an abstract simplicial complex be turned into a top space?
What choice do we make?
- The metric topology using l_2 metric on R^V - not very natural
- The weak/coherent topology on |K|. Each geometric simplex lies in a copy of R^n+1. Give the simplex the standard topology from R^n+1 (metric). Define a subset A in |K| to be open [closed] if its intersection with each simplex is open [closed]. “paste together subspace topologies”
We use weak topology!
When is a function f: |K| –> X continuous? where K is an abstract simplicial complex
Is |K| Hausdorff?
iff its restriction to every simplex is cts
yes, can pull back a separation of points from |K| –> |K|_d
Prove: Any subset A of |K| contains a discrete subset with exactly one point from each open simplex meeting A
Every compact subset of |K| meets only finitely many open simplices
pg 14
Define: Locally finite (in context of simplicial complexes)
Locally finite means each vertex is in only finitely many simplices - in this case the metric topology on |K| matches the weak topology
Define: triangulated space
Examples of triangulations? Can all topological spaces be triangulated?
A top space is triangulated if X is homeomorphic to |K| for some abstract simplicial complex K
Polyhedra - built from the basic building blocks of lines, triangles, tetrahedra, n-simplices
The triangulation conjecture—first formulated by Kneser in 1924—claimed that every manifold was triangulable. The conjecture turned out to be false in general, although it is true for manifolds of dimension up to 3, and also for all differentiable manifolds
Floer homology - Manolescue 2016
Discuss calc 3 motivation to homology
pgs 17-18
When are two orderings of vertices in a simplex equivalent? What are equivalence classes called?
Equivalent if they differ by an even permutation - called orientations [v0, v1, … , vn]
pg 18
Define: group of (oriented) p-chains on a simplicial complex K
C_p(K) = abelian group gen by all oriented p-simplices with relations o = -o’ if o and o’ are opposite orientations of same simplex - “integer linear combinations of a bunch of simplices of order p”
free abelian with basis given by choosing an orientation for each p-simplex
Define: boundary map for simplicial complex K
Most important property? Proof?
Examples?
del_p:C_p(K) –> C_p-1(K) is defined on each oriented simplex, then extended linearly
del_p[v0, … , vp] = sum_{i=0}^p (-1)^i [v0, …, vi hat, … vp]
key property: del^2 = 0
these alternating sums must have something to do with differential forms
pgs 20-22
Define: p-cycles, p-boundaries, pth homology group of K
K a simplicial complex
p-cycles: Z_p(K) = p-chains without boundary
p-boundaries: B_p(K) = p-chains that are a boundary of a p+1-chain
Z_p(K) = ker del_p
B_p(K) = im del_p+1
del^2 = 0 implies B_p(K) < Z_p(K) – every boundary is a cycle!
H_p(K) = Z_p(K) / B_p(K) = cycles/boundaries
pgs 23-24
Do examples of homology groups via simplicial homology for loop graph and filled in square
pgs 24-25
Define: homologous p-cycles, homologous to zero
chain c carried by a subcomplex L
p-cycles are homologous if they differ by a p-boundary. They represent the same element in quotient group H_p(K). i.e. c -c’ = boundary_p+1(d) for some p+1 chain d
homologous to 0 if boundary_p+1(d) = c. Also say “c bounds”
Compute homology of T^2 using simplicial complex
pg 29 - 31 Show we can push off to outer edge
H2 = Z, H1 = Z^2 H0 = Z
Munkres 34 - 36
Prove: The decomposition of simplicial complex |K| into components {K_alpha} gives an isomorphism H_p(K) = direct sum over alpha H_p(K_alpha)
What about singular homology?
Simplicial: pg 33
Singular: Look at path components of X.
Delta^n path connected, so image must be path connected so must be contained in path component of X. boundary also lies in the same path component
pg 114
Discuss H_0(K) in simplicial setting. Proof?
What about in singular homology?
If |K| connected, then H_0(K) = Z. In general free abelian with one factor for each component.
pgs 34-35
Munkres 41
Singular: In general have direct sum of Z’s, one for each path component
pg115
Define: Chain complex
A sequence of abelian groups with boundary homomorphisms satisfying boundary^2 = 0. Homology groups H_p(C) = im boundary_p=1 / ket boundary_p
pg35
Define: reduced homology
simplicial vs. singular?
We let epsilon: C_0(K) —> Z take sum n_i*v_i —> sum n_i
epsilon is called the “augmentation map” of C_0(K) and define the “augmented chain complex” which just has epsilon tacked on at the end.
Confirm epsilon after boundary = 0.
Reduced homology groups are just H tilda _p(K) = H_p(K) if k>0 and ker(epsilon)/im(bdy) if p =0
The only thing that changes is 0 dimensional homology group
singular on pg 115
What is the relationship between reduced homology and standard homology?
Only dim 0 changes. H_0(K) = H_0 tilda(K) + Z
H_0(K) = 0 when K is connected - closer to measuring 0 dim holes - disconnection
Define: join of simplicial complex
Examples?
Let K_1 and K_2 be simplicial complexes. The join K_1*K_2 is a simplicial complex with vertx set Vert(K_1) disjoint union Vert(K_2) with simplices sigma_1 disjoint union sigma_2 for sigma_i a simplex of K_i
pg 38-40
Cone and suspension
What are the homology groups of a cone? Proof?
Define: bracket operation
A cone has trivial reduced homology - i.e. is acyclic
pg 41-42
For bracket see Munkres 45
Compute the homology of (n-1) - sphere using boundary of simplex
pg 43 -44
Munkres 46
Define: pair of simplicial complexes
(K,L) K = simplicial complex, L = subcomplex
Define: relative chains for simplicial pair (K,L)
C_p(K,L) = C_p(K) / C_p(L) i.e. the free abelian group with basis elements sigma^p + C_p(L) where sigma^P is in K - L.
cosets are represented by chains not involving L
pg 45
Define: relative homology groups
Discuss also for singular homology
First, we define a relative chain complex by observing boundary maps are well-define on quotient groups and boundary^2 = 0. Define H_p(K/L) = ker(boundary_p)/ im(boundary_p+1) = Z_p(K,L) / B_p(K,L)
pg 46, 56
pg 127
Discuss relative cycles and relative boundaries - draw example
pgs 47 - 48
What is the Excision Theorem - simplicial setting - intuition?
Proof?
Now discuss in singular setting, what changes?
The main idea is that H_p(K,L) ignores everything inside L (treats it as 0) so then modifying K inside of L does not change H_p(K,L)
Excision Thm. Let A and L be subcomplexes of K with A union L = K. Let B = A intersect L. Inclusion induces an isomorphism H_p(A,B) —> H_p(K,L).
Proof. Show the two chain complexes are isomorphic so that their homology groups are isomorphic – pg 50
Singular: Much more complex…source of nearly all explicit computations of singular homology groups of topological spaces.
Suppose B < A < X with B closure < int A. Then (X - B, A - B) —> (X, A) induces an isomorphism of singular homology groups. pgs 129 - …
Strategy:
- Define subdivision sd
- sd is chain homotopic to identity
- Iterated subdivision leads to arbitrarily small singular simplicies
- Hi(X) is determined by the “small” simplices
With this in hand, the excision thm is easy to prove: pg148
Hatcher 117
How does a simplicial map induce a chain map? - simplicial
vs.
How does a continuous map induce a chain map? - singular
simplicial: F_sharp [v0, … , vp] = [f(v0) … f(vp)] where [f(v0) … f(vp)] = 0 if any of the vertices are repeated pg 51
singular: f_sharp Sn(X) –> Sn(Y) is composition of maps f after sigma pg 117
The key property of chain maps is
boundary f# = f# boundary
Prove: A simplicial map f: K —> L induces a homomorphism f_* : H_p(K) —> H_p(L)
Same for continuous map inducing a homomorphism
51 and 117
Discuss simplicial and singular homology categorically
Simplicial: Covariant functor from category of simplicial complexes and simplicial maps to category of abelian groups and homomorphisms
Singular: Covariant functor from category of topological spaces and continuous maps to category of abelian groups and homomorphisms
Define: chain homotopy simplicial vs. singular
A chain homotopy is a homomorphis between f_# and g_#
D:C_p(K) —>C_p+1(L) st boundary(D sigma) = g_#(sigma) - f_#(sigma) - D(boundary sigma)
singular: 120
Simplicial: When do 2 simplicial maps induce the same homology map? Proof?
Singular: When do 2 continuous maps induce the same homology map? Proof. Corollaries?
If f and g are chain homotopic. See book
If f and g are homotopic. So the homology maps depend only on the homotopy class of f. pg 119
Cor. If f: X –> Y is a homotopy equivalence, then f_* : Hn(X) —> Hn(Y) is an isomorphism - i.e singular homology depends only on homotopy type
Cor. If X is contractible, then H_n(X) tilda = 0 for all n - acyclic.
Define: exact sequence of abelian groups
A sequence of homomorphisms … –> A_n+1 —> A_n —> A_n-1 —> … is exact at A_n if im alpha_n+1 = ker alpha_n
Exact sequence means exact at each term
Give examples of exact sequences
- surjection
- injection
- isomorphism
- s.e.s
Define: split short exact sequence
Examples/ nonexamples
Let 0–>A–>B–>C–>0 be a s.e.s. The following are equivalent:
- The short exact sequence splits
- Exists j:C–>B s.t, b after j = Id_C
- B = A + C
- Exists i:B–>A s.t. a after i = id_A
Z x2 map
Prove: If 0–>A–>B–>C–>0 is s.e.s. and C is free abelian, then the s.e.s. must split
Can easily define j:C –> B by mapping basis of C to preimage pg 54
Define: homomorphism of chain complexes
pg 55 A sequence of homomorphisms making diagram of two chain complexes commute
-also called a chain map
What is The Long Exact Sequence of a Pair?
Simplicial vs. singular?
Proof?
If (K,L) is a simplicial pair, the following sequence is exact:
… –> H_p+1(K,L) –> H_p(L) –> H_p(K) –> H_p(K,L) –> H_p-1(L) –> …
Discuss in terms of short exact sequence of chain complexes.
Boundary is connecting homomorphism
Proof:
- Define boundary*
- Show boundary* well-defined
- Show boundary* is homomorphism
- Check exactness via diagram chase
pg 57 - 63
pg 127 same proof works for either…if we have good pair can prove something stronger - replace H(X,A) with H(X/A)
Define: reduced homology exact sequence of pair (K,L)
Simplicial vs. singular?
Same as Long Exact Sequence of Pair but we use augmented chain complexes
pg 64
Compute homology of S^n using long exact sequence of pair - simplicial
vs.
singular
pg 66
pg 123
Show that oridinary and reduced homology are both special cases of relative homology
pg 67
What is Exact Homology sequence of a triple (K, L2, L1)?
L1 < L2
pg 68
In what sense is the Homology Exact Sequence Natural?
Simplicial vs. singular?
Given a homomorphism from one ses of chain maps to another, we get a homomorphism of homology long exact sequences. pg69-71
What is Five Lemma? Proof?
5 maps, 4 known to be isomorphisms, then 5th is…
pg 72
Define: cokernal of abelian group homomorphism
If f: G –> H then cokernal is H/f(G)
What is a free abelian group?
An abelian group with a basis as a Z-module–> each g in G can be written uniquely as finite sum n_i g_i with n an integer
Define: torsion subgroup, torsion free, internal direct sum of abelian groups
The set of all elements of finite order in G is the torsion subgroup. torsion free if torsion subgroup = 0.
If there is a collection of subgroups of G, G_i st each g in G can be written uniquely as a finite sum of gi, the G is internal direct sum of the G_i
Discuss attaching spaces along a subspace.
Define: mapping cylinder
Importance?
Define: mapping cone
Start with a space X0 and another space X1 that we wish to attach to X0 by identifying points in a subspace A in X1 with points of X0. The data needed is a map f:A –> X0 for then we can form the quotient space of the disjoint union of X0 and X1 by identifying each point of a in A with its image f(a) in X0. Hatcher 13
Let f: X –> Y, then the mapping cylinder M_f is the quotient space if the disjoint union (X x I) U Y obtained by identifying each (x,1) in X x I with f(x) in Y. We see this is a special case of the attaching construction…
Def retracts onto Y
Hatcher pg 2
Do all deformation retracts arise as mapping cylinders?
No, consider def retract of X onto point
Do all retractions come from deformation retractions?
No. A space X always retracts onto any point x in X via constant map but a space that deformation retracts onto x must be path connected since def retract gives path joining each x in X to x_0
More generally, spaces with nontrivial fundamental group not homotopy equivalent to point - no def retract
Prove: Two spaces X and Y are homotopy equivalent <=> there exists a third space Z containing both X and Y as deformation retracts
<= easy
=> Take mapping cylinder of homotopy equivalence f:X –> Y. We know this def retracts onto Y. Need to show def retracts onto X. Hatcher 16-17
What is a mapping cylinder neighborhood? Relation to homotopy extension property?
Example?
Hatcher 15. Let (X, A) be a pair.
A mapping cylinder neighborhood is a closed neighborhood N containing a subspace B, thought of as the boundary of N, with N - B an open neighborhood of A, s.t. there exists a map f: B –> A and a homeomorphism h : M_f –> N with h| A U B = identity
A pair (X,A) has the homotopy extension property if A has a mapping cylinder neighborhood in X
Example: Thick letters
Define: acyclic
A complex whose reduced homology vanishes in all dimensions is stb acyclic
Define: singular n-simplex
If X is a topological space, a singular n-simplex in X is a map sigma: delta^n –> X
where delta^n is the standard n-simplex {(t0, …, tn) in R^n+1 | sum ti =1 and ti in [0,1]}
Only requirement is that sigma is cts. Need not be injective etc. Could be very complicate - 1-simplex is just [0,1] so all cts paths allowed - space filling curves etc
Define: singular n-chain
S_n(X) = free abelian group with basis all singular n-simplicies. Elements are finite sums ni sigma_i ni in Z, sigma a singular n-simplex
mirror simplicial bit everything singular now. Crazy uncountable basis
pg 112
Define: boundary map
in singular setting
Key property?
del_n : Sn(X) —> Sn-1(X)
del_n(sigma) = sum_i=0^n (-1)^i sigma|[v0, …, vi hat , … vn]
we canonically identify the face with delta^n-1 by a linear map that preserves order of vertices
Again, the key property is boundary^2 = 0
pg 113
Define: singular homology groups
H_n(X) = ker del_n / im del_n+1. Possible to define since d^2 = 0 i.e. im d_n+1 < ker d_n
pg 113
Prove: If X is a single point, then reduced homology is trivial
pg 116
Define: good pair
Examples?
A pair (X,A) is a good pair if A has a nbd that def retracts onto A
Example: All CW complexes - go a little out into cell tofind neighborhood
Prove: If A < X is a retract, then i* is injective and r* is surjective
Example?
Follows easily from functorial properties of homology
No retraction r : D^n –> S^n-1. Just plug into commutative diagram
pg 124
Prove: Brower Fixed Point Theorem
Thm: Every map D^n –> D^n has a fixed point
Pf. Existence of such a map would imply D^n retracts onto S^n-1
Hatcher 113, notes
Prove: Perron-Frobenius
Let A be an nxn matrix with all entries real and positive. Then A has a positive eigenvalie and a corresponding positive eigenvector (all entries real and postive
Pf. Define a cts map and appeal to Brower fixed point
pg 126
Prove: If (X, A) is a good pair, then there exists an exact sequence … -> H_n(A) –> H_n(X) –> H_n(X/A) –> H_n(A) –> …
on reduced homology.
notice this is different then l.e.s. of pair - have quotient space, not relative homology.
This follows from the prop. If (X, A) is a good pair, then (X,A) –> (X/A, A/A) induces an isomorphism H_n(X, A) = H_n(X/A, A/A) = H_n(X/A) reduced
pg 149 -150
Moral: H_n(X,A) is the same as H_n(X/A) reduced as long as you have a good pair
Discuss the homology of a wedge sum of spaces. Proof?
Be sure to take reduced homology, then wedge sum of spaces yields direct sum of homology groups
Good pair lets us move from relative homology to quotient space
What is the relationship between singular and simplicial homology? Proof? Finite dimensional vs. infinite dimensional?
When both defined, they are equal
There is a canonical map from simplicial to singular homology groups taking a simplex to its inclusion map into X.
Thm. The canonical map is an isomorphism for any simplicial pair (K,L).
If it is known for single spaces, then 5 lemma gives for simplicial pairs.
pg 154
Define: degree of a map f: S^n –> S^n
Examples?
Recall: H_n(S^n) = Z for n >= 1. f: S^n –> S^n induces f_* : H_n(S^n) —> H_n(S^n) i.e from Z –> Z. Must be multiplication by some d in Z.
The degree of a map f: S^n –> S^n is the number d s.t. f_*: Z –> Z is multiplication by d. deg(f) = d
Examples
S^1 –> S^1
f(z) = z^d had degree d f(z) = complex conjugation - degree -1
Triangulate…
pg 159-160
Properties of degree?
- deg I = 1
- deg f = 0 if f not surjective
- If f, g homotopic, then deg(f) = deg(g)
- deg(fg) = deg(f)deg(g) — multiplicative — implies homotopy equivalence S^n to S^n has degree +- 1
- deg(f) = -1 if f is a reflection
- The antipodal map f(x) = -x is a composition of n+1 reflections, hence has degree (-1)^n+1
- If f has no fixed point, then deg(f) = (-1)^n+1 — homotopic to antipodal map
pg 161 - 162
Prove: The Hairy Ball Thm
Thm. S^n admits a continuous nonzero vector field <=> n is odd.
Note: Then S^n not parallelizable for even n, can’t be Lie group
If n is odd, consider unit sphere in C^K. Let v(z1, … zk) =
If n even, use vector field to define a homotopy from 1 to -1 –> a contradicition
pg162-163
Define local homology and prove invariance of dimension
The local homology of space X at a point x in X is H_n(X, X- {x}). Only depends on the shape of small neighborhood of x.
If U and V are open sets of R^m and R^n that are homeomorphic, then m = n.
This follows from excision:
Define: Short exact sequence of chain complexes
Hatcher 114
Prove: A short exact sequence of chain complexes yields a long exact sequence of Homology groups
Purely algebraic result. Hatcher 114 - 115
Diagram chasing
Define: the group of cellular n-chains for a CW complex
If X is a CW complex, the group of cellular n-chains is H_n(X^n, X^n-1) — the nth singular homology group of the pair (X^n, X^n-1)
What is a basis for H_k(X^n, X^n-1)?
if k = n, this is free abelian with basis the n-cells of X
if k != n, 0
Key: this is a good pair, so reduced homology is the same as X^n/X^n-1.
X^n/X^n-1 is a wedge sum of n-spheres, one for each n-cell of X
Prove: H_k(X^n) = 0 if k > n
l.e.s. of pair … pg 170
Prove: H_k(X^n) —> H_k(X) is an isomorphism if k < n
l.e.s of pair. Need to distinguish finite dim complex from infinite dim complex pg 171
Define: Cellular chain complex
The idea is to weave together the homology exact sequences for (X^n, X^n-1) and (X^n-1, (X^n-2)
pg 172
Prove: Cellular homology = singular homology
pg 173 chase around diagram
How do you compute with cellular homology? Use to compute homology groups of closed surfaces.
Very nice for computations. Get a basis for cellular n-chain group by choosing an orientation for each n-cell and taking free group on these n-cells.
The boundary map goes from n-cell to (n-1)-cells. Boundary of n-cell is S^n-1 this is attached into X^n-1. To understand value of boundary map on a specific (n-1)-cell e, collapse everything in X^(n-1) except e, get S^n-1. Then boundary is just degree of this map
pgs 174 - 178
Compute the homology groups of RP^n
pgs 178 - 181
Different for n even vs. n odd
Define: Euler characteristic
Prove invariant of homotopy type
Let c_n be the number of n-cells in a CW complex. The Euler characterstic is the alternating sum a0 - a1 + a2 - a3 …
What is Euler characteristic of closed surfaces?
Recall familiar case for polyhedron v - e + f
Maybe say a little about Gauss-Bonnet theorem - integrating Gaussian curvature of Riemannian manifold over entire manifold always equals 2 pi * Euler characteristic
pg 181 - 182
What is Meyer-Vietoris Thm? Proof?
Suppose X is any space, A,B < X. Suppose X = int(A) U int(B). Then there exists a l.e.s. –> H_n(A intersect B) —> H_n(A) + H_n(B) —> H_n(X) —> H_n-1(A intersect B) —> …
Natural with respect to continuous maps of spaces.
Proof. Has close connection to excision thm. Use a nice s.e.s. of chain complexes
pg 183 -184
What does the connecting homomorphism do in the Mayer-Vietoris sequence?
pg 185
A cycle c in X can be written as a sum of a chain in A and a chain in B, c = a + b. 0 = Boundary c (since cycle) = boundary a + boundary b. So boundary a = -boundary b is in A intersect B. The connecting hom sends c = a+b —> boundary a = -boundary b
What happens to Meyer-Vietoris in reduced homology?
As long as A intersect B != 0 this works
pg 186
Use Mayer-Vietoris to calculate reduced homology of S^n
Use 2 hemispheres and induction
pg 186
What is Alexander’s Theorem? -not really sure if this is Alexander?
Special cases?
Proof?
Let h:S^k –> S^n be any embedding. Then reduced homology of S^n - h(S^k) = Z if i = n-k-1, 0 otherwise
Special cases:
- Jordan curve theorem: h: S^1 –> S^2
- Jordan-Brower separation theorem: h: S^n-1 –> S^n
Both of these just say the complement of a codimension 1 sphere has 2 components. Each with the homology of a point.
Now in the case of S^1 —> S^2, all embeddings are topologically equivalent - in particular both components of S^2 - S^1 are contractible
However, if dim n > 2, then S^n-1 may separate S^n into two components, one of which is not contractible - Alexander’s Horned Sphere. — this can’t be seen using homology.
Proof. We first show that if h: D^k –> S^n is an embedding, then S^n - h(D^k) is acyclic in reduced homology. Removing a disc makes homology trivial - induction on k
pgs 187 - 193
Compare Mayer-Vietoris and Van Kampen
Think about…
What is invariance of domain?
Thm. If U < R^n is open and h:U –> R^n is a continuous injection, then h(U) is open in R^n
So every cts injection of U into R^n is an open mapping
pg 193 - 194
Discuss homology with coefficients in G
We can use any abelian group for coefficients. Formally everything works out the same. Good choice of coefficients may yield richer supply of algebraic invariants
On the other hand H_n(X, G) is algebraically completely determined by knowledge of the homology groups of X with integer coefficients - Universal Coefficients Theorem
Pros and cons of homology?
Alternatives?
Pros
- highly computable - linear algebra
- rich combinatorial/algebraic structure
Cons
- intuition less clear - what is a ‘cycle’ geometrically?
- incomplete invariant - not enough to pin down homotopy type
Cobordism - replace simplicies with manifolds
Homotopy groups - complete invariant
Whitehead Thm: If f:X –> Y induces an iso on pi_n for all n, then f is a homotopy equivalence.
Hurewicz Thm - If X is path-connect, then H_1(X, Z) is the abelianization of p_1(X)
ss pg 1 - 4
Define: perfect group
relation to field theory?
A group with trivial abelinization. Any non-abelian simple group e.g. A_5 since no nontrivial normal subgroups - loose info in abelianization
Define: path homotopy
A homotopy f_t : I –> X s.t. f_t(0) = x0 and f_t(1) = x1 for all t - the endpoints are glued down. i.e. a homotopy rel to boundary I = {0,1}
[f] = path homotopy class of f
pg 6
How can you compose paths?
If f(1) = g(0) then glue together
f . g (s) = f(2s) if s <= 1/2
g(2s-1) if s>= 1/2
Define: the fundamental group and prove it is a group
As a set = all path-homotopy classes of loops in X based at x0.
Define group operation [f][g] = [f . g]
Check: well-defined and group axioms
The group axiom checks come down to fact that reparameterization does not impact homotopy type
pgs 7 - 9
Prove: If X deformation retracts to a point x0, then pi_1(X, xo) = identity
Let r_t be a def retract onto x0, then for any loop f based at x0, r_t after f is a path homotopy from f to c, [f] = 1
pg 9
What impact does change of basepoint have on pi_1?
Say x0 and x1 are joined by a path h, then B_h[f] = [h . f . h bar] is an isomorphism of groups
prove: well-defined, homomorphism, bijection - think conjugation of groups = isomorphism
pg 10 - 11
Define: simply connected
X simply connected if path-connected and pi_1(X) = 1
Prove: X is simply connected <=> for all x0, x1 in X, there exists a unique homotopy class of paths from x0 to x1
pg 12 -13
Define: covering space
Examples?
A covering space of X is a space X hat together with a map p: X hat –> X (the covering projection) s.t. X has an open cover {U_alpha} by evenly covered set U_alpha
Evenly covered - U_alpha has preimage a disjoint union of open sets in X hat each mapped homeomorphically to U_alpha via p
Examples
- exp: C –> C - {0}
- x –> z^2 in C
- R - S^1
pg 13 - 16
Define: group action on space X, covering space action, orbit space
Examples?
An action of a group G on a space X is a homomorphism G –> Homeo(X)
A covering space action is an action of G on X s.t. each x in X has a nbd U whose G-translates are pairwise disjoint
i.e. points in orbit are separated - spaced out. g1(U) intersect g2(U) != 0 => g1 = g2 or equivalently U intersect g(U) != 0 =>g = identity
The orbit space is the quotient X/G where X/~ where x ~ g(x) for all x, g with quotient topology. That is points of X/G are orbits Gx = {gx | g in G}
Z action on R –> S^1
Z^2 action on R^2 –> T^2
pg 16-20
Prove: The quotient map X –> X/G of a covering space action is a covering space projection
So X is a covering space of the orbit space X/G
The disjoint translates {g(U) | g in G} are identified to a single open set in X/G. Preimage is then clearly disjoint set of open neighborhoods homeomorphic to U. Every point has a nbd like this by def of covering space action
Is covering space action free?
yes g(x) = x for any g,x implies g = 1. No fixed points
Say X simply connected and G has covering space action on X. What can we say about pi_1(X/G)? Proof?
pi_1(X/G) = G
First we define phi : G –> pi_1(X/G). Choose some x0 in X to act as basepoint. Now since X simply connected, there exists a unique homotopy class of paths gamma from x0 to g(x0) for each g in G. Then p gamma is a loop in X/G based at p(x0). Define phi(g) = [p gamma]. Now show phi is group isomorphism.
pg 21 -24
Let p: X hat –> X be a covering space and f: Y –> X. What is a lift of f? What can be said about lifts of paths? homotopies?
A map f hat : Y –> X hat s.t. f = p f hat. A map upstairs that projects nicely downstairs
- Path Lifting: {x} x I –> X
- Homotopy Lifting: I x I –> X
pg 22
These are special cases of the Homotopy Lifting Theorem (pg25)
Given a space Y, a covering space p: X hat –> X, a map F : Y x I –> X, and a lift F hat : Y x {0} –> X hat, then F hat has a unique extension to a map F hat : Y x I –> X hat lifting F.
i.e. if we can lift one end of the map F, Y x {0} end, then the rest of the map lifts uniquely
Define: free group, letter word, reduced, rank, basis
pg 32 -33
Define: Cayley graph, circuit, tree
Examples?
Given group G and generating set S, define a graph (directed, labeled)
vertices: elements of G
edges : (g, gs) for all g in G, s in S
i.e. use generating set S to move around the graph - reach different elements of G
A circuit in a graph is a closed edge path with no repeated vertex
A tree is a connected graph with no circuit
Examples
- Z
- Z^2
- Free group –> tree
Prove: Trees are contractible, simply connected
idea is to pick some root x0, then look at level 1, level 2, etc and def retract edges back to previous level. No loops so this all works nicely…pg 36-37
Prove in two ways: For any set S, vS^1 over s in S, then pi_1(vS^1) = F(S)
- The key is that F(S) acts on its Cayley graph, a tree via covering space action - quotient is the wedge sum of circles. Since trees are simply connect pi_1(T / F(S) ) = F(S)
pg 37 - Van Kampen pgs 67-68
What is pi_1(X x Y)? Proof?
pi_1(X) x pi_1(Y)
38
Discuss functorial properties of pi_1
Covariant functor from pointed Top spaces, pointed cts maps to Groups, homomorphism
What is induced hom?
pgs 39 - 40
Prove: IF phi: X –> Y is a homotopy equivalence, then phi_* : pi_1(X, xo) –> pi_1(Y, phi(x0)) is an isomorphism for all x0 in X
The big problem we are addressing here is that fundamental group always has issues with basepoint.
pg 40 -42
What is the fundamental group of a connected graph?
Choose a maximal tree T in graph, if necessary appealing to Zorn’s lemma. Then pi_1 is free group with basis one element for each edge not contained in T.
Just retract maximal tree to get a wedge of circles
pg 43-44
What is Hurewicz Theorem? (1-dim case)?
Hurewicz map?
Proof?
If X is any path-connect space, then H_1(X, Z) is the abelianization of pi_1(X)
The Hurewicz map h: pi_1(X, xo) –> H_1(X) takes the path homotopy class [f] to the homology class [f] (notice any loop may be considered a singular 1-simplex - f(0) = f(1) so this is actually a cycle boundary f = 0)
h is a homomorphism of groups, if X path-connected, h induces an isomorphism [pi_1(X, x0)]ab –> H_1(X)ab = H_1(X)
pgs 45 - 51
Define: free product of groups, reduced
Examples?
If {G_a} is a family of groups, the free product *_a G_a is given as set of words g1…gm with each g_i in some G_a.
A word is reduced if
- No g_i is identity
- g_i and g_i+1 come from different groups
Operation: concatenate and simplify
Examples:
- Free groups - free product of Z’s
- Z_2 * Z_2
pg 52-53
What is the universal mapping property of free products?
Given any group H and homomorphisms phi_a : G_a –> H, there exists a unique homomorphism phi : *_a G_a –> H induced by phi_a’s s.t. g1 … gm —> phi_a1(g1)…phi_am(gm)
pg 53
Discuss the infinite dihedral group. Relation to free products?
The group of isometries of R generated by two reflections a(x) = -x and b(x) = 1-x
isomorphic to Z_2 * Z_2
pg 54
fundamental group of wedge sum of RP^2 pg 69
What is van Kampen Theorem? Proof?
Define: factorization
(a) IF X is the union of path-connected open sets {A_a} each containing x0 and s.t. each A_a intersect A_b is path-connected, THEN every loop in X based at x0 is homotopic to a product of loops each from some A_a.
INTERPRETATION: Each inclusion A_a –> X induces pi_1(A_a) –> pi_1(X). These induce a map phi from the free product of pi_1(A_a) to pi_1(X). (a) states that this map is surjective. The loops in pi_1(A_a)’s generate pi_1(X).
(b) phi is often not injective. Ker(phi) consists of “relations” between words in the A_a’s. The kernel of phi is the normal subgroup N generated by all elements i_ab(w) i_ba(w)^-1 for all w in pi_1(A_a intersect A_b) for all a,b
pg 56-57, pg 62-66
Prove: pi_1(S^n) = 1 if n >= 2 and pi_1(RP^n) = Z_2
2 ways
- Van Kampen on S^n using open cover S^n - {N} and S^n - {S} North and South poles. pg 58-59, 67
- Using CW structure point and n-cell - generator, no relators - pg 70
Prove: If n is even, Z_2 is the only nontrivial finite group acting freely on S^n
First observe that since S^n is compact, the only groups that can have covering space actions on S^n are finite groups. Use degree of maps S^n –> S^n. deg: G –> {+ - 1}
pg 59-61
Discuss the fundamental group of a CW complex
For CW complex we have a natural presentation for pi_1 where:
1-cells = generators
2-cells = relators
n-cells (n>2) have no effect at all!
Thm. (a) If Y is formed from X by attaching 2-cells via attaching paths phi_a : S^1 –> X, let N = normal subgroup generated by the loops {phi_a}. Then pi_1(X) –> pi_1(Y) [map induced by inclusion] is surjective with kernel = N i.e. pi_1(Y) = pi_1(X) / N
(b) If Y is formed from X by attaching n-cells for some n > 2, then pi_1(X) –> pi_1(Y) [induced by inclusion] is an isomorphism
(c) A cell complex Y with 2 skeleton X satisfies pi_1(X) –> pi_1(Y) [induced by inclusion] is an isomorphism.
pgs 70, 73 - 76
Find fundamental groups of closed surfaces
- M_g has 2g generators = 1 cells and 1 relator:
F(a1, b1, … , ag,bg) / < < [a1, b1] … [ag, bg] > > - N_g has g generators and 1 relator:
F(a1, … , ag) / < < a1^2 … ag^2 > >
What is the presentation 2-complex?
For any group G with presentation G = < s_a | r_b > there is a 2-complex X with pi_1(X) = G.
1 vertex
1-cell for each generator
2-cell for each relator
pg 77
Discuss knots and links - in particular torus knots and strength of pi_1 vs. homology groups
Every knot is an embedding of S^1 so the knots themselves are homeomorphic, what is different is how S^1 is embedded into S^3. Thus, we study the complement of the knot.
We use S^3 to compactify R^3. Normally just think of R^3 and work locally.
Can realize S^3 as the union of two solid tori - glued together along boundary torus.
Recall by Alexander’s Thm, homology cannot distinguish different knots. pi_1 can actually be useful here. Have nice family of torus knots K_m,n where (m,n) = 1
Computation of this is an involved use of Van Kampen
Get pi_1(X_m,n) = < a , b | a^m = b^n > … to see that we get distinct groups, mod out by center < a^m = b^n>, get < a , b | a^m = b^n = 1 > = Z_m * Z_n.
pgs 78 - 86
5 key facts about covering spaces?
A based cover p: (X hat, xo hat) —> (X, xo) corresponds to a subgroup of pi_1(X, x0)
- The map p_* : pi_1(X hat) –> pi_1(X) is always injective. p_*(pi_1(X hat)) is a subgroup of pi_1(X)
- The index of the subgroup p_*(pi_1(X hat)) in pi_1(X) equals the number of sheets of the cover
- Change of basepoint gives conjugate subgroups
- If the symmetries of X hat act transitively on the vertices, then p_*( pi_1(X hat) ) is a normal subgroup N of pi_1(X)
- Every subgroup of pi_1(X) corresponds to some based cover X hat –> X “Galois correspondence” i.e. the lattice of subgroups is in 1-to-1 correspondence with the lattice of based covers. The trivial subgroup 1 corresponds to the simply connected cover - called the universal cover
Discuss the universal cover of closed surfaces
Consider only orientable - If genus g > 0, then this is homeomorphic to R^2.
Not all conformally equivalent - relate to unformization thm which says open disc, complex plane, or Riemann sphere are the options.
Also relate to hyperbolic geometry which gives S^2, R^2, hyperbolic plane
pg 93
Define: Isomorphism of covering spaces
An isomorphism of covering spaces is a homeomorphism f: X1 hat –> X2 hat s.t. p1 = p2f - draw diagram
pg 95
Prove: If p: (X hat, x0 hat) —> (X, x0) is a covering space, then the induced map on fundamental groups is always injective.
Simple application of homotopy lifting property
pg 96
Describe what p_*(pi_1(X hat, x0 hat) ) looks like inside pi_1(X, x0). Proof?
Consists of all homotopy classes of loops in X at x0 whose lift to X hat starting at x0 hat is a loop.
Some loops in X lift to a path - unravel. Others stay a loop. All we can do in moving up is unravel loops
pg 96-97
Prove: The number of sheets of a cover p: (X hat, x0 hat) –> (X, x0) equals the index of the subgroup [pi_1(X, xo) : p_*( pi_1(X hat, x0 hat) ) ] assuming X and X hat are path-connected
Recall: # sheets = #p^-1(x) if X, X hat path connected, then this number is constant.
Index = number of right cosets
pg 97 -99
Let p: (X hat, x0 hat) —> (X, x0) be a cover and choose x1 hat a different basepoint in p^-1(x0). What is the relationship between p_( pi_1(X hat, x0 hat) ) and p_( pi_1(X hat, x1 hat) )?
proof?
conjugate subgroups
pg 100
Given a covering space (X hat, x0 hat) —> (X, x0) and a map f: (Y, y0) —> (X, x0), discuss existence and uniqueness of lifts of f to X hat. Proof?
Relationship to homotopy lifting property?
EXISTENCE: Assume Y is path-connected and locally path-connected. f lifts <=> im(f_) < im(p_)
UNIQUENESS: Two lifts that agree at one point are equal at every point.
Note: Our proofs relied on: path lifting, connected, locally path-connectedness of Y.
pg 101 - 106
Prove: Neilson-Schreier Thm - Every subgroup of a free group is free
Free group has covering space action on tree T = Cayley(F,S) which is simply connected. Let H be a subgroup of F. H acts on T with quotient T/H a graph. H = pi_1(T/H) is a free group - already saw fundamental group of any graph is free
“Any cover of a graph is a graph” its fundamental group is free
pg 110
Define: deck transformation
This is an “automorphism” of covers - permute sheets of cover - “shuffle the deck” a map f: X’ –> X’ s.t. p = pf
“Label preserving symmetry”
The set of all deck transformations is a group acting on X’ by a covering space action
pg 110-111
Prove: If X has a simply connected cover, it satisfies Galois Correspondence
What if we don’t require base-points to be preserved?
Prop: Two path-connected covers p, p’ of X are base-point preserving isomorphic <=> im(p_) = im(p’_)
=> Easy
<= Follows from lifting property
This shows that the Galois correspondence is well-define and one-to-one. The set of base-point preserving isomorphism classes of covers of X injects into the set of subgroups of pi_1(X, x0). It remains to show this is surjective - i.e. there is a cover corresponding to every subgroup of pi_1(X, x0).
If we ignore base-points, covers p, p’ isomorphic <=> im(p_) and im(p’_) are conjugate in pi_1(X, x0).
pg 106 -108
Prop: If X is the orbit space of a covering space action of G on a simply connected space X tilda, then every subgroup of pi_1(X, x0) has a corresponding cover - i.e. surjective
Recall: G = pi_1(X, x0). If H < G, then we have a covering space action of H on X tilda. The map X tilda –> X tilda / G factors as X tilda –> X tilda / H —> X tilda / G
pg 109
Prop. Let X be path-connect and locally path-connected. Assume X has a simply connected cover X’ –> X. Then every subgroup of pi_1(X, x0) is realized as im(p_*) for some cover p: X hat –> X.
Pf. Consider the deck transformation group G(X’) -acts transitively by lifting criterion since s.c.
pg 112- 113
Define: universal cover, semilocally simply connected
A universal cover is just simply connected cover of X
X is semilocally simply connected if each point x in X has a nbd U s.t. i_* : pi_1(U,x) –> pi_1(X, x) is the trivial map i.e. all loops in U are nullhomotopic in X
When does X have a simply connected cover? Proof sketch?
Let X be path connect and locally path connected. Then X has a simply connected cover <=> X is semilocally simply connected
=> Every point has a evenly covered nbd U…
<= Notice if we have a simply connected cover X’ –> X, there is a 1-1 correspondence:
{points of X’} {homotopy classes of paths in X starting at x0}
Thus, we define a universal covering space for X has a set { [gamma] | gamma is a path in X starting at x0} the set of all homotopy classes of paths starting at x0. Define p :x’ –> X by [gamma] –> gamma(1)
Now define topology and check s.c.
pgs 115 - 119
Define: regular/normal covering space
A covering space p: X’ –> X is regular or normal if for each x in X and all x1’, x2’ in p^-1(x), there exists a deck transformation taking x1’ –> x2’
Let p: (X’, x0’) —> (X, x0) be a covering and H = im(p_*) < pi_1(X,x0)
When is a covering space normal?
What can we say about the group of deck transformations of a cover X’ —> X?
Normal <=> H is a normal subgroup
Group of deck transformations G(X’) = N(H) / H the normalizer of H in pi_1(X,x0) - notice if H normal then G(X’) = pi_1(X, x0) / H.
Define: local orientation, orientation, orientable
Notice by excision, H_n(M, M - {x} ) - the local homology at x - is isomorphic to Z for every point x of a manifold M. (locally looks like R^n).
A LOCAL ORIENTATION of M at x is a choice of one of the possible generators for H_n(M, M - {x} ) = +1 or -1. Two ways to wrap around the hole.
An ORIENTATION of M is a function x –> mu_x assigning a local orientation to each point of M which is LOCALLY CONSISTENT - Each x has an open neighborhood B s.t. all of the chosen mu_y’s for y in B are images of the same generator of H_n(M, M-B) = H_n( R^n, R^n - B) = Z under inclusion: (R^n, R^n - B) –> (R^n, R^n - {y}). i.e. within B, all generators “wind” in same direction
M is ORIENTABLE if an orientation exists.
pg 128-130
Discuss Poincare Duality,
fundamental class
If M is a CLOSED, ORIENTABLE manifold of dimension n, then H_k(M, Z) = H^n-k(M, Z).
If nonorientable, can use Z_2 coefficients and get same result.
n-dimensional manifold = Hausdorff, second countable, locally homeo to R^n
Closed = compact without boundary
A FUNDAMENTAL CLASS is a generator of H_n(M, Z) whose image in H_n(M, M-{x}) is the chosen orientation
pg 136
Give examples of orientable non orientable manifolds.
How can you work with something orientable when dealing with non-orientable manifolds?
When is cover orientable?
Consider connected M.
Every manifold M has an orientable 2-sheeted covering space.
M is orientable <=> double cover has two components.
M nonorientable <=> double cover is connected
It follows that if M is simply connected, M is orientable - can’t have connected double cover.
Further, if pi_1(M) does not contain an index 2 subgroup, then M is orientable - no 2-sheeted covering
Examples
- S^n –> RP^n if n is even
- T^2 –> Klien bottle
pg 130-131
Discuss Dual polyhedra
This provides the intuition for Poincare Duality.
Draw picture of dual cell structures on S^2 as a cube…
0-cells 2-cells
1-cells 1-cells
2-cells 0-cells
Pairing of cells and dual cells gives an identification C_0 = C_2, C_1 = C_1, C_2 = C*_0 of cellular chain groups
Use Z_2 coefficients to avoid orientation issues
Boundary map for C_i becomes coboundary map for C*_2-i
The argument works for any closed n-manifold with dual cell structures.
Define: K(G,1) - Space
Examples?
aka Eilenburg-MacLane Space - any path connected space X with contractible universal cover X tilda and pi_1(X) = G.
Recall - universal cover normally only simply connected, extra strength of contractible gives much more powerful structure
Examples
- S^1 is a K(Z, 1) space
- Any closed surface except S^2 and RP^2 is a K(G,1) space - note if M_g has g >2, then M_g covers M_2, so they both have same universal cover
- RP^2 not K(G,1) space
pgs 138-139
Prove: S^inf is contractible
Recall S^inf is union of all S^n = {(x1,x2,x3,…) in R^inf | only finitely many terms nonzero and sum xi^2 = 1}
Key is the shift map
pg 140
Discuss: infinite dimensional lens spaces
Using the fact that S^inf is contractible, we view S^inf as unit sphere in C^inf which gives an easy action of Z_n on S^inf by scalar mult by nth root of unity. The quotien S^inf / Z_n has fundamental group Z_n and contractible universal cover S^inf so is a K(Z_n, 1)-space
Prove: If X and Y are K(G,1) CW-complex for the same group G, then X and Y are homotopy equivalent
This says the homotopy type of a K(G,1) space depends only on the group G - so can be seen as a topological model of the group
The key to the proof is the following proposition:
Prop. Let X be a connected CW complex and Y be a K(G,1) space. Then every homomorphism of groups pi_1(X, x0) –> pi_1(Y, y0) is induced by a map of spaces (X, x0) –> (Y, y0) which is unique up to homotopy preserving both basepoints
Pf of prop.
Let H = pi_1(X, x0) and G = pi_1(Y, y0)
Key: A map X –> Y inducing phi: H –> G is equivalent to a phi-equivariant map X tilda –> Y tilda.
Then since Y tilda contractible, we really have no obstruction to building any map we like
pg 142 - 145
Discuss 3-dimensional lens spaces L(m,l)
Compute fundamental group and homology groups
These provide simplest examples of the following phenomena:
- Manifolds which agree on pi_1 and all homology groups but are not homotopy equivalent - L(5, 1) and L(5, 2)
- Manifolds which agree on all homotopy groups but are not homeomorphic - L(7, 1) and L(7, 2)
We require gcd(m,l) = 1. Then view S^3 as { (z, w) in C^2 : |z|^2+|w|^2 = 1}. Z_m = <a> acts on S^3 via (z, w) –> (az, z^l w) where a is primative mth root of unity. This is a covering space action with quotient a closed 3-manifold L(m, l) = S^3 / Z_m.</a>
If m=2 we get RP^3
pi_1(L(m,l) ) = Z_m
H_0 = Z H_1 = Z_m H_2 = 0 H_3 = Z
KEY: The fundamental group and homology groups can’t see the value of l. Our algebraic invariants can’t distinguish these spaces.
These spaces are homeomorphic <=> m’ = m and l’ = l^+-1 mod m
Homotopy equivalent <=> m = m’ and l’ = +-k^2 l mod m for some k in Z
These are not K(G, 1) spaces. In fact, Z_m does not admit a finite dimensional K(G, 1) space. It then follows that if X is any finite dimensional K(G, 1) CW complex, then G is torsion free.
pg 146 - 153</a>
Discuss action of SU(2) on itself by left multiplication and conjugation.
pg 154 - 159
How do we get a homomorphism of SU(2) to SO(3, R)?
By conjugation
Define: Binary icosahedral group
I*_120 = the preimage of the icosahedral group I_60 < SO(3, R) under the hom SU(2) –> SO(3, R)
Note I_60 is isomorphic to A_5 - simple of order 60
and we can prove I*_120 is perfect - has trivial abelianization
What is the Poincare Homology sphere? Importance?
This is the first example of a HOMOLOGY SPHERE - a closed n-manifold with the same integral homology as S^n - which is not homeomorphic to S^n.
Sigma^3 = S^3 / I*_120
Where I*_120 acts by left multiplication.
See homology is same as S^3 using abelinization, universal coeficients, and Poincare
See not S^3 since pi_1 is I*_120
How can you get entire homology for 3-manifold just from pi_1?
Hurwicz theorem gives H_1 = pi_1 abelianized
universal coefficients + Poincare duality fill in H^0 –> H_3 and H^1 —> H_2
H_0 = Z since connected
Define: pi_n(X, x0)
What does this look like for n = 0, 1, 2, …?
The homotopy class of maps f: (I^n, boundary I^n) –> (X, x0) where the homotopies must map boundary I^n to x0 for all t.
Equivalent to mapping (S^n , s0) –> (X, x0) but easier to compute with.
n= 0 : pi_0(X, x0) I^0 is a point, boundary of I^0 is empty set. So we map ({x}, empty set) –> (X, x0) - gives set of path components of X - no natural group structure
n=1 : covered heavily a group - often non-abelian
n > 1 : an abelian group
pg 166
Prove: If n > 1, pi_n(X, x0) has the structure of an abelian group.
First define + on set by copying pi_1 concatenation definition in just the first coordinate of f. Interpret also using spheres… pg 168
Group structure is easy, commutativity proof relies on 2 + dimensions to move around maps in
pg 167
How can we view pi_n as a pi_1 module?
For any gamma path in X from x0 to x1, we can define gamma f ; (I^n, boundary I^n) —> (X, x0) by following gamma on each radial line segment.
Then if [f] in pi_1(X, x0) –> [gamma f] is an automorphjism of pi_n(X, x0) - gives action of pi_1 on pi_n
What is the relationship between covering space and homotopy groups? Proof?
A covering space projection p : (X tilda, x0 tilda) —> (X, x0) induces isomorphisms p_* on pi_n for all n > 1.
Hatcher 351
How do the homotopy groups behave w.r.t products of spaces?
pi_n of a product of spaces is isomorphic to product of pi_n for each individual space
Hatcher 352
Prove: Consider a map f: S^n –> Y. Choose a basepoint s0 in S^n. TFAE
(a) f is null homotopic [allowing basepoint to move]
(b) f can be extended to a map f:D^n+1 –> Y
(c) f is null homotopic rel s0
pg 169
Define: aspherical, n-connected
Examples?
X is aspherical if X is path-connected and pi_n(X) = 0 for all n >= 2
X is n-connected if pi_n(X) = 0 for all k <= n
Examples
- K(G, 1) -spaces are aspherical
- 0-connected = path connected
- 1-connected = simply connected
pg 169
Define: relative homotopy groups
I^n = unit cube [0,1]^n
I^n-1 = the set {(s1, ... , sn) | sn = 0} in I^n
J^n-1 = closure of (boundary I^n) - I^n-1 = all faces of boundary I^n except I^n-1
Now p_n(X, A, x0) = homotopy classes of maps f : (I^n, boundary I^n, J^n-1) —> (X, A, x0) with homotopies of the same type (X, A, x0)
Prove: Given f: (D^n. S^n-1. so) —> (X, A, x0).
[f] = 0 in pi_n(X, A, x0) <=> f is homotopic rel S^n-1 to a map with image in A.
The trivial elements of relative homotopy are the balls that can be pushed entirely into A without moving boundary (think 2-disks). A disk in A can be homotoped to x0 easily.
=> We are given a homotopy f_t : (D^n, S^n-1, s0) —> (X, A, x0) from f to a constant map c(y) = x0. Inside the domain of f_t, D^n x I, there is a family of discs starting at D^n x {0} and ending at (D^n x {1} ) union (S^n-1 x I) all sharing same boundary S^n-1 x {0}. This gives us a homotopy D^n x I which pushes f into A keeping f fixed on S^n-1
<= Now suppose f is homotopic rel S^n-1 to g with image in A. Clearly [f] = [g] in pi_n(X, A, xo). We just need to show [g] = 0 in pi_n. g: (D^n, S^n-1, s0) –> (X, A, x0) is a map with image in A. D^n def retracts to s0 (straight-line homotopy) so we can just compose g with the d.r. to get a homotopy shrinking D^n to a point without leaving A.
pg 171 - 172
What is the homotopy exact sequence of the pair (X, A, x0)?
Proof?
Very simple compared to homology exact sequence of pair.
First, note pi_n(X, {x0}, x0) = pi_n(X, x0) (X, x0)
(X, x0) = (X, {x0}, x0) –> (X, A, x0)
Boundary map induced by restriction of
(D^n, S^n-1, s0) –> (X, A , x0) to (S^n, s0) –> (A, x0)
Can very easily see what is happening on boundary - roughly matches the intuition for homology exact sequence
Thm. For any pointed pair of spaces (X, A, x0) where x0 in A < X, the following sequence is exact:
… –> pi_n(A, x0) –> pi_n(X, x0) –> pi_n(X, A, x0) –> pi_n-1(A, x0) –> … –> pi_1(X, x0) –> pi_1(X, A, x0) –> pi_0(A, x0) –> pi_0(X, x0)
Notice above pi_1(X, x0) is the last group in this l.e.s.
Need to check boundary^2 = 0 and exactness
pgs 173 - 176
What is Cellular Approximation Theorem? Proof?
Thm. Every map f: X –> Y of CW complexes is homotopic to a cellular map [i.e. maps X^(n) to Y^(n) for all n].
If f is already cellular on a subcomplex A < X, then exists homotopy rel A.
Notice this is part of a long line of results like this where we show a continuous map between spaces in some category can be homotoped to the desired map from that category. i.e. cts between smooth manifolds homotopic to smooth etc.
The trickiest part here is dealing with the possibility of space filling curves/surfaces. We need to show any lower dimensional cell which fills up a higher dimensional cell is homotopic to one that does not fill up the higher dimensional cell - slightly more general proof pgs 180 - 183. Basic idea is that we can pull a map of a lower dimensional space taught to piecewise linear.
With this handled, Cellular approximation is proved on pgs 183 - 184. The key is that since we can homotope a space filling map to miss a point, we can further push it all the way down onto boundary of cell…
Prove: pi_k(S^n) = 0 if k < n
This follows immediately from cellular approximation. We use a cell structure on S^n : one 0-cell, one n-cell.
Then the k-skeleton is a point
Define: order of open cover, mesh(open cover), refinement of open cover
The ORDER of an open cover U of a space X is the largest k s.t. U has k+1 sets with nonempty intersection
If X is a metric space, MESH(U) = sup (diam u) for u in U
A REFINEMENT of U is an open cover V s.t. for all v in V, there exists a u in U s.t. v < u.
Discuss how order of an open cover can be used to give some concept of dimension
The minimum order of an open cover provides some measure of dimension - this was how Lesbgue thought of it. Notice for any epsilon > 0 I^n has an open cover of order <= n and mesh < epsilon. Any compact K in R^n with open cover U has a finite refinement of order <=n
So the order really is capturing dimension in R^n - combinatorial complexity of how different sets can overlap each other
pg 179
What is Brouwer-Hopf Degree thm? Proof
Recall, a map f:S^n –> S^n induces f_* : H_n(S^n) –> H_n(S^n) DEGREE of f = d means f_* is multiplication by d.
Thm. Two maps S^n –> S^n (n > 0) are homotopic <=> they have the same degree.
=> Easy. Homotopic maps induce the same map on homology.
<= Induction on n. For n =1, we already know pi_1(S^1) = Z (degree) by covering space theory.
Prop 1. Let x1, … , xm, y1, …, ym in S^n be distinct points (n>1). Then there exists a homeomorphism of S^n homotopic to 1 that maps x_i to y_i for all i.
Prop 2. For n > 1 every map S^n –> S^n is homotopic to one that preserves hemispheres i.e. the closed upper [lower] hemisphere maps to itself (forces equator to go to equator)
Proof of Thm. Recall from fall semester that a map f: S^n –> S^n has a suspension Sf : S^n+1 –> S^n+1 s.t. deg(f) = deg(Sf). We claim each map f: S^n –> S^n is homotopic to the iterated suspension of the map S^1 to S^1 taking z –> z^d for some d in Z. Prove by induction on n. Base n=1 is done.
pg 185 - 191
Prove pi_n(S^n) = H_n(S^n)
Simple application of Brouwer-Hopf Degree thm.
Define: Fiber bundle, total space, base space, fiber, local trivializations
A FIBER BUNDLE is a map p: E –> B s.t. “all fibers p^-1(b) are homeomorphic top the same space F.” Locally looks like a product. Each point of B has a nbd U s.t. p^-1(U) = U x F. Have commutative diagram… pg 192
TOTAL SPACE = E the big fibered space
BASE SPACE = B the space we project onto
FIBER = F the preimage of a point of B
LOCAL TRIVIALIZATIONS = the maps h: p^-1(U) –> U x F
F –> E –> B
Examples of fiber bundles?
- Covering spaces: A bundle with a discrete fiber is the same as a covering space with constant number of sheets
- Mobius band: I = fiber, Mobius = total, S^1 = base
- Klein bottle: S^1 = fiber, Klien = total, S^1 = base
- Projective spaces: S^0 –> S^n –> RP^n
S^1 –> S^2n+1 –> CP^n
S^3 –> S^4n+3 –> HP^n
pgs 193 - 195
Discuss the Hopf fibrations
There are only 4 finite dimensional real division algebras: R, C, H, O of dimensions 1, 2, 4, 8 over R. We get a Hopf fibration for R, C, H, O but can’t go further. A theorem of Adams states that 4 Hopf fibrations are the only fiber bundles F –> E –> B with F, E, B all spheres.
These just come from sphere coverings of the one dimensional projective spaces which just compactify R, C, H, O to S^1, S^2, S^4, S^8 respectively by adding point at infinity.
- Real :S^0 –> S^1 –> S^1 = RP^n
- Complex: S^1 –> S^3 –> S^2 = CP^1 the projection from S^3 to S^2 just takes (z,w) –> z/w the slope of the line
- Quaternionic: S^3 –> S^7 –> S^4 = HP^1
- Octonian: S^7 –> S^15 –> S^8 = OP^1
pgs 196 - 199
What is long exact sequence of fiber bundle? Examples?
… –> pi_n(F) –> pi_n(E) –> pi_n(B) –> pi_n-1(F) –> … –> pi_0(B).
Examples
1. If F = discrete and B = path connected, then we are looking at covering spaces. Since pi_n(F) = pi_n-1(F) for n>1, we recover the isomorphism of pi_n(E) with pi_n(B)
- If F = path-connected and B = aspherical, then we get a s.e.s. of pi_1
0 –> pi_1(F) –> pi_1(E) –> pi_1(B) –> 0 - The Hopf fibration pi_n(S^3) –> pi_n(S^2) is an isomorphism if n > 2. In particular pi_3(S^2) = Z! The first nontrivial example of a higher homotopy group pi_n+k(S^n) - very different than homology - still an open problem to understand.
- HOMOGENEOUS SPACES - let G be a Lie group acting by diffeomorphisms on a manifold M. If the action is transitive, M is a homogeneous space. M = G / H for H = Stab(x0) orbit/stabalizer thm
Then H –> G –> M = G/H is a fiber bundle
O(n) acts on S^n-1 transitively with H = O(n-1) for all n >0 we get a bundle O(n-1) –> O(n) –> S^n-1 so pi_i(O(n-1) ) = pi_i( O(n) ) for i < n-2. In particular, if we hold i constant and take n–> inf, we see that pi_i stabilizes for large n. Does not depend on n.
pgs 200 - 202
Discuss homotopy lifting in the context of fiber bundles
As with covering space theory, fiber bundle theory relies heavily on our ability to lift homotopies from base space to total space.
We say a map p: E –> B has the HOMOTOPY LIFTING PROPERTY w.r.t. a space X if any homotopy X x I –> B with a lift X x {0} –> E extends to a lift X x I –> E
Note: covering spaces have HLP w.r.t any X.
The most important/useful case is when X is a disk. We say p: E –> B is a (Serre) FIBRATION if it has HLP w.r.t. X = D^n for all n.
pg 203
Prove: Every fiber bundle is a fibration
Example on pg 204 shows B x F –> B is a fibration so locally a fiber bundle is a fibration. We just need to show this local property can be extended to a global fibration.
pg208-209
What is compression lemma? Proof
Let (X, A) be a CW-pair, (Y,B) any topological pair B != 0. Assume for each n s.t. X-A has cells of dim n pi_n(Y, B, y0) = 0 for all y0 in B. Then every map f : (X, A) –> (Y, B) is homotopic rel A to a map X –> B.
Be sure to discuss corollary that (Y, X) CW pair and X –> Y induces isos on pi_n for all n, then X is a def retract of Y.
pgs 209-210
What is Whitehead’s Theorem? Proof
A map f: X –> Y of connected CW complexes that induces isomorphisms on pi_n for all n is a homotopy equivalence.
PF. By cellular approximation f is homotopic to a cellular map g: X –>Y. Consider mapping cylinder of g: X–>Y.
- Since g cellular, Mg is a CW complex and (Mg, X) is a CW pair
- Mg def retracts to Y
- By corollary of compression lemma, Mg d.r. onto X.
Hence X and Y are homotopy equivalent.
NOTE: It is not enough to show pi_n(X) = pi_n(Y) for all n. There mist be a map f:X –> Y inducing isomorphisms f_*
An important corollary:
A CW complex is contractible <=> pi_n(X) = 0 for all n.
Just apply Whitehead to the constant map X –> point.
pg 211 -212
Is it possible for two CW complexes to have all the same homotopy groups, but not be homotopic? Examples
Yes! Whitehead requires the existence of a map f: X –> Y inducing the isomorphisms of homotopy groups.
- 3 -dim Lens spaces L(5,1) vs. L(5,2)
- S^2 and S^3 x CP^inf
Define: Weak homotopy equivalence
A weak homotopy equivalence f: X –> Y is a map inducing isomorphisms on pi_n for all n.
For CW complexes, a weak homotopy equivalence is a homotopy equivalence by Whitehead.
Prove: A weak homotopy equivalence also induces isomorphisms on H_n and H^n
By universal coefficients, inducing isos on H_n => inducing isos on H^n. Replacing Y with the mapping cylnder Mf, it suffices to consider the special case X < Y and f: X –> Y is the inclusion.
By l.e.s. in pi_n for (X,Y) inducing iso on pi_n <=> pi_n(Y, X) = 0
By l.e.s. in H_n for (X,Y) inducing iso on H_n <=> H_n(Y,X) = 0.
So we just need to show p_n(Y,X) = 0 for all n => H_n(Y,X) = 0 for all n.
pg 215-216
What is a CW approximation? Does every space have one? Proof?
Given a topological space X, a CW approximation to X is a CW complex Z and a weak homotopy equivalence f:Z –>X.
THM. Every topological space has a CW approximation.
For studying pi_n, H_n, H^n of any space, we can simply pass to a CW approximation and work within the category of CW complexes and cellular maps. Much easier to deal with.
pgs 217 -
Does every group have a K(G,1) -complex?
Yes. Start with a presentation 2-complex then attach 3 cells to kill all elements of pi_2(X^(2) ), 4 cells to kill … so X has pi_n(X) = 0 for all n > 1. Then X tilda has pi_n zero for all n so X tilda is contractible by Whitehead
We already saw that any two K(G,1) complexes for the same space are homotopy equivalent. Thus we can study groups using topology!
pg 219 -222
