Algebraic Topology Flashcards

1
Q

Poincare Duality

A

Let M be an n-dim closed, oriented manifold. Then H_k(M;Z) = H^(n-k)(M;Z)

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2
Q

Alexander Duality

A

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)

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3
Q

Cohomology

A

Replace Cn with its dual cochain group, Cn* = Hom(Cn,G).

Replace d with d* where d*(p) = p(d)

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4
Q

Universal Coefficient Theorem

Cohomology

A

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

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5
Q

Ext Functor Properties

A

Ext(H + H’, G) = Ext(H,G) + Ext(H’,G)
Ext(H,G) = 0 If H is free
Ext(Zn, G) = G/nG

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6
Q

Group Cohomology

A

The cohomology of a group G is given by Hn(K(G,1)), usually written Hn(G)

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7
Q

Eilenberg-MacLane Space

A

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.

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8
Q

Degrees of maps

A

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

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9
Q

Mayer Vietoris Sequence

A

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

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10
Q

Covering Space

A

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

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11
Q

Homotopy Lifting Property

A

Given f0’: Y -> Z’ there exists a unique f’t:Y -> X’ that lifts the homotopy ft

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12
Q

Homotopy lifting criterion

A

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.

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13
Q

Covering Spaces and pi_1

A

p*: pi1(X’) -> pi1(X) is injective

The index of p*(pi1(X’)) in pi1(X) is the number of sheets of X’

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14
Q

Universal Cover

A

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.

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15
Q

Deck Transformations

A

The isomorphisms X’ to X’ are called deck transformations. They form a group G(X’) under composition.

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16
Q

Normal Covering Space

A

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)

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17
Q

Deck transformations of Universal Cover

A

Let X’ be the universal cover of X, then the group of deck transformations of X’ is pi1(X).

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18
Q

Homology and path components

A
Hn(X) = sum Hn(Xi) where the Xi's are path components of X
H0 = sum Z for each path component of X
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19
Q

Chain Map

A

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

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20
Q

Induced maps and homotopy

A

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

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21
Q

Homotopy Groups

A

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

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22
Q

Covering maps and Higher homotopy groups

A

A covering map induces isomorphisms between pi_n of the covering space and base space for all n>1.

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23
Q

Cup Product

A

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.

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24
Q

Induced Cup Product

A

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)

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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.
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Retract
A mapping of a space X to a subspace A where the map is the identity on all points in A.
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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
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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.
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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.
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Homotopy Extension Property
(X,A) has the HEP if and only if X*{0} union A * I is a deformation retract of X*I If (X,A) has the HEP and A is contractible then X -> X/A is homotopy equivalence.
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Fundamental Group
pi_1(X) is the classes of all paths up to homotopy. It is a group with respect to [f][g] = [fg].
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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.
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Cap Product
Similar to Cup product but C_k * C^l -> C_(k-l). Get an induced cap product similarly.
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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.
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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.
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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)
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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)
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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. ```
39
Euler Characteristic
Given a space X with homology Hn(X) the Euler Characteristic is given by SUM_n (-1)^n rank(Hn(X)).
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Cellular Homology
Cn= cells of dimension n dell = attaching map for cells Homology of this sequence is cellular homology
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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) ->...
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Exact Sequence
A sequence ...->An+1 ->(an+1) An -> (an) An-1 ->... is exact if ker(an) = Im(an+1)
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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).
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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)
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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 .
46
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).
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Path Connected
X is path connected if there is a path connecting every pair of points in X. X,Y path connected means pi1(X*Y) = pi1(X)*pi1(Y)
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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)
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Contractible
X is contractible if the identity map is nullhomotopic. | "Can be shrunk to a point"
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Semi locally Simply Connected
If for every x in X there exists a neighborhood U such that pi(U) -> pi(X) is trivial
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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.
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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.
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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.
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Homology of RP2
0: Z 1: Z2 else = 0
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Homology of CPn
Z in even dimensions, 0 in odd
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Homology of Torus
0: Z 1: Z + Z 2: Z else = 0
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Homology of Sn
0: Z n: Z else = 0
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Homology of Klein bottle
0: Z 1: Z + Z2 else = 0