Hungry, Hungry Homology
|
|
- Jerome Reeves
- 5 years ago
- Views:
Transcription
1 September 27, 2017
2 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of C by A. Clearly, the answer is yes as if we take E = A C, then A E and E /A = C.
3 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of C by A. Clearly, the answer is yes as if we take E = A C, then A E and E /A = C.
4 Motiving Problem: Algebra Problem Given groups A, C, how many extensions E exist?
5 Motiving Problem: Topology Problem How do we tell spaces apart?
6 Motiving Problem: Topology C = {e 2πin : n [0, 1)} D = {(r cos θ, r sin θ): r [0, 1], θ [0, 2π)} T = (2 + 2 cos θ) cos φ, (2 + 2 cos θ) sin φ, 2 sin θ): 0 θ, φ 2π
7 Motiving Problem: Topology C = {e 2πin : n [0, 1)} D = {(r cos θ, r sin θ): r [0, 1], θ [0, 2π)} T = (2 + 2 cos θ) cos φ, (2 + 2 cos θ) sin φ, 2 sin θ): 0 θ, φ 2π
8 Movie Time
9 Motiving Problem: Analysis Problem Find a set X 3 such that... V := {F : 3 \ X 3 : F = 0} W := {F : F = g} dim(v /W ) = 8
10 Algebra
11 Theorem (First Isomorphism Theorem) If φ : G H is a homomorphism of groups, then ker φ G and G/ ker φ = φ(g). Corollary If φ : G H is a surjective homomorphism of groups, then G/ ker φ = H.
12 Theorem (First Isomorphism Theorem) If φ : G H is a homomorphism of groups, then ker φ G and G/ ker φ = φ(g). Corollary If φ : G H is a surjective homomorphism of groups, then G/ ker φ = H.
13 Definition (Exact) A pair of R-homomorphisms M f N g P is called exact (at N) if im f = ker g. An infinite sequence of homomorphisms is exact if it is exact in each spot. A sequence of the form is called a short exact sequence. 0 M f N g P 0
14 Example (i) The sequence 0 M f N is exact if and only if f is injective. (ii) The sequence N g P 0 is exact if and only if g is surjective.
15 Example (i) The sequence 0 M f N is exact if and only if f is injective. (ii) The sequence N g P 0 is exact if and only if g is surjective. (iii) The sequence 0 M f N 0 is exact if and only if f is an isomorphism.
16 Example (i) The sequence 0 M f N is exact if and only if f is injective. (ii) The sequence N g P 0 is exact if and only if g is surjective. (iii) The sequence 0 M f N 0 is exact if and only if f is an isomorphism. (iv) The sequence 0 M f N g P 0 is exact if and only if f is injective, g is surjective, and im f = ker g.
17 Example (i) The sequence 0 M f N is exact if and only if f is injective. (ii) The sequence N g P 0 is exact if and only if g is surjective. (iii) The sequence 0 M f N 0 is exact if and only if f is an isomorphism. (iv) The sequence 0 M f N g P 0 is exact if and only if f is injective, g is surjective, and im f = ker g.
18 Example (v) The sequence 0 ker f M f N coker f 0 is an exact sequence. (vi) If P N M is a tower of submodules, there is an exact sequence 0 N/P M/N M/P 0.
19 Example (v) The sequence 0 ker f M f N coker f 0 is an exact sequence. (vi) If P N M is a tower of submodules, there is an exact sequence 0 N/P M/N M/P 0. i (vii) The sequence 0 M M N π N 0 is an exact sequence. Therefore, an extension of N by M always exists. To see an explicit example, take the following: 0 i ( /n ) π /n 0
20 Example (v) The sequence 0 ker f M f N coker f 0 is an exact sequence. (vi) If P N M is a tower of submodules, there is an exact sequence 0 N/P M/N M/P 0. i (vii) The sequence 0 M M N π N 0 is an exact sequence. Therefore, an extension of N by M always exists. To see an explicit example, take the following: 0 i ( /n ) π /n 0 However, this is not the only possible extension of /n by : 0 n π /n 0
21 Example (v) The sequence 0 ker f M f N coker f 0 is an exact sequence. (vi) If P N M is a tower of submodules, there is an exact sequence 0 N/P M/N M/P 0. i (vii) The sequence 0 M M N π N 0 is an exact sequence. Therefore, an extension of N by M always exists. To see an explicit example, take the following: 0 i ( /n ) π /n 0 However, this is not the only possible extension of /n by : 0 n π /n 0
22 Definition (Homomorphism of Exact Sequence) If (M n, f n ) n and (P n, g n ) n are exact sequences, then a homomorphism of exact sequences is a collection of maps (β n ) n, β n : M n P n such that for all n and m M n, g n β n (m) = β n 1 f n (m); that is, the following diagram commutes f n+1 M n+1 M n M n 1 f n f n 1 β n+1 β n β n 1 g n+1 g n g n 1 P n+1 P n P n 1
23 Example (i) Suppose m, n + and n m. Let k = m/n. Then (abusing notation by letting π to denote the obvious projection maps in each case) the following diagram commutes. 0 /n 0 π n n 0 /k /m /n 0 π π π 1
24 Example (ii) The following is a non-canonical morphism of exact sequences: 0 /2 /2 /2 /2 0 1 i 1 0 /2 /2 /2 /2 0 s π 2 i 1 π 1 where i j is the inclusion into the jth component, π i is the projection onto the ith component, and s swaps components. Note defining maps the obvious way would not give a commutative diagram. 1
25 Proposition (The Short 5 Lemma) If the rows in the following commutative diagram are exact and α, γ are isomorphisms, then β is an isomorphism. 0 A B C 0 α f r 0 M N P 0 β g s γ
26 Proposition (The 5 Lemma) Consider the following commutative diagram with exact rows: f 1 f 2 M 1 M 2 M 3 M 4 M 5 f 3 f 4 h 1 h 2 h 3 h 4 h 5 g 1 g 2 g 3 g 4 N 1 N 2 N 3 N 4 N 5 (a) if h 2, h 4 are surjective and h 5 is injective, then h 3 is surjective. (b) if h 2, h 4 are injective and h 1 is surjective, then h 3 is injective. (c) if h 1, h 2, h 4, and h 5 are isomorphisms, then h 3 is an isomorphism.
27 Theorem (Mitchell Embedding Theorem) If is an abelian category, there is an associative unital ring R and a fully faithful exact functor F : Mod(R).
28 Definition (Chain Complex) A sequence of R-modules, (C n, d n ) n, with maps (called boundary maps) d n : C n C n 1 is said to be a chain complex if im d n+1 ker d n for all n. dn+3 d n+2 d n+1 d n d n 1 d n 2 C n+2 C n+1 C n C n 1 C n 2
29 Definition (Homology) Given a sequence of R-modules (C n, δ n ) n, the nth homology group of the complex is H n := ker δ n / im δ n+1. Definition (Chain Map) Given two chain complexes (C n, δ n ) n and (C n, δ n ) n, a map of chain complexes is a family of R-homomorphisms (f n : C n C n ) n so that δ n f n = f n 1 δ n for all n ; that is, that the diagram below commutes. δ n+1 C n+1 C n C n 1 δ n δ n 1 f n+1 f n f n 1 δ n+1 δ n δ n 1 C n+1 C n C n 1
30 Definition (Homology) Given a sequence of R-modules (C n, δ n ) n, the nth homology group of the complex is H n := ker δ n / im δ n+1. Definition (Chain Map) Given two chain complexes (C n, δ n ) n and (C n, δ n ) n, a map of chain complexes is a family of R-homomorphisms (f n : C n C n ) n so that δ n f n = f n 1 δ n for all n ; that is, that the diagram below commutes. δ n+1 C n+1 C n C n 1 δ n δ n 1 f n+1 f n f n 1 δ n+1 δ n δ n 1 C n+1 C n C n 1
31 Lemma (Snake Lemma) Given the following commutative diagram of R-modules with exact rows f M N P 0 α 0 M N P there is an exact sequence β g f g ker α f ker β g ker γ δ coker α f coker β g coker γ γ
32 0 0 0 ker α ker β ker γ M f N g P 0 α β γ 0 M N P f g coker α coker β coker γ 0 0 0
33 0 0 0 ker α ker β ker γ δ M f N g P 0 α β γ 0 M N P f g coker α coker β coker γ 0 0 0
34 Lemma (9 Lemma or 3 3 Lemma) If the following is a commutative diagram with exact columns and the bottom two rows exact, then the top row is exact f 0 A B C 0 α β γ f 0 A B C 0 0 A B C 0 g α β γ g f g 0 0 0
35 We can recognize all group extensions as a cohomology group using delooping from Topoi Theory. H 2 Grp (C, A) = H2 (BC, A) = H(BC, B 2 A) Or better yet using Ext groups. 0 A E C 0
36 We can recognize all group extensions as a cohomology group using delooping from Topoi Theory. H 2 Grp (C, A) = H2 (BC, A) = H(BC, B 2 A) Or better yet using Ext groups. 0 A E C 0 F 2 F 1 F 0 C 0
37 We can recognize all group extensions as a cohomology group using delooping from Topoi Theory. H 2 Grp (C, A) = H2 (BC, A) = H(BC, B 2 A) Or better yet using Ext groups. 0 A E C 0 F 2 F 1 F 0 C 0 0 Hom R (F 0, A) Hom R (F 1, A) Hom R (F 2, A) There is a bijective correspondence between the equivalence class of extensions of C by A and the elements of Ext 1 R (C, A).
38 We can recognize all group extensions as a cohomology group using delooping from Topoi Theory. H 2 Grp (C, A) = H2 (BC, A) = H(BC, B 2 A) Or better yet using Ext groups. 0 A E C 0 F 2 F 1 F 0 C 0 0 Hom R (F 0, A) Hom R (F 1, A) Hom R (F 2, A) There is a bijective correspondence between the equivalence class of extensions of C by A and the elements of Ext 1 R (C, A).
39 Topology
40 Definition (Path Homotopy) A homotopy of paths f 0, f 1 (from x 0 to x 1 ) in a space X is a family f t : I X, 0 t 1 such that (i) f t (0) = x 0 and f t (1) = x 1 are independent of t. (ii) the associated map F : I I X defined by F (s, t) = f t (s) is continuous. If f 0 and f 1 are connected by a homotopy f t, we say that f 0 and f 1 are homotopic and denote this f 0 f 1. That is, f 0 and f 1 are path homotopic if we can continuously deform the path f 0 into the path f 1, as in the figure below.
41 Definition (Path Homotopy) A homotopy of paths f 0, f 1 (from x 0 to x 1 ) in a space X is a family f t : I X, 0 t 1 such that (i) f t (0) = x 0 and f t (1) = x 1 are independent of t. (ii) the associated map F : I I X defined by F (s, t) = f t (s) is continuous. If f 0 and f 1 are connected by a homotopy f t, we say that f 0 and f 1 are homotopic and denote this f 0 f 1. That is, f 0 and f 1 are path homotopic if we can continuously deform the path f 0 into the path f 1, as in the figure below.
42 Definition (Fundamental Group) The fundamental group of a space X at a point x 0 X is the set of equivalence classes of path homotopic loops at x 0, denoted π 1 (X, x 0 ).
43
44 Definition (Fundamental Group) The fundamental group of a space X at a point x 0 X is the set of equivalence classes of path homotopic loops at x 0, denoted π 1 (X, x 0 ). Theorem Every subgroup of a free group is free.
45 Definition (Fundamental Group) The fundamental group of a space X at a point x 0 X is the set of equivalence classes of path homotopic loops at x 0, denoted π 1 (X, x 0 ). Theorem Every subgroup of a free group is free. Problem Find all index 2 subgroups of the free group F (2) = a, b.
46 Definition (Fundamental Group) The fundamental group of a space X at a point x 0 X is the set of equivalence classes of path homotopic loops at x 0, denoted π 1 (X, x 0 ). Theorem Every subgroup of a free group is free. Problem Find all index 2 subgroups of the free group F (2) = a, b. Theorem Let F (n) denote the free group on n generators. Then for n = 2, 3,..., F (n) F (2).
47 Definition (Fundamental Group) The fundamental group of a space X at a point x 0 X is the set of equivalence classes of path homotopic loops at x 0, denoted π 1 (X, x 0 ). Theorem Every subgroup of a free group is free. Problem Find all index 2 subgroups of the free group F (2) = a, b. Theorem Let F (n) denote the free group on n generators. Then for n = 2, 3,..., F (n) F (2).
48
49 ab 1 = b 1 a
50 ab 1 = b 1 a a b = ( b) + a
51 ab 1 = b 1 a a b = ( b) + a a b + c d = (a c) + (b d) = (a d) + (b c)
52
53
54 0 δ 2 C 1 = = a δ 1 C 0 = = v δ 0 0
55
56 0 δ 2 C 1 = 2 = a, b δ 1 C 0 = 2 = v, w δ 0 0
57 0 δ 3 C 2 = = U δ 2 C 1 = 2 = a, b δ 1 C 0 = 2 = v, w δ 0 0
58 Analysis
59 Problem If f = (f 1, f 2 ) : U 2 is a smooth function on an open set U 2, is there a smooth function F : U so that F x = f 1 and F y = f 2, i.e. F = (f 1, f 2 )?
60 Since F is smooth function, we have 2 F f 1 y = f 2 x. y x = 2 F x y and hence
61 Clearly, this condition is necessary but is it sufficient?
62 Let f : 2 \ {(0, 0)} 2 be given by f (x, y) = y x 2 + y 2, x x 2 + y 2 It is routine to check that f 1 y = f 2 x. However, there is no function F : 2 \ {(0, 0)} so that F = f.
63 Let f : 2 \ {(0, 0)} 2 be given by f (x, y) = y x 2 + y 2, x x 2 + y 2 It is routine to check that f 1 y = f 2 x. However, there is no function F : 2 \ {(0, 0)} so that F = f.
64 If there were then 2π 0 d F (cos θ, sin θ) dθ = F (1, 0) F (1, 0) = 0 dθ However, the chain rule gives d F F (cos θ, sin θ) = dθ x ( sin θ) + F y cos θ = f 1 (cos θ, sin θ) sin θ + f 2 (cos θ, sin θ) cos θ = 1
65 If there were then 2π 0 d F (cos θ, sin θ) dθ = F (1, 0) F (1, 0) = 0 dθ However, the chain rule gives d F F (cos θ, sin θ) = dθ x ( sin θ) + F y cos θ = f 1 (cos θ, sin θ) sin θ + f 2 (cos θ, sin θ) cos θ = 1 Therefore, no such F can exist. Note that the open set U where f (x, y) is defined contains a hole namely (0, 0). However for a nice open region, the question we asked as an affirmative solution.
66 If there were then 2π 0 d F (cos θ, sin θ) dθ = F (1, 0) F (1, 0) = 0 dθ However, the chain rule gives d F F (cos θ, sin θ) = dθ x ( sin θ) + F y cos θ = f 1 (cos θ, sin θ) sin θ + f 2 (cos θ, sin θ) cos θ = 1 Therefore, no such F can exist. Note that the open set U where f (x, y) is defined contains a hole namely (0, 0). However for a nice open region, the question we asked as an affirmative solution.
67
68 Definition (Star-Shaped) A subset X n is star-shaped with respect to x 0 X if the line segment {tx 0 + (1 t)x : t [0, 1]} is contained in X for all x X. Proposition If U 2 is an open star-shaped region, then any smooth function f = (f 1, f 2 ) : U 2 with f 1 y = f 2 x, there is a function F : U with F = f.
69 Definition (Star-Shaped) A subset X n is star-shaped with respect to x 0 X if the line segment {tx 0 + (1 t)x : t [0, 1]} is contained in X for all x X. Proposition If U 2 is an open star-shaped region, then any smooth function f = (f 1, f 2 ) : U 2 with f 1 y = f 2 x, there is a function F : U with F = f.
70 Let U 3 be an open set and C (U, n ) be the set of smooth functions φ : U n on U. Taking U 3, we define the following maps: Grad : C (U, ) C (U, 3 ); Curl : C (U, 3 ) C (U, 3 ); Div : C (U, 3 ) C (U, ); φ φ φ φ φ φ
71 Let U 3 be an open set and C (U, n ) be the set of smooth functions φ : U n on U. Taking U 3, we define the following maps: Grad : C (U, ) C (U, 3 ); Curl : C (U, 3 ) C (U, 3 ); Div : C (U, 3 ) C (U, ); φ φ φ φ φ φ Now Grad, Rot, and Div are linear operators, Curl Grad = 0 (gradient fields are irrotational), and Div Rot = 0 (curls are incompressible) so that im Grad is a subspace of ker Curl and im Rot is a subspace of ker Div.
72 Let U 3 be an open set and C (U, n ) be the set of smooth functions φ : U n on U. Taking U 3, we define the following maps: Grad : C (U, ) C (U, 3 ); Curl : C (U, 3 ) C (U, 3 ); Div : C (U, 3 ) C (U, ); φ φ φ φ φ φ Now Grad, Rot, and Div are linear operators, Curl Grad = 0 (gradient fields are irrotational), and Div Rot = 0 (curls are incompressible) so that im Grad is a subspace of ker Curl and im Rot is a subspace of ker Div. Therefore, we define the following quotients: H 0 (U) := ker Grad H 1 (U) := ker Curl/ im Grad H 2 (U) := ker Div/ im Curl
73 Let U 3 be an open set and C (U, n ) be the set of smooth functions φ : U n on U. Taking U 3, we define the following maps: Grad : C (U, ) C (U, 3 ); Curl : C (U, 3 ) C (U, 3 ); Div : C (U, 3 ) C (U, ); φ φ φ φ φ φ Now Grad, Rot, and Div are linear operators, Curl Grad = 0 (gradient fields are irrotational), and Div Rot = 0 (curls are incompressible) so that im Grad is a subspace of ker Curl and im Rot is a subspace of ker Div. Therefore, we define the following quotients: H 0 (U) := ker Grad H 1 (U) := ker Curl/ im Grad H 2 (U) := ker Div/ im Curl
74 Proposition An open set U k is connected if and only if H 0 (U) =. In fact, this can be extended to show that dimh 0 (U) is the number of connected components of U. By Proposition 4.1 above, we have H 1 (U) = 0 whenever U 2 is star-shaped. Furthermore from our previous work, we know that H 1 ( 2 \ {(0, 0)} 0. Thus for a star-shaped region, we have
75 Proposition An open set U k is connected if and only if H 0 (U) =. In fact, this can be extended to show that dimh 0 (U) is the number of connected components of U. By Proposition 4.1 above, we have H 1 (U) = 0 whenever U 2 is star-shaped. Furthermore from our previous work, we know that H 1 ( 2 \ {(0, 0)} 0. Thus for a star-shaped region, we have Proposition If U 3 is an open star-shaped region, then H 0 (U) = and H 1 (U) = H 2 (U) = 0.
76 Proposition An open set U k is connected if and only if H 0 (U) =. In fact, this can be extended to show that dimh 0 (U) is the number of connected components of U. By Proposition 4.1 above, we have H 1 (U) = 0 whenever U 2 is star-shaped. Furthermore from our previous work, we know that H 1 ( 2 \ {(0, 0)} 0. Thus for a star-shaped region, we have Proposition If U 3 is an open star-shaped region, then H 0 (U) = and H 1 (U) = H 2 (U) = 0.
77 Definition (Alternating Map) A k-linear map ω : V k is alternating if ω(v 1,..., v k ) = 0 whenever v i = v j for i j. The vector space of alternating k-linear maps is denoted Alt k (V ). Definition (Exterior Product) For ω 1 Alt p (V ) and ω 2 Alt q (V ), define (ω 1 ω 2 )(v 1, v 2,..., v p+q ) := sgn(σ) ω 1 (v σ(1),..., v σ(p) ) σ S(p,q) ω 2 (v σ(p+1),..., v σ(p+q) ) where S(p, q) is all the permutations of {1, 2,..., p + q} with σ(1) < < σ(p) and σ(p + 1) < < σ(p + q).
78 Definition (Alternating Map) A k-linear map ω : V k is alternating if ω(v 1,..., v k ) = 0 whenever v i = v j for i j. The vector space of alternating k-linear maps is denoted Alt k (V ). Definition (Exterior Product) For ω 1 Alt p (V ) and ω 2 Alt q (V ), define (ω 1 ω 2 )(v 1, v 2,..., v p+q ) := sgn(σ) ω 1 (v σ(1),..., v σ(p) ) σ S(p,q) ω 2 (v σ(p+1),..., v σ(p+q) ) where S(p, q) is all the permutations of {1, 2,..., p + q} with σ(1) < < σ(p) and σ(p + 1) < < σ(p + q).
79 This product has the following properties: (i) ω 1 ω 2 is a (p + q)-linear map. (ii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 Alt p+q (V ).
80 This product has the following properties: (i) ω 1 ω 2 is a (p + q)-linear map. (ii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 Alt p+q (V ). (iii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 = ( 1) pq ω 2 ω 1.
81 This product has the following properties: (i) ω 1 ω 2 is a (p + q)-linear map. (ii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 Alt p+q (V ). (iii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 = ( 1) pq ω 2 ω 1. (iv) if ω 1 Alt p (V ), ω 2 Alt q (V ), and ω 3 Alt r (V ), then ω 1 (ω 2 ω 3 ) = (ω 1 ω 2 ) ω 3.
82 This product has the following properties: (i) ω 1 ω 2 is a (p + q)-linear map. (ii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 Alt p+q (V ). (iii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 = ( 1) pq ω 2 ω 1. (iv) if ω 1 Alt p (V ), ω 2 Alt q (V ), and ω 3 Alt r (V ), then ω 1 (ω 2 ω 3 ) = (ω 1 ω 2 ) ω 3. Let U n be an open set, {e 1, e 2,..., e n } denote the standard basis for n, and {ε 1, ε 2,, ε n } the dual basis of Alt 1 ( n ).
83 This product has the following properties: (i) ω 1 ω 2 is a (p + q)-linear map. (ii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 Alt p+q (V ). (iii) if ω 1 Alt p (V ) and ω 2 Alt q (V ), then ω 1 ω 2 = ( 1) pq ω 2 ω 1. (iv) if ω 1 Alt p (V ), ω 2 Alt q (V ), and ω 3 Alt r (V ), then ω 1 (ω 2 ω 3 ) = (ω 1 ω 2 ) ω 3. Let U n be an open set, {e 1, e 2,..., e n } denote the standard basis for n, and {ε 1, ε 2,, ε n } the dual basis of Alt 1 ( n ).
84 Definition (Differential p-form) A differential p-form on U is a smooth map ω : U Alt p ( n ). The vector space of all such maps is denoted by Ω p (U). For example, if p = 0 then Alt 0 ( ) and Ω 0 (U) is just the vector space of smooth real-valued functions on U so that Ω 0 (U) = C (U, ). Let Dω denote the ordinary derivative of a smooth map ω : U Alt p ( n ) and its value at x by D x ω.
85 Definition (Differential p-form) A differential p-form on U is a smooth map ω : U Alt p ( n ). The vector space of all such maps is denoted by Ω p (U). For example, if p = 0 then Alt 0 ( ) and Ω 0 (U) is just the vector space of smooth real-valued functions on U so that Ω 0 (U) = C (U, ). Let Dω denote the ordinary derivative of a smooth map ω : U Alt p ( n ) and its value at x by D x ω.
86 Definition (Exterior Differential) The exterior differential d p : Ω p (U) Ω p+1 (U) is the linear operator p+1 d p x ω(v 1,..., v p+1 ) = ( 1) i 1 D x ω(v i )(v 1,..., v i,..., v p+1 ) i=1 where D x ω(e j ) = ω I x (x)ε j I, j = 1,..., n, ε I = ε i1 ε ip, and I runs over all sequences with 1 i 1 < i 2 < < i p n. Now d p x ω Altp+1 ( n ) and the composition d 2, Ω p (U) Ω p+1 (U) Ω p+2 (U), is zero. This should make one think that we can define a homology on these spaces and this is indeed the case.
87 Definition (Exterior Differential) The exterior differential d p : Ω p (U) Ω p+1 (U) is the linear operator p+1 d p x ω(v 1,..., v p+1 ) = ( 1) i 1 D x ω(v i )(v 1,..., v i,..., v p+1 ) i=1 where D x ω(e j ) = ω I x (x)ε j I, j = 1,..., n, ε I = ε i1 ε ip, and I runs over all sequences with 1 i 1 < i 2 < < i p n. Now d p x ω Altp+1 ( n ) and the composition d 2, Ω p (U) Ω p+1 (U) Ω p+2 (U), is zero. This should make one think that we can define a homology on these spaces and this is indeed the case.
88 Definition (de Rham Cohomology) The pth de Rham cohomology group is the quotient vector space H p (U) = ker d p / im d p 1 Then in particular H 0 (U) is the kernel of d : C (U, ) Ω 1 (U) the vector space of maps f C (U, ) with vanishing derivative, i.e. the space of locally constant maps. The case where U is again a star-shaped region, not surprisingly, have rigid restrictions on their de Rham cohomology groups.
89 Definition (de Rham Cohomology) The pth de Rham cohomology group is the quotient vector space H p (U) = ker d p / im d p 1 Then in particular H 0 (U) is the kernel of d : C (U, ) Ω 1 (U) the vector space of maps f C (U, ) with vanishing derivative, i.e. the space of locally constant maps. The case where U is again a star-shaped region, not surprisingly, have rigid restrictions on their de Rham cohomology groups.
90 Theorem (Poincaré s Lemma) If U n is an open star-shaped region, then H p (U) = 0 for p > 0 and H 0 (U) =. Theorem If an open set U n is covered by convex open sets U 1,..., U r, then H p (U) is finitely generated.
91 Theorem (Poincaré s Lemma) If U n is an open star-shaped region, then H p (U) = 0 for p > 0 and H 0 (U) =. Theorem If an open set U n is covered by convex open sets U 1,..., U r, then H p (U) is finitely generated.
92 Theorem If C n is a closed subset, then H p+1 ( n+1 \ C) = H p ( n \ C), p 1 H 1 ( n+1 \ C) = H 0 ( n \ C)/ H 0 ( n+1 \ C) = Theorem If U n is an open contractible set, then H p (U) = 0 for p > 0 and H 0 (U) =.
93 Theorem If C n is a closed subset, then H p+1 ( n+1 \ C) = H p ( n \ C), p 1 H 1 ( n+1 \ C) = H 0 ( n \ C)/ H 0 ( n+1 \ C) = Theorem If U n is an open contractible set, then H p (U) = 0 for p > 0 and H 0 (U) =.
94 Theorem (Mayer-Vietoris) If U 1, U 2 n are open subsets, there is an exact sequence of cohomology of vector spaces H p (U 1 U 2 ) H p (U 1 ) H p (U 2 ) H p (U 1 U 2 ) δ H p+1 (U 1 U 2 )
95 We now return to the original problem: V = {F : 3 \ X 3 : F = 0} W = {F : F = g} dim(v /W ) = 8
96 Notice then we can phrase our original question as follows: find a set X so that dim H 1 ( 3 \ X ) = 8. Take X to be the set of 8 lines parallel to the z-axis, distributed equally along the unit circle.
97 Notice then we can phrase our original question as follows: find a set X so that dim H 1 ( 3 \ X ) = 8. Take X to be the set of 8 lines parallel to the z-axis, distributed equally along the unit circle.
98
99 Notice then we can phrase our original question as follows: find a set X so that dim H 1 ( 3 \ X ) = 8. Take X to be the set of 8 lines parallel to the z-axis, distributed equally along the unit circle. Observe that 3 \ X deformation retracts to the plane minus the 8 intersections of these lines with the plane. It is a routine exercise to show that, p = 0, 1 H p ( 2 \ {(0, 0)}) = 0, p 2
100 Notice then we can phrase our original question as follows: find a set X so that dim H 1 ( 3 \ X ) = 8. Take X to be the set of 8 lines parallel to the z-axis, distributed equally along the unit circle. Observe that 3 \ X deformation retracts to the plane minus the 8 intersections of these lines with the plane. It is a routine exercise to show that, p = 0, 1 H p ( 2 \ {(0, 0)}) = 0, p 2 It is then a matter of induction using Mayer-Vietoris to show that H 1 ( 2 \ (X 2 )) = 8. But then we have H 1 ( 3 \ X ) = 8, as desired. Generally, the dimension of H 1 will give the number of 1-dimensional holes in the space and the dimension H 2 gives the number 0-dimensional holes in the space.
101 Notice then we can phrase our original question as follows: find a set X so that dim H 1 ( 3 \ X ) = 8. Take X to be the set of 8 lines parallel to the z-axis, distributed equally along the unit circle. Observe that 3 \ X deformation retracts to the plane minus the 8 intersections of these lines with the plane. It is a routine exercise to show that, p = 0, 1 H p ( 2 \ {(0, 0)}) = 0, p 2 It is then a matter of induction using Mayer-Vietoris to show that H 1 ( 2 \ (X 2 )) = 8. But then we have H 1 ( 3 \ X ) = 8, as desired. Generally, the dimension of H 1 will give the number of 1-dimensional holes in the space and the dimension H 2 gives the number 0-dimensional holes in the space.
Hungry, Hungry Homology
Hungry, Hungry Homology Caleb McWhorter Introduction The way mathematics textbooks and courses are structured, one would think Mathematics is an extraordinary linear topic with perhaps a few paths leading
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationMath 530 Lecture Notes. Xi Chen
Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary
More informationLECTURE 3: RELATIVE SINGULAR HOMOLOGY
LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationTopology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2
Topology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2 Andrew Ma August 25, 214 2.1.4 Proof. Please refer to the attached picture. We have the following chain complex δ 3
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationTree-adjoined spaces and the Hawaiian earring
Tree-adjoined spaces and the Hawaiian earring W. Hojka (TU Wien) Workshop on Fractals and Tilings 2009 July 6-10, 2009, Strobl (Austria) W. Hojka (TU Wien) () Tree-adjoined spaces and the Hawaiian earring
More informationNOTES ON BASIC HOMOLOGICAL ALGEBRA 0 L M N 0
NOTES ON BASIC HOMOLOGICAL ALGEBRA ANDREW BAKER 1. Chain complexes and their homology Let R be a ring and Mod R the category of right R-modules; a very similar discussion can be had for the category of
More informationPatrick Iglesias-Zemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category
More informationA TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor
A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it
More informationThe Universal Coefficient Theorem
The Universal Coefficient Theorem Renzo s math 571 The Universal Coefficient Theorem relates homology and cohomology. It describes the k-th cohomology group with coefficients in a(n abelian) group G in
More informationAn Outline of Homology Theory
An Outline of Homology Theory Stephen A. Mitchell June 1997, revised October 2001 Note: These notes contain few examples and even fewer proofs. They are intended only as an outline, to be supplemented
More informationHOMOLOGY AND COHOMOLOGY. 1. Introduction
HOMOLOGY AND COHOMOLOGY ELLEARD FELIX WEBSTER HEFFERN 1. Introduction We have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationA duality on simplicial complexes
A duality on simplicial complexes Michael Barr 18.03.2002 Dedicated to Hvedri Inassaridze on the occasion of his 70th birthday Abstract We describe a duality theory for finite simplicial complexes that
More informationNOTES ON CHAIN COMPLEXES
NOTES ON CHAIN COMPLEXES ANDEW BAKE These notes are intended as a very basic introduction to (co)chain complexes and their algebra, the intention being to point the beginner at some of the main ideas which
More informationMTH 428/528. Introduction to Topology II. Elements of Algebraic Topology. Bernard Badzioch
MTH 428/528 Introduction to Topology II Elements of Algebraic Topology Bernard Badzioch 2016.12.12 Contents 1. Some Motivation.......................................................... 3 2. Categories
More informationMath Homotopy Theory Hurewicz theorem
Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S
More informationDe Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)
II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the
More informationHomework 3: Relative homology and excision
Homework 3: Relative homology and excision 0. Pre-requisites. The main theorem you ll have to assume is the excision theorem, but only for Problem 6. Recall what this says: Let A V X, where the interior
More informationGeneralized Alexander duality and applications. Osaka Journal of Mathematics. 38(2) P.469-P.485
Title Generalized Alexander duality and applications Author(s) Romer, Tim Citation Osaka Journal of Mathematics. 38(2) P.469-P.485 Issue Date 2001-06 Text Version publisher URL https://doi.org/10.18910/4757
More informationGeometry and Topology, Lecture 4 The fundamental group and covering spaces
1 Geometry and Topology, Lecture 4 The fundamental group and covering spaces Text: Andrew Ranicki (Edinburgh) Pictures: Julia Collins (Edinburgh) 8th November, 2007 The method of algebraic topology 2 Algebraic
More informationS n 1 i D n l S n 1 is the identity map. Associated to this sequence of maps is the sequence of group homomorphisms
ALGEBRAIC TOPOLOGY Contents 1. Informal introduction 1 1.1. What is algebraic topology? 1 1.2. Brower fixed point theorem 2 2. Review of background material 3 2.1. Algebra 3 2.2. Topological spaces 5 2.3.
More informationEXT, TOR AND THE UCT
EXT, TOR AND THE UCT CHRIS KOTTKE Contents 1. Left/right exact functors 1 2. Projective resolutions 2 3. Two useful lemmas 3 4. Ext 6 5. Ext as a covariant derived functor 8 6. Universal Coefficient Theorem
More informationSECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS
SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon
More informationMATH 215B HOMEWORK 5 SOLUTIONS
MATH 25B HOMEWORK 5 SOLUTIONS. ( marks) Show that the quotient map S S S 2 collapsing the subspace S S to a point is not nullhomotopic by showing that it induces an isomorphism on H 2. On the other hand,
More informationAlgebraic Topology exam
Instituto Superior Técnico Departamento de Matemática Algebraic Topology exam June 12th 2017 1. Let X be a square with the edges cyclically identified: X = [0, 1] 2 / with (a) Compute π 1 (X). (x, 0) (1,
More information1 Introduction Category Theory Topological Preliminaries Function Spaces Algebraic Preliminaries...
Contents 1 Introduction 1 1.1 Category Theory........................... 1 1.2 Topological Preliminaries...................... 5 1.3 Function Spaces............................ 9 1.4 Algebraic Preliminaries.......................
More informationMAT 530: Topology&Geometry, I Fall 2005
MAT 530: Topology&Geometry, I Fall 2005 Problem Set 11 Solution to Problem p433, #2 Suppose U,V X are open, X =U V, U, V, and U V are path-connected, x 0 U V, and i 1 π 1 U,x 0 j 1 π 1 U V,x 0 i 2 π 1
More informationThe Fundamental Group and Covering Spaces
Chapter 8 The Fundamental Group and Covering Spaces In the first seven chapters we have dealt with point-set topology. This chapter provides an introduction to algebraic topology. Algebraic topology may
More informationMATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS
Key Problems 1. Compute π 1 of the Mobius strip. Solution (Spencer Gerhardt): MATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS In other words, M = I I/(s, 0) (1 s, 1). Let x 0 = ( 1 2, 0). Now
More informationEILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY
EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define
More information3.2 Modules of Fractions
3.2 Modules of Fractions Let A be a ring, S a multiplicatively closed subset of A, and M an A-module. Define a relation on M S = { (m, s) m M, s S } by, for m,m M, s,s S, 556 (m,s) (m,s ) iff ( t S) t(sm
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationMath 440 Problem Set 2
Math 440 Problem Set 2 Problem 4, p. 52. Let X R 3 be the union of n lines through the origin. Compute π 1 (R 3 X). Solution: R 3 X deformation retracts to S 2 with 2n points removed. Choose one of them.
More informationAn introduction to derived and triangulated categories. Jon Woolf
An introduction to derived and triangulated categories Jon Woolf PSSL, Glasgow, 6 7th May 2006 Abelian categories and complexes Derived categories and functors arise because 1. we want to work with complexes
More informationMath 752 Week s 1 1
Math 752 Week 13 1 Homotopy Groups Definition 1. For n 0 and X a topological space with x 0 X, define π n (X) = {f : (I n, I n ) (X, x 0 )}/ where is the usual homotopy of maps. Then we have the following
More informationLie Algebra Cohomology
Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of R-modules is a family {C n } n Z of R-modules, together with R-modul maps d n : C n C n 1 such that d d
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More information7. Homotopy and the Fundamental Group
7. Homotopy and the Fundamental Group The group G will be called the fundamental group of the manifold V. J. Henri Poincaré, 895 The properties of a topological space that we have developed so far have
More informationQualifying Exam Syllabus and Transcript
Qualifying Exam Syllabus and Transcript Qiaochu Yuan December 6, 2013 Committee: Martin Olsson (chair), David Nadler, Mariusz Wodzicki, Ori Ganor (outside member) Major Topic: Lie Algebras (Algebra) Basic
More informationNOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY
NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY 1. Closed and exact forms Let X be a n-manifold (not necessarily oriented), and let α be a k-form on X. We say that α is closed if dα = 0 and say
More informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationMath 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).
More informationNOTES IN COMMUTATIVE ALGEBRA: PART 2
NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationSECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES
SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)
More informationLecture on Equivariant Cohomology
Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationDirect Limits. Mathematics 683, Fall 2013
Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry
More informationPart II. Algebraic Topology. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0
More informationMATH8808: ALGEBRAIC TOPOLOGY
MATH8808: ALGEBRAIC TOPOLOGY DAWEI CHEN Contents 1. Underlying Geometric Notions 2 1.1. Homotopy 2 1.2. Cell Complexes 3 1.3. Operations on Cell Complexes 3 1.4. Criteria for Homotopy Equivalence 4 1.5.
More informationHomological Methods in Commutative Algebra
Homological Methods in Commutative Algebra Olivier Haution Ludwig-Maximilians-Universität München Sommersemester 2017 1 Contents Chapter 1. Associated primes 3 1. Support of a module 3 2. Associated primes
More informationMATH 547 ALGEBRAIC TOPOLOGY HOMEWORK ASSIGNMENT 4
MATH 547 ALGEBRAIC TOPOLOGY HOMEWORK ASSIGNMENT 4 ROI DOCAMPO ÁLVAREZ Chapter 0 Exercise We think of the torus T as the quotient of X = I I by the equivalence relation generated by the conditions (, s)
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationTHE GHOST DIMENSION OF A RING
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 THE GHOST DIMENSION OF A RING MARK HOVEY AND KEIR LOCKRIDGE (Communicated by Birge Huisgen-Zimmerman)
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationp,q H (X), H (Y ) ), where the index p has the same meaning as the
There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore
More informationHomework 3 MTH 869 Algebraic Topology
Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }
More informationNotes for Math 535 Differential Geometry Spring Francis Bonahon. Department of Mathematics, University of Southern California
Notes for Math 535 Differential Geometry Spring 2016 Francis Bonahon Department of Mathematics, University of Southern California Date of this version: April 27, 2016 c Francis Bonahon 2016 CHAPTER 1 A
More informationarxiv: v1 [math.kt] 18 Dec 2009
EXCISION IN HOCHSCHILD AND CYCLIC HOMOLOGY WITHOUT CONTINUOUS LINEAR SECTIONS arxiv:0912.3729v1 [math.kt] 18 Dec 2009 RALF MEYER Abstract. We prove that continuous Hochschild and cyclic homology satisfy
More informationAlgebraic Topology M3P solutions 2
Algebraic Topology M3P1 015 solutions AC Imperial College London a.corti@imperial.ac.uk 3 rd February 015 A small disclaimer This document is a bit sketchy and it leaves some to be desired in several other
More informationAlgebraic Topology II Notes Week 12
Algebraic Topology II Notes Week 12 1 Cohomology Theory (Continued) 1.1 More Applications of Poincaré Duality Proposition 1.1. Any homotopy equivalence CP 2n f CP 2n preserves orientation (n 1). In other
More informationContinuous Cohomology of Permutation Groups on Profinite Modules
Continuous Cohomology of Permutation Groups on Profinite Modules D M Evans and P R Hewitt Abstract. Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups
More informationSolution: We can cut the 2-simplex in two, perform the identification and then stitch it back up. The best way to see this is with the picture:
Samuel Lee Algebraic Topology Homework #6 May 11, 2016 Problem 1: ( 2.1: #1). What familiar space is the quotient -complex of a 2-simplex [v 0, v 1, v 2 ] obtained by identifying the edges [v 0, v 1 ]
More informationNonabelian Poincare Duality (Lecture 8)
Nonabelian Poincare Duality (Lecture 8) February 19, 2014 Let M be a compact oriented manifold of dimension n. Then Poincare duality asserts the existence of an isomorphism H (M; A) H n (M; A) for any
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More informationSTABLE MODULE THEORY WITH KERNELS
Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationFrom singular chains to Alexander Duality. Jesper M. Møller
From singular chains to Alexander Duality Jesper M. Møller Matematisk Institut, Universitetsparken 5, DK 21 København E-mail address: moller@math.ku.dk URL: http://www.math.ku.dk/~moller Contents Chapter
More informationAlgebraic Topology Homework 4 Solutions
Algebraic Topology Homework 4 Solutions Here are a few solutions to some of the trickier problems... Recall: Let X be a topological space, A X a subspace of X. Suppose f, g : X X are maps restricting to
More informationMath 210B. Profinite group cohomology
Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the
More informationCOHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9
COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We
More informationLecture 7. This set is the set of equivalence classes of the equivalence relation on M S defined by
Lecture 7 1. Modules of fractions Let S A be a multiplicative set, and A M an A-module. In what follows, we will denote by s, t, u elements from S and by m, n elements from M. Similar to the concept of
More informationDerivations and differentials
Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,
More information6 Axiomatic Homology Theory
MATH41071/MATH61071 Algebraic topology 6 Axiomatic Homology Theory Autumn Semester 2016 2017 The basic ideas of homology go back to Poincaré in 1895 when he defined the Betti numbers and torsion numbers
More informationAn Introduction to Spectral Sequences
An Introduction to Spectral Sequences Matt Booth December 4, 2016 This is the second half of a joint talk with Tim Weelinck. Tim introduced the concept of spectral sequences, and did some informal computations,
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More information58 CHAPTER 2. COMPUTATIONAL METHODS
58 CHAPTER 2. COMPUTATIONAL METHODS 23 Hom and Lim We will now develop more properties of the tensor product: its relationship to homomorphisms and to direct limits. The tensor product arose in our study
More informationAPPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY
APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then
More informationHomotopy and homology groups of the n-dimensional Hawaiian earring
F U N D A M E N T A MATHEMATICAE 165 (2000) Homotopy and homology groups of the n-dimensional Hawaiian earring by Katsuya E d a (Tokyo) and Kazuhiro K a w a m u r a (Tsukuba) Abstract. For the n-dimensional
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationFUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents
FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen
More informationThe group C(G, A) contains subgroups of n-cocycles and n-coboundaries defined by. C 1 (G, A) d1
18.785 Number theory I Lecture #23 Fall 2017 11/27/2017 23 Tate cohomology In this lecture we introduce a variant of group cohomology known as Tate cohomology, and we define the Herbrand quotient (a ratio
More informationMath 6510 Homework 10
2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained
More informationMath 225B: Differential Geometry, Final
Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of
More informationExercises for Algebraic Topology
Sheet 1, September 13, 2017 Definition. Let A be an abelian group and let M be a set. The A-linearization of M is the set A[M] = {f : M A f 1 (A \ {0}) is finite}. We view A[M] as an abelian group via
More informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More informationSpectral sequences. 1 Homological spectral sequences. J.T. Lyczak, January 2016
JT Lyczak, January 2016 Spectral sequences A useful tool in homological algebra is the theory of spectral sequences The purpose of this text is to introduce the reader to the subject and proofs are generally
More informationDeligne s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities
Deligne s Mixed Hodge Structure for Projective Varieties with only Normal Crossing Singularities B.F Jones April 13, 2005 Abstract Following the survey article by Griffiths and Schmid, I ll talk about
More informationL E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S
L A U R E N T I U M A X I M U N I V E R S I T Y O F W I S C O N S I N - M A D I S O N L E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S i Contents 1 Basics of Homotopy
More information1 Whitehead s theorem.
1 Whitehead s theorem. Statement: If f : X Y is a map of CW complexes inducing isomorphisms on all homotopy groups, then f is a homotopy equivalence. Moreover, if f is the inclusion of a subcomplex X in
More informationAlgebraic Topology Lecture Notes. Jarah Evslin and Alexander Wijns
Algebraic Topology Lecture Notes Jarah Evslin and Alexander Wijns Abstract We classify finitely generated abelian groups and, using simplicial complex, describe various groups that can be associated to
More information