Algebraic Topology Flashcards
Poincare Duality
Let M be an n-dim closed, oriented manifold. Then H_k(M;Z) = H^(n-k)(M;Z)
Alexander Duality
Let K be a compact, locally contractible, nonempty proper subspace of S^n, then H_i(S^n-K;Z) = H^(n-i-1)(K,Z)
Cohomology
Replace Cn with its dual cochain group, Cn* = Hom(Cn,G).
Replace d with d* where d*(p) = p(d)
Universal Coefficient Theorem
Cohomology
For a chain complex C with homology groups Hn(C) we have the split exact sequence:
0 -> Ext(Hn-1(C), G) -> H^n(C;G) -> Hom(Hn(C),G) -> 0
Ext Functor Properties
Ext(H + H’, G) = Ext(H,G) + Ext(H’,G)
Ext(H,G) = 0 If H is free
Ext(Zn, G) = G/nG
Group Cohomology
The cohomology of a group G is given by Hn(K(G,1)), usually written Hn(G)
Eilenberg-MacLane Space
Let G be a group, n integer. K(G,1) is an Eilenberg MacLane space if it has nth homotopy group = G, all others trivial, and has contractible universal cover.
Degrees of maps
Let f: S^n -> S^n, then there is an induced map f: Z -> Z which must be of the form f(a) = d*a for some integer d. d is the degree.
We know deg(id) = 1
Homotopic maps have the same degree
Mayer Vietoris Sequence
Let A and B be subsets of X, such that the union of their interiors gives X. Then we have the long exact sequence:
-> Hn(A cap B) -> Hn(A) + Hn(B) -> Hn(X) -> Hn-1(A cap B) -> … -> H0(X) -> 0
Covering Space
A covering space of a space X is a space X’ together with map p: X’ -> X such that U open over of X and p^(-1)(U) is a disjoint union of open sets in X’ that are mapped by p homomorphically to U
Homotopy Lifting Property
Given f0’: Y -> Z’ there exists a unique f’t:Y -> X’ that lifts the homotopy ft
Homotopy lifting criterion
Let p: X’ -> X and f:Y -> X where Y is path connected and locally path connected. Then a lift f’ of f exists if and only if f(pi1(Y)) subset p(pi1(X’))
If two lifts of f agree on one point of Y, then if Y is connected they must agree on all of Y.
Covering Spaces and pi_1
p*: pi1(X’) -> pi1(X) is injective
The index of p*(pi1(X’)) in pi1(X) is the number of sheets of X’
Universal Cover
A simply connected covering space of X is its universal cover. This is unique up to isomorphism.
X has a universal cover if it is semi locally simply connected.
Deck Transformations
The isomorphisms X’ to X’ are called deck transformations. They form a group G(X’) under composition.
Normal Covering Space
A covering space is normal for all x in X if for each of lifts x1’ and x2’ of x, there is a deck transformation taking x1’ to x2’.
X’ is normal if and only if p(pi1(X’)) normal in p1(X)
Deck transformations of Universal Cover
Let X’ be the universal cover of X, then the group of deck transformations of X’ is pi1(X).
Homology and path components
Hn(X) = sum Hn(Xi) where the Xi's are path components of X H0 = sum Z for each path component of X
Chain Map
A map f: X ->Y induces a chain map
f#:Cn(X) -> Cn(Y). Chain map satisfies f#d = df# and sends cycles to cycles and boundaries to boundaries
A chain map induces homomorphisms between homology groups
Induced maps and homotopy
Homotopic maps induce the same homomorphism on homology.
f* induced by homotopy equivalence f are isomorphisms.
Chain homotopic maps induce the same homomorphism on homology
Homotopy Groups
pi_n is the set of homotopy classes of maps from S^n to X.
Base point does not matter is X is path connected.
pi_n of a product is the product of pi_n’s
Covering maps and Higher homotopy groups
A covering map induces isomorphisms between pi_n of the covering space and base space for all n>1.
Cup Product
Given two cochains a in C^k and b in C^l, the cup product of a and b lives in C^(k+l). It is given by taking a on the first k vertices of a simplex, and b on the last l vertices.
a cup b = b cup a with a possible sign change when my ring is commutative.
Induced Cup Product
We have a map from H^k * H^l -> H^(k+l) induced by the cup product. It satisfies f(a cup b) = f(a) cup f*(b)
Cross Product
External Cup Product
H^k * H^l -> H^(k+l) given by a * b -> p1(a) cup p2(b) where pi is the projection map.
Retract
A mapping of a space X to a subspace A where the map is the identity on all points in A.
Deformation Retract
A DR of X onto A is a family of maps f_t: X -> X such that f0 = id and f1=A and ft is the identity on A always.
Think of continually shrinking X to A
Homotopy
A homotopy is a family of maps f_t: X -> Y such that F: X * I -> Y given by F(x,t) = ft(x) is continuous.
Two maps are homotopic if there is a homotopy between them.
Homotopy Equivalence
f:X -> Y is a homotopy equivalence if there exists g: Y -> X such that f composed with g is the identity.
This is an equivalence relation.
Homotopy Extension Property
(X,A) has the HEP if and only if
X{0} union A * I is a deformation retract of XI
If (X,A) has the HEP and A is contractible then X -> X/A is homotopy equivalence.
Fundamental Group
pi_1(X) is the classes of all paths up to homotopy. It is a group with respect to [f][g] = [fg].
Whitehead’s Theorem
If a map f between connected CW complexes induces an isomorphism f* on pi_n’s, for all n, then f is a homotopy equivalence.
If f is the inclusion of a sub complex X to Y, then X is a deformation retract of Y.
Cap Product
Similar to Cup product but
C_k * C^l -> C_(k-l).
Get an induced cap product similarly.
Betti Numbers
If we write H_n(X) as the direct sum of cyclic groups, then the number of Z summands is the nth Betti number.
Torsion Coefficients
If we write H_n(X) as the direct sum of cyclic groups, then the integers specifying the orders of the finite cyclic summands are torsion coefficients.
Covariant Functor
Let C and D be categories. A covariant functor F: C to D associates each object X in C an object F(X) in D and associates to each morphism f:X to Y in C a morphism F(f) from F(X) to F(Y) in D.
F(id) = id, and F(g circ f) = F(g) circ F(f)
Contravariant Functor
Let C and D be categories. A covariant functor F: C to D associates each object X in C an object F(X) in D and associates to each morphism f:X to Y in C a morphism F(f) from F(Y) to F(X) in D.
F(id) = id, and F(g circ f) = F(g) circ F(f)
Simplicial Homology
Let Delta_n(X) be a free abelian group with the basis of open n-simplicies of X. The elements of this group are n-chains. Simplicial Homology is the homology of this Simplicial homology is only defined for simplicial complexes.
Euler Characteristic
Given a space X with homology Hn(X) the Euler Characteristic is given by
SUM_n (-1)^n rank(Hn(X)).
Cellular Homology
Cn= cells of dimension n
dell = attaching map for cells
Homology of this sequence is cellular homology
Homology of a pair
Let A be a deformation retract of a neighborhood in X, then (X,A) is a good pair.
Then we have the exact sequence:
…-> Hn(A) -> Hn(X) -> Hn(X/A) -? Hn-1(A) ->…
Exact Sequence
A sequence
…->An+1 ->(an+1) An -> (an) An-1 ->…
is exact if ker(an) = Im(an+1)
Short Exact Sequence
0 -> A -> (alpha) B -> (beta) C -> 0 exact if and only if alpha is injective, beta is surjective and ker(beta) = Im(alpha). So we have isomorphism C = B/Im(alpha).
Relative Homology
Given a space X and a subspace A let Cn(X,A) = Cn(X)/Cn(A). Relative homology is the homology of this sequence.
Get LES
…-> Hn(A) -> Hn(X) -> Hn(X,A) -> Hn-1(A)->…0
Hn(X,A) = 0 implies Hn(A) = Hn(X)
Excision Theorem
Given subspaces Z ⊂ A ⊂ X such that the closure of Z is contained in the interior of A, then the inclusion (X − Z, A − Z) into (X, A) induces isomorphisms Hn (X − Z , A − Z )→Hn (X , A) for all n .
Van Kampen’s Theorem
If X is the union of open path connected An’s with the basepoint in the intersection of all of these, and each pairwise intersection is path connected, then phi: *pi_1(An) -> pi_1(X) is surjective.
If each 3 way intersection is path connected then pi(X) = *pi_1(A)/N where N is ker(phi).
Path Connected
X is path connected if there is a path connecting every pair of points in X.
X,Y path connected means pi1(XY) = pi1(X)pi1(Y)
Simply Connected
X is simply connected if it is path connected an has pi_1(X) = 0.
Contractible -> simply connected but not the other way around (Think sphere)
Contractible
X is contractible if the identity map is nullhomotopic.
“Can be shrunk to a point”
Semi locally Simply Connected
If for every x in X there exists a neighborhood U such that pi(U) -> pi(X) is trivial
Induced Homomorphisms
Let phi: X -> Y then phi induces homomorphism phi; pi_1(X) -> pi_1(Y) where phi*[f] = [phif] for loops f based at the base point.
Singular Homology
A singular n-simplex is a map sigma: Delta -> X. Cn(X) is the free abelian group with basis os singular simplices.
Singular homology group is given by this.
Reduced Homology
Reduced homology group goes from C0 -> Z -> 0.
Reduced homology_0 + Z = H_0 normal. And the rest of the groups are the same.
Homology of RP2
0: Z
1: Z2
else = 0
Homology of CPn
Z in even dimensions, 0 in odd
Homology of Torus
0: Z
1: Z + Z
2: Z
else = 0
Homology of Sn
0: Z
n: Z
else = 0
Homology of Klein bottle
0: Z
1: Z + Z2
else = 0
