Algebraic Topology

Question 1

Poincare Duality

Answer 1

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

Question 2

Alexander Duality

Answer 2

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)

Question 3

Cohomology

Answer 3

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

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

Question 4

Universal Coefficient Theorem

Cohomology

Answer 4

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

Question 5

Ext Functor Properties

Answer 5

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

Question 6

Group Cohomology

Answer 6

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

Question 7

Eilenberg-MacLane Space

Answer 7

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.

Question 8

Degrees of maps

Answer 8

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

Question 9

Mayer Vietoris Sequence

Answer 9

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

Question 10

Covering Space

Answer 10

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

Question 11

Homotopy Lifting Property

Answer 11

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

Question 12

Homotopy lifting criterion

Answer 12

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.

Question 13

Covering Spaces and pi_1

Answer 13

p*: pi1(X’) -> pi1(X) is injective

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

Question 14

Universal Cover

Answer 14

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.

Question 15

Deck Transformations

Answer 15

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

Question 16

Normal Covering Space

Answer 16

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)

Question 17

Deck transformations of Universal Cover

Answer 17

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

Question 18

Homology and path components

Answer 18

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

Question 19

Chain Map

Answer 19

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

Question 20

Induced maps and homotopy

Answer 20

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

Question 21

Homotopy Groups

Answer 21

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

Question 22

Covering maps and Higher homotopy groups

Answer 22

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

Question 23

Cup Product

Answer 23

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.

Question 24

Induced Cup Product

Answer 24

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)

Question 25

Cross Product

External Cup Product

Answer 25

H^k * H^l -> H^(k+l) given by a * b -> p1(a) cup p2(b) where pi is the projection map.

Question 26

Retract

Answer 26

A mapping of a space X to a subspace A where the map is the identity on all points in A.

Question 27

Deformation Retract

Answer 27

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

Question 28

Homotopy

Answer 28

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.

Question 29

Homotopy Equivalence

Answer 29

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.

Question 30

Homotopy Extension Property

Answer 30

(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.

Question 31

Fundamental Group

Answer 31

pi_1(X) is the classes of all paths up to homotopy. It is a group with respect to [f][g] = [fg].

Question 32

Whitehead’s Theorem

Answer 32

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.

Question 33

Cap Product

Answer 33

Similar to Cup product but
C_k * C^l -> C_(k-l).

Get an induced cap product similarly.

Question 34

Betti Numbers

Answer 34

If we write H_n(X) as the direct sum of cyclic groups, then the number of Z summands is the nth Betti number.

Question 35

Torsion Coefficients

Answer 35

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.

Question 36

Covariant Functor

Answer 36

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)

Question 37

Contravariant Functor

Answer 37

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)

Question 38

Simplicial Homology

Answer 38

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.

Question 39

Euler Characteristic

Answer 39

Given a space X with homology Hn(X) the Euler Characteristic is given by
SUM_n (-1)^n rank(Hn(X)).

Question 40

Cellular Homology

Answer 40

Cn= cells of dimension n
dell = attaching map for cells
Homology of this sequence is cellular homology

Question 41

Homology of a pair

Answer 41

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) ->…

Question 42

Exact Sequence

Answer 42

A sequence
…->An+1 ->(an+1) An -> (an) An-1 ->…
is exact if ker(an) = Im(an+1)

Question 43

Short Exact Sequence

Answer 43

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).

Question 44

Relative Homology

Answer 44

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)

Question 45

Excision Theorem

Answer 45

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 .

Question 46

Van Kampen’s Theorem

Answer 46

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).

Question 47

Path Connected

Answer 47

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)

Question 48

Simply Connected

Answer 48

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)

Question 49

Contractible

Answer 49

X is contractible if the identity map is nullhomotopic.

“Can be shrunk to a point”

Question 50

Semi locally Simply Connected

Answer 50

If for every x in X there exists a neighborhood U such that pi(U) -> pi(X) is trivial

Question 51

Induced Homomorphisms

Answer 51

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.

Question 52

Singular Homology

Answer 52

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.

Question 53

Reduced Homology

Answer 53

Reduced homology group goes from C0 -> Z -> 0.

Reduced homology_0 + Z = H_0 normal. And the rest of the groups are the same.

Question 54

Homology of RP2

Answer 54

0: Z
1: Z2
else = 0

Question 55

Homology of CPn

Answer 55

Z in even dimensions, 0 in odd

Question 56

Homology of Torus

Answer 56

0: Z
1: Z + Z
2: Z
else = 0

Question 57

Homology of Sn

Answer 57

0: Z
n: Z
else = 0

Question 58

Homology of Klein bottle

Answer 58

0: Z
1: Z + Z2
else = 0