Algebraic Topology
DRAFT 2013 - U.T.
Contents
0 Introduction 2
1 Chain complexes 2
2 Simplicial homology 4
4 Chain homotopy and homotopy invariance 8 4.1 Chain homotopy ...... 8 4.2 Homotopy invariance ...... 9
5 The Snake Lemma and relative homology 10
6 The Five Lemma and excision 12 6.1 Atool ...... 12 6.2 Excision ...... 13 6.3 Locality ...... 14 6.4 Mayer-Vietoris sequence ...... 15
7 The degree of a map of spheres 15
8 Cellular homology 17
9 Cohomology and the Universal Coefficient Theorem 20 9.1 Cohomology ...... 20 9.2 Universal Coefficient Theorem ...... 21
1 0 Introduction
We study topological spaces by associating algebraic objects which are invariant under homotopy (deformation).
Sets: number of path components π0 Numbers: Euler characteristic χ winding number w γ Groups: fundamental group π1 ( ) Graded abelian groups: Homology H∗ Graded rings: Cohomology H∗ This course will introduce and study homology and cohomology.
Why? • Nonabelian groups are difficult to work with
• π1 only tells us something about the 2-skeleton of a space n • higher dimensional analogues, πn {homotopy classes of based maps from S to the space k of interest}, are very hard to compute. Even πn S are not all know for n k. = Homology, H∗, is a refinement of the Euler characteristic and still relatively easy> to compute.
References Everything contained in this course is covered in • chapters 2 and 3 of ‘Algebraic Topology’ by Allen Hatcher, available online at http://www.math.cornell.edu/∼hatcher/
1 Chain complexes
Definition 1.1. Let C0,C1,... be abelian groups (or vector spaces) and let ∂n Cn Cn−1 be homomorphisms (or linear maps). Assume that ∂n ∂n+1 0 for all n. Then ∶ → ∂3 ∂2 ∂1 ∂0 0 . ○ = = C●, ∂● C2 C1 C0 0 is a chain complex. Furthermore( ) define= ⋯ ÐÐ→ ÐÐ→ ÐÐ→ ÐÐÐ→
• n-chains: Cn . • n-cycles: Zn . ker ∂n
. • n-boundaries:=Bn ( im) ∂n+1
Sometimes we refer to a= chain( complex) as C● for simplicity.
Note 1.2. As ∂n ∂n+1 0, Bn Zn. ○ = ⊆
2 Definition 1.3. The n-th homology of a chain complex C●, ∂● is
. Hn C●, ∂● Zn Bn ker ∂n ( im ∂n)+1 .
C● is called exact at Cn if Hn( 0, i.e.) = if ker ∂n = im( ∂n)+1 . (C● is) called exact if it is exact at Cn for all n 0. = ( ) = ( ) Example 1.4.≥ Let α β 0 A B C 0 be a short exact sequence of abelianÐ→ groups,ÐÐ→ thenÐÐ→ Ð→ ker α im 0 0 α is injective ker β im α A ( ) = ( ) = { } Ô⇒ ker 0 C im β β is surjective ( ) = ( ) = Thus C is isomorphic to the quotient( ) = =B A(.) Note thatÔ⇒ the group B is not determined by A and C. For example, when A Z and C Z 2Z then B could be Z or Z Z 2Z. On the otherhand, if in a short exact sequence of vector spaces~ where A, B and C are finite dimensional and all maps are linear, B is determined= by=A and~ C upto isomorphism: dim⊕ B~ dim A dim C .
Definition 1.5. A chain map φ C●, ∂● C●, ∂● is a collection of homomorphisms( ) = ( ) +φn C(n ) Cn such that the following diagram commutes. ∶ ( ) → ̃ ̃ ∶ → ̃ ∂2 ∂1 ∂0 / C2 / C1 / C0 / 0
⋯ φ2 φ1 φ0 / ∂̃2 / ∂̃1 / ∂̃0 / C2 C1 C0 0 ⋯ ̃ ̃ ̃ That is, such that φn−1 ∂n ∂n φn for all n 0.
Proposition 1.6. φ C○● =C̃● induces○ a homomorphism≥ φ∗ Hn C● Hn C● for all n 0 by φ x φn x where x x Bn Zn Bn Hn C . ∗ ∶ → ̃ ● ∶ ( ) → ̃ ≥ Proof.([We]) show= [ ( that)] the map[ ] is= well-defined:+ ∈ = ( ) Let x x′ , then x x′ b for some b Bn. So φn x φn x′ b φn x′ φn b . We need φn b Bn, then φn x φn x′ . [ ] = [ ] = + ∈ ( ) = ( + ) = ( ) + ( ) ( ) ∈ ̃ [ ( )] = [ ( )] ∂n+1 ∂n Since b B , there is a b′ C such that ∂ b′ b. C / C / C n n+1 n+1 n+1 n n−1 Then, by commutativity, b′ / b _ ∈ ∈ ( ) = φn+1 φn φn−1 ′ φn b φn ∂n+1 b ∂̃n+1 ∂̃n ′ C / C / C ∂n+1 φn+1 b n+1 n n−1 ( ) = ( ) φn b Bm im ∂n 1 . ̃ ̃ ̃ = ̃ ○ + ( ) ∈ ̃ = ̃ Finally, one checks that φn x Zn: as x is a cycle, ∂n φn x φn−1 ∂n x 0. ̃ ̃ We will study three different( ) notions∈ of homology groups○ (H)n =X for○ a topological( ) = space X, which will agree when defined. ( )
3 Simplicial Homology easy to compute and good geometric intuition Singular Homology good for theoretical work Cellular Homology better for computations and applications Through out the course we will use freely the following result from algebra. Theorem 1.7. Any finitely generated abelian group is isomorphic to
r n1 n2 nk Z Z p1 Z Z p2 Z Z pk Z for some r Z≥0, k Z≥0, each⊕ pi is prime ⊕ (possibly ⊕ with ⋯ ⊕pi pj for i j) and each ni Z>0. Moreover, r, k, pi and ni are unique up to reordering. ∈ ∈ = ≠ ∈ Example 1.8. (1) Z 6Z Z 2Z Z 3Z (2) Z 9Z Z 3Z Z 3Z ≅ ⊕ ≇ ⊕ 2 Simplicial homology
Definition 2.1. The standard n-simplex ∆n is
n+1 t0, . . . , tn R ti 1 and ti 0 i .
Example 2.2. ( ) ∈ ∶ Q = ≥ ∀
∆0 e2 = e1 e1 ∆2 ∆1 =
= e0 e0
n k Definition 2.3. Let v0, . . . , vn R + be such that v1 v0, . . . , vk v0 are linearly independent. The n-simplex spanned by v0, . . . , vn is { } ⊆ − − tivi ti 0 i and ti 1 .
Its vertices are v0, . . . , vn . LetQv0, . . . ,∶ vn ≥denote∀ theQn-simplex= spanned by v0, . . . , vn together with the given ordering of its vertices. The ordering induces an orientation of its edges vi, vj from vi to vj. { } [ ] [ ] For any nonempty subset of the vertices, the simplex that they span is a face of v0, . . . , vn . The vertices of this face inherit an ordering. [ ] There is a canonical homeomorphism
n ∆ v0, . . . , vn
t0, . . . , tn t0v0 tnvn. Ð→ [ ] Definition 2.4. A ∆-complex is( obtained) asz→ follows:+ ⋯ +
(i) Start with an indexing set In for each n Z≥0.
∈4 n (ii) For each α In take a copy σα of the standard n-simplex. (iii) Form the disjoint union of all these simplices over all n : σn. ∈ Z≥0 n∈Z≥0 α∈In α n (iv) We require that for each n 1 -dimensional face of each n∈-simplex∐ σα there∐ is an associated n−1 n 1 -simplex σβ for some β In−1. ( − ) n (v) (Form− ) the quotient space by identifying∈ each n 1 -dimensional face of each σα with the n−1 associated simplex σβ using the canonical homeomorphism. These homeomorphisms preserve the ordering of the vertices. ( − ) Example 2.5. The torus admits the structure of a ∆-complex with one 0-simplex three 1-simplices two 2-simplices
Note 2.6. The following is not a well-defined ∆-complex because the vertices of each 2-simplex are not totally ordered.
Remark 2.7. Any simplicial complex is a ∆-complex, but in a ∆-complex an n-simplex may not be uniquely determined by its vertices as in Example 2.5. ∆ Definition 2.8. Let X● be a ∆-complex. The n-th chain group Cn X● of X● is the free abelian group generated by the set of n-simplices X of X . An element of C∆ X is c σn, where n ● n ● α∈In α α each cα Z and only finitely many of the cα’s are non-zero. The boundary( ) homomorphism is ( ) ∑ ∂ C∆ X C∆ X ∈ n n ● n−1 ● i v0, . . . , vn 1 v0,..., vˆi, . . . , vn ∶ ( ) Ð→ i ( ) [ ] z→ Q(− ) [ ] . where v0,..., vˆi, . . . , vn v0, . . . , vi−1, vi+1, . . . , vn . Example[ 2.9. ] = [ ]
v2 v1 v1 1 1 ∂ ∂2 1 ⎛ ⎞ v0 v ⎛ × − ⎞ ⎛ ⎞ ⎛ + 0 ⎞ ⎜ v v ⎟ 1 × ⎜ 0 1 ⎟ = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ − ⎠ ⎝ ⎠ ⎝ × ⎠ Key Lemma 2.10. ∂n−1 ∂n 0 n
Proof. ○ = ∀
i ∂n−1 ∂n v0, . . . , vn ∂n−1 1 v0,..., vˆi, . . . , vn i i j ○ ([ ]) = 1Q(−1 ) [v0 ..., vˆj,..., vˆi, .] . . , vn ji 0. + Q(− ) (− ) [ ]
=
5 ∆ Definition 2.11. For n 0 the n-th simplicial homology group Hn X● is the n-th homology group of the chain complex ≥ ( ) ∂ ∂ C∆ X n+1 C∆ X n C∆ X n+1 ● n ● n−1 ● 1 1 Example 2.12. X● ⋯SÐ→S ( ) ÐÐ→ ( ) ÐÐ→ ( ) Ð→ ⋯
= ∨ / ∆ / ∆ ∂1 / ∆ / C2 X● C1 X● C0 X● 0 a b ⋯ v (0 ) Z ( Z ) Z( ) generated generated by v by a⊕and b
∂1 a v v and ∂1 b v v so
( ) = − ( ) H= 0 −X● ker ∂0 im ∂1 Z 0 Z H X ker ∂ im ∂ 0 1( ●) = ( 1) ( 1) = Z{ }Z= Z Z H X 0 for n 2. n( ●) = ( ) ( ) = ( ⊕ ){ } = ⊕
Example 2.13. X● torus( T) = ≥ v a v = 1 2 / ∆ / ∆ ∂2 / ∆ ∂1 / ∆ / f 2 C3 X● C2 X● C1 X● C0 X 0 b b c ⋯ ( ) ( ) ( ) ( ) g 0 Z ZZ Z Z Z 0 generated generated generated by v 0 1 by f and g by a, b, c v a v ⊕ ⊕ ⊕
∂1 a ∂1 b ∂1 c v v 0 and ∂2 f ∂2 g a b c so
( ) = H0( X) =● (ker) =∂0 − im= ∂1 Z (0) = Z ( ) = + − H X ker ∂ im ∂ im ∂ generated by a and b 1( ●) = ( 1) ( 2) = Z{ }Z= Z 2 Z Z H X ker ∂ im ∂ ker ∂ 0 generated by f g 2( ●) = ( 2) ( 3) = ( ⊕ n ⊕ ) Z( ) ≅ ⊕ H X 0 for n 3. n( ●) = ( ) ( ) = ( ){ } ≅ − Example 2.14.( ) =A simplicial≥ complex with only 0-simplices (vertices) and 1-simplices (edges) is a graph G. If G is connected and finite then its first homology is free of rank equal the number of edges minus the number of vertices plus one.
3 Singular homology
Singular homology is a theoretical tool. It is not practical for computations. Our immediate goals are to show the following:
• continuous maps induce a map on H∗ (this section)
• H∗ is invariant under homotopy (section 4)
6 n Definition 3.1. Let X be a topological space. A continuous map σ ∆ e0, . . . , en X is called a singular n-simplex. The singular n-chains are the elements of the free abelian group generated by the n-simplices: ∶ = [ ] →
. n Cn X . aασα aα Z with only finitely many 0, σα ∆ X α The boundary operator( ) = Qis defined∶ on∈ generators by ≠ ∶ → ¡
∂n Cn X Cn−1 X n σ ∂ σ . 1 iσ ∶ ( ) Ð→ n ( ) [e0,...,eˆi,...,en] i=0 z→ = Q(− ) S Lemma 3.2. ∂n−1 ∂n 0 n
Proof. Heuristically,○ ∂n restricts= ∀ a singular simplex σ to its boundary ∂n e0, . . . , en . The bound- ary has no boundary. So ∂n−1 ∂nσ restricts σ restricted to the empty set and ∂n−1 ∂n e0, . . . , en 0. For a formal proof imitate the proof of Key Lemma in section 2. [ ] ( ) ( [ ]) = . Definition 3.3. Singular homology Hn X Hn C● X .
Remark 3.4. For a simplicial complex X( ●,) the= map( that( )) assigns to an n-simplex α its canon- ical homeomorphism σ ∆n α induces a chain map C∆ X C X which induces an α ● ● ● isomorphism of homology groups as we will see later. ∶ → ( ) → ( ) Example 3.5. X pt. For each n there is only one map cn ∆n X. n = ∂ cn 1 icn ∶ → n [e0,...,eˆi,...,en] i=0 ( ) = Q0(− ) if Sn is odd n 1 ⎧c − if n is even ⎪ = ⎨ ⎪ ⎩⎪ 1 0 1 0 0 C● Z Z Z Z 0 Z n 0 Ô⇒ Hn=pt⋯ Ð→ Ð→ Ð→ Ð→ Ð→ ⎧0 n 0 ⎪ = Ô⇒ ( ) = ⎨ ⎪ ⎩⎪ ≠ Naturality. Let f X Y be a continuous maps. Define
∶ → f♯ Cn X Cn Y σ f σ ∆n X Y. ∶ ( ) Ð→ ( )
Note that ∂n f♯ σ f♯ ∂n σ . Hencez→f♯ induces○ ∶ a→ map→ of chain complexes and hence a homomorphism of homology groups: ○ ( ) = ○ ( ) f∗ Hn X Hn Y for all n. Furthermore, if g Y Z is another∶ continuous( ) Ð→ map( ) then
∶ →g f ♯ g♯ f♯ and g f ∗ g∗ f∗ idX id and idX idH for all n ( ○ )♯ = C○●(X) ( ○ )∗ = ○n where idX X X is( the identity) = map. ( ) =
∶ → 7 Aside 3.6.( Optional) This says that homology is a functor from the category of topological spaces and continuous maps to the category of graded abelian groups and homomorphisms that preserve the grading:
X H∗ X Hn X . n≥0 z→ ( ) = ? ( ) 4 Chain homotopy and homotopy invariance
We will first establish a criterion (chain homotopy) for when two chain maps induce the same homomorphism in homology. Then we will show that a homotopy between two continuous maps induces such a chain homotopy and hence the two maps induce the same homomorphism in homology.
4.1 Chain homotopy
Definition 4.1. Two maps of chain complexes φ●, ψ● C●, ∂● C●, ∂● are chain homotopic if there exist homomorphisms hn Cn Cn 1 such that + ∶ ( ) → ̃ ̃ ∂ h h ∂ φ ψ . n∶ +1 →ñ n−1 n n n h is called a chain homotopy. ● ̃ ○ + ○ = −
Proposition 4.2. If φ● and ψ● are chain homotopic then
φ∗ ψ∗ Hn C● Hn C● . ̃ Proof. Let z Zn ker ∂n. Then = ∶ ( ) Ð→ φ z φ z ∂ h h ∂ z as z Z ∈ = n n n+1 n n−1 n n ∂ h z B im ∂ . ( ) − ( ) = ̃n+1 ○ n + ○n ( ) n+1 ∈ Hence, φ z φ z ψ z ψ z . ∗ n n = ̃∗ ○ ( ) ∈ ̃ = ̃ / ∂n / ([ ]) = [ ( )] = [ (C)]n+1= ([ ]) Cn Cn−1 ss s hn ss sss ss φ−ψ ss ss sshn−1 sy s y ss / s / Cn+1 Cn Cn−1 ∂̃n+1 ̃ ̃ ̃ Main Example 4.3. Let i0, i1 X X 0, 1 be the inclusions x x, 0 and x x, 1 . Then there exists a chain homotopy h● between ∶ → × [ ] ↦ ( ) ↦ ( ) i0 ♯ and i1 ♯ C● X C● X 0, 1 . n n n Construction 4.4. Let ∆ 0 v0, . . . , vn , ∆ 1 w0, . . . , wn and divide ∆ 0, 1 ( ) ( ). ∶ ( ) Ð→ ( × [ ]) into n 1 n 1 -simplices of the form si . v0, . . . , vi, wi, . . . , wn where i ranges from 0 to n. Let Γ . n+1 1 is C ∆n × {0,}1= [, then ] × { } = [ ] × [ ] n i=0 i n+1 + ( + ) = [ ] ∂Γ ”∂∆n 0, 1 ∆n ∂ 0, 1 ” = ∑ (− ) ∈ n ( × [ ]) n+1 i i j = × [1 ] O1 v0×,...,[ vˆj,] . . . , vi, wi, . . . , wn i=0 j=0 = Q Qn+(1−n+)1 (− ) [ ] i j+1 1 1 v0, . . . , vi, wi,..., wˆj, . . . , wn i=0 j=i + Q Q(− ) (− ) [ ] 8 n Terms i j cancel except for w0, . . . , wn v0, . . . , vn . Terms i j are ”Γ ∂∆ 0, 1 ”. w w = 0 [ 1 ] − [ ] ≠ ( × [ ]) Γ1 v0, w0, w1 v0, v1, w1
= [ ] − [ ] ÿ ∂Γ1 w0, w1 v0, w1 v0, w0 v1, w1 v0, w1 v0, v1 v0 ⟲ v1 = [ ] − [ ] + [ ] − [ ] + [ ] − [ ] n n Let σ ∆ X Cn X and σ 1 ∆ 0, 1 X 0, 1 , x, t σ x , t . Define
( ∶ → ) ∈ ( ) h×Cn∶ X × [ C]n→+1 X× [ 0, 1] ( ) ↦ ( ( ) ) σ σ 1 Γn ∶ ( ) Ð→ ( ♯ × [ ]) Then z→ ( × ) ( )
∂h σ ∂ σ 1 ♯ Γn σ 1 ∂ Γn ( ) = ( × ♯) ( ) i1 σ h∂ σ i0 σ = ( ×♯ ) ( ) ♯ and so ∂●h● h●∂● i1 ♯ i0 ♯. = ( ) ( ) − ( ) − ( ) ( ) + = ( ) − ( ) 4.2 Homotopy invariance
Definition 4.5. Two continuous maps f0, f1 X Y are homotopic if there exists a continuous map ∶ → F x, 0 f0 x F X 0, 1 Y such that F x, 1 f1 x . ( ) = ( ) ∶ × [ ] Ð→ F is called a homotopy and we write f0 f1. If A is a subset( of) =X (and) f0 A f1 A and the homotopy F satisfies F x, t f0 x f1 x for all x A and all t 0, 1 , then f0 and f1 are said to be homotopic relative to A. ∼ S = S ( ) = ( ) = ( ) ∈ ∈ [ ] Note 4.6. Homotopy is an equivalence relation. Definition 4.7. Two spaces X and Y are homotopy equivalent if there exist maps
f X Y g f idX such that g Y X f g idY . ∶ Ð→ ○ ∼ X is called contractible if it is∶ homotopyÐ→ equivalent to a point.○ ∼ n n n Example 4.8. R is contractible. f R , g 0 R . Then f g id∗ and n n F R 0, 1 ∶R → ∗ ∶ ∗ → F∈ x, 0 0 g ○f =x with x, t tx F x, 1 x id n x ∶ × [ ] Ð→ ( ) = = ○R ( )
Theorem 4.9 (Homotopy invariance)( ) z→ . If f0 f1, then (f0 ∗) = f1= ∗ H(n X) Hn Y . ∼ ( ) = ( ) ∶ ( ) → ( )
9 Geometric intuition Y
X f1 z cycle F
f0
f1 z f0 z is a boundary.
( ) − ( ) Proof. Let F be a homotopy between f0 and f1. Then f0 F i0, f1 F i1 and
f1 ♯ f0 ♯ F♯ i1♯ i0♯ by naturality= ○ = ○ F ∂ h h ∂ by Main Example 4.3 ( ) − ( ) = ♯( ● −● )● ● ∂ F h F h ∂. = ●( ♯ ● + ♯ )● So F h is a chain homotopy between f and f , and so f f ♯ ● = ( 0♯) + ( 1♯ ) 0∗ 1∗ Corollary( ) 4.10. If X and Y are homotopy equivalent then Hn=X Hn Y for n 0. Proof. By definition, there exists ( ) ≅ ( ) ≥
f X Y g f idX such that g Y X f g idY . ∶ Ð→ ○ ∼ So g f g f id id and f g id . Hence f H X H Y is an ∗ ∗ ∗ X ∗ ∶ Ð→Hn(X) ∗ ∗ Hn(Y )○ ∼ ∗ n n isomorphism. ○ = ( ○ ) = ( ) = ○ = ∶ ( ) → ( ) Z for n 0 Corollary 4.11. If X is contractible then Hn X Hn ⎧0 for n 0. ⎪ = ( ) ≅ (∗) ≅ ⎨ ⎪ ⎩⎪ > 5 The Snake Lemma and relative homology
This is a computational tool: ‘Divide and conquer, Part I’ Definition 5.1. The sequence of chain complexes
i j 0 A● B● C● 0 ( ) is short exact if for all n 0 Ð→ ÐÐ→ ÐÐ→ Ð→ ∗
i j ≥ 0 An Bn Cn 0 is short exact. Ð→ ÐÐ→ ÐÐ→ Ð→
10 Theorem 5.2 (The Snake Lemma). Given a short exact sequence of chain complexes ( ) there exist connecting homomorphisms ∗
δ Hn C● Hn−1 A● such that the following sequence is long∶ exact:( ) Ð→ ( )
δ i∗ j∗ δ Hn A● Hn B● Hn C● Hn−1 A● H0 B● H0 C● 0
Proof.⋯ ÐÐ→ Definition( ) ÐÐ→ of δ: ( ) ÐÐ→ ( ) ÐÐ→ ( ) Ð→ ⋯ Ð→ ( ) Ð→ ( ) Ð→
Let c Zn C● . Since j is a surjection there exists b Bn such that j b c. Then ∂b Bn−1 and j ∂b ∂j b ∂c 0. By exactness there exists a An−1 such that i a ∂b. Define δ c . ∈ a .( ) ∈ ( ) = ∈ ( ) = ( ) = = ∈ ( ) = Well-defined:[ ] = [ ] (1) a is a cycle: i ∂a ∂ i a ∂ ∂b 0, and so ∂a 0 as i is injective.
′ (2) Independence( of choice) = ( ( of))c:= assume( ) =c c = [ ] = [ ] c′ c ∂c′′ and c′′ j b′′ b′′ / b b′ i e ∂b′′ b b′ i e and as ∂∂b′′ 0 Ô⇒ − = = ( ) − + ( ) ∂i e i ∂e ∂b ∂b′ ia ia′ Ô⇒ = − + ( ) = ′′ / ′ ′ c c c a a Hn 1 A . Ô⇒ ( )) = ( ) −= ●− = − − Ô⇒ [ ] = [ ] ∈ ( ) Exactness at Hn C● : im j∗ ker δ: let c im j∗, then there exists b such that j∗ b c . In particular ∂b 0 so if i a ∂b then a 0 so δ c 0. ( ) ⊆ [ ] ∈ [ ] = [ ] im j∗ ker δ: let δ c a 0 = ( ) = = [ ] = a′ ∂a′ a ⊇ [a′] = [ ] /=a ∂ia′ i∂a′ ia ∂b where jb c Ô⇒ ∃ ∶ = put b′ b ia′ b / ∂b Ô⇒ = = = = note jb′ jb jia′ jb c = − ∂b′ ∂b ∂ia′ ∂b ∂b 0 = − = = c / 0 c jb′ j b′ . Ô⇒ = − ∗ = − = Exactness at Hn A● and Hn B● follow in a similar way. Ô⇒ [ ] = [ ] = [ ] Note 5.3. If A● (B●)and C● (B●)A● with ∂ b An ∂b An−1 then δ b An ∂b . Let X,A be a⊂pair of spaces= , i.e. a space (X +and) a= subspace+ A. Then[ the+ inclusion] = [ ] A X induces an inclusion of chain complexes C● A C● X . Define the relative homology of X,A as ( ) ↪ .( ) ↪ ( ) ( ) Hn X,A Hn C● X C● A .
. Note 5.4. A cycle in C● X,A C( ● X) =C● A is( a) chain ( x) Cn X with ∂x Cn−1 A and δ x ∂x . ( ) = ( ) ( ) ∈ ( ) ∈ ( ) [ ] = [ ] A
11 It will be convenient to introduce the reduced homology of a space, which is defined to be . Hn X . ker Hn X Hn pt . Note that Hn X Hn X for n 0. Furthermore, a continuous map f X Y defines a map of reduced homology groups. ̃ ( ) = ( ( ) → ( )) ̃ ( ) = ( ) > Corollary 5.5. There∶ → is a long exact sequence in reduced homology δ H1 A H1 X H1 X,A H0 A H0 X H0 X,A 0.
Proof.⋯ByÐ→ thẽ Snake( ) Ð→ Lemmã ( ) Ð→ ̃ ( ) ÐÐ→ ̃ ( ) Ð→ ̃ ( ) Ð→ ( ) Ð→
δ H1 X,A H0 A H0 X H0 X,A 0
≅ is exact. It is an⋯ exerciseÐ→ ( to show) ÐÐ→ that(H)0 Ð→A (H0)XÐ→, and( that) Ð→A X maps Z Z in H0 A H0 A Z and H0 X H0 X Z. So the sequence in reduced homology is exact. ̃ ( ) → ̃ ( ) ↪ Ð→ ( ) = ̃ ( ) ⊕ ( ) = ̃ ( ) ⊕ k k 1 k Example 5.6. X,A D ,S − . Note D and so Hn X 0 for all n 0 so
k k−1 k̃−1 ( ) = ( H)n D ,S ≃ ∗Hn−1 S ( ) = ≥ for all n 0. ̃ ( ) ≅ ̃ ( )
Let X,A≥ and Y,B be two pairs of spaces. A map of pairs of spaces is a continuous map f X Y such that f A B. ( ) ( ) Proposition∶ → 5.7. A map( ) ⊂ of pairs of spaces induces a map of long exact sequences. Proof. This is left as an exercise.
6 The Five Lemma and excision
Excision distinguishes homology from homotopy and makes the former more accessible to com- putations.
6.1 A tool
Five Lemma 6.1. In the diagram below, if the rows are exact then γ is an isomorphism.
A / B / C / D / E
α ≅ β ≅ γ δ ≅ ε ≅ A′ / B′ / C′ / D′ / E′
Proof. Diagram chasing. (1) β, δ surjective, ε injective γ surjective: Let c′ C′, ∂c′ d′ δd.
∂d′ ∂∂c′ 0 ∂ δdÔ⇒ ε ∂d 0 ∂∈c′ γc = 0 = b′ ∂b′ c′ γc ∂d 0 b ∂βb c′ γc = = Ô⇒ ( ) = ( ) = ( − ) = Ô⇒ ∃ ∶ = − c ∂c d. γ ∂b c′ γc Ô⇒ = Ô⇒ ∃ ∶ = − c′ γ ∂b c . Ô⇒ ∃ ∶ = Ô⇒ ( ) = − Ô⇒ = ( + ) 12 (2) β, δ injective, α surjective γ injective: similarly.
β Ô⇒/ α / / / Example 6.2. Assume 0 G H Ui K 0 is split exact with β γ id. Ap- _γ i plying the Five Lemma to ○ =
/ / / / 0 G HO K 0 α⋅γ 0 / G / G K / K / 0 gives H G K. ⊕
≃ ⊕ 6.2 Excision
Definition 6.3. Let A X be a subspace. A map r X X is a retract onto A if r X A 2 and r A idA (so r r). If r is a homotopy equivalence (rel A) then r is a deformation retract. ⊂ ∶ → ( ) = S = = A pair X,A is called good if A is a non-empty, closed subset of X such X that it is a deformation retract of some neighbourhood V of A in X. A ( ) V Define the quotient X A as the quotient under the equivalence relation x y iff (x A and y A) or (x y). ∼ ∈ 1 1 1 Example∈ 6.4.= X S S A S V X A with A Example 6.5. X = Dk∨ A ⊇ Sk=−1 V = X A Sk with≃ Sk−1 [a point.] =
Theorem 6.6 (Excision= ⊇ and= relative homology= as the homology ≃ of a space)[ . Let] =A V X be subspaces of X such that A¯ V˚. Then the map of pairs X A, V A X,V induces an ≅ ⊂ ⊂ isomorphism Hn X A, V A Hn X,V for all n 0. ⊂ ( ∖ ∖ ) → ( ) Corollary 6.7. (For∖ a good∖ pair) Ð→X,A( , the) quotient map≥ X,A X A, induces an iso- morphism H X,A ≅ H X A, H X A for all n 0. n n ( ) n ( ) → ( ∗) Proof of corollary.( As) Ð→ X,A( is good∗) = ̃ there( exists) V A≥ such that A V is a homotopy equivalence. Consider the following commutative diagram. ( ) ⊃ ↪ ≅ ≅ Hn X,A / Hn X,V o Hn X A, V A
( q ) ( q ) ( ∖ q ∖ ) ≅ ≅ Hn X A, A A / Hn X A, V A o Hn X A A A, V A A A
Using Proposition( 5.7 and ) the Five Lemma( one shows) that( the left∖ horizontal arrows∖ ) are isomor- phisms since A ≃ V and A A ≃ V A are homotopy equivalences. The right horizontal arrows are isomorphisms by excision and the right vertical arrow is the identity as X A A A X A. The diagram commutesÐ→ and thereforeÐ→ also the other two vertical arrows are isomorphisms. ∖ = ∖ k k 1 k k 1 k k 1 k Example 6.8. D ,S − is good Hn D ,S − Hn D S − Hn S .
( ) Ô⇒ ( ) ≅ ̃ ( ) ≅ ̃ ( )
13 k Z n k Claim 6.9. Hn S ⎧0 n k ⎪ = ̃ ( ) = ⎨ ⎪ ≠ k k−1 k−1 Proof. Recall from section⎩ 5, Hn D ,S Hn−1 S . So
̃ ( ) ≅ ̃ ( ) n k k k k−1 k−1 1 0 Z Hn S Hn D ,S Hn 1 S Hn k 1 S Hn k S − − + − 0 n k. ⎪⎧ ̃ ̃ ̃ ̃ ̃ ⎪ = ( ) ≅ ( ) ≅ (n ) ≅ ⋯k≅ k 1 ( ) ≅ ( ) = ⎨ Remark 6.10. The generator of Hn S Hn D ,S − Z can be represented⎪ by the≠ n-cycle n n n n ⎩ n ∆1 ∆2 where S ∆1 ∆2 ∂∆1 ∂∆2 or by the relative cycle ∆1 D . By excision n n n n ( ) ≅ ( ) ≅ Hn ∆1 , ∂∆1 Hn S , ∆2 . − = ( ∐ )( = ) ≃ ( ) ≅ ( ) 6.3 Locality
Let U be a collection of open subsets of X with X U˚ . Define CU X C X i i∈I i∈I i ● ● n to be the subcomplex generated by the n-simplices σ with σ ∆ Ui for some i I. U = { } = ⋃ ( ) ⊆ ( ) Theorem 6.11 (Locality). H CU X ≅ H C X H X for all n 0. n ● n ● ( n ) ⊂ ∈
Sketch. ( ( )) ÐÐ→ ( ( )) = ( ) ≥ (1) Given an n-simplex σ ∆n X, we can subdivide ∆n into smaller n-simplices such that the image is contained in some Ui by applying repeatedly the barycentric subdivision: ∶ → Barycentric subdivision: S ∆1 S ∆2
( ) = ( ) = (2) S defines a chain map C● X C● X , σ σ S with ∂S S∂. (3) S is chain homotopic to the( ) identity:→ ( ) ↦ ○ =
T Cn X Cn+1 X n proj n σ ∶ ( σ) Ð→ T σ ∆( ) I ∆ X ∆1 I ∆2 I and ∂T T ∂ S id z→ ∶ × ÐÐ→ ÐÐ→ × = × = Proof of Theorem 6.6+ (Excision).= − Let A V X, put U X A and U, V . Then U
U Cn X⊆ A⊆ Cn V A= ∖Cn U CUn=V{ U } Cn X Cn V . A V There( is∖ a map) of( long∖ exact) = sequences( ) ( ∩ ) ≅ ( ) ( )
H V / H CU / H CU C V / H V / H CU X n n ● n ● ● n−1 n−1 ●
( ) ≅ ( ) ( ) ≅ ( ) / / / / Hn V Hn X Hn C● X C● V Hn−1 V Hn−1 X
( ) ( ) Hn( X,V) ( ) ( ) ( )
( )
14 So by the Five Lemma:
H X A, V A H CU C V H X,V . n n ● ● n ( ∖ ∖ ) ≅ ( ) ≅ ( ) 6.4 Mayer-Vietoris sequence
B ‘Divide and conquer: Part II’ Let X be a topological space and A, B X with A˚ B˚ X. A ⊂ ∪ ⊇ Theorem 6.12. With A, B and X as above there is a long exact sequence
Hn A B Hn A Hn B Hn X Hn−1 A B H0 A B H0 A H0 B H0 X 0. ⋯ Ð→ ( ∩ ) Ð→ ( ) ⊕ ( ) Ð→ ( ) Ð→ ( ∩ ) Ð→ ⋯ Similarly, there is a long exact sequence⋯ Ð→ in( reduced∩ ) Ð→ homology.( ) ⊕ ( ) Ð→ ( ) Ð→
Proof. Applying locality (Theorem 6.11) to A, B ,
CU X C X ● U = { ● } induces an isomorphism in homology. Consider( ) Ð→ the short( ) exact sequence
0 / C A B / C A C B / CU X / 0 ● ● ● ● σ / σ, σ ( ∩ ) ( ) ⊕ ( ) ( ) σ, µ / σ µ ( − ) and apply the Snake Lemma. ( ) + Note 6.13. δ σ µ ∂σ ∂µ . Definition 6.14. The cone on X is the quotient space CX . [ + ] = [ ] = [− ] X 0, 1 where x, s y, t if and only if x, s y, t or s t 1. = × [ ] ∼ ≃ ∗ ( ) ∼ ( ) ( ) = ( ) . The =suspension= on X is the quotient space ΣX . X 0, 1 where x, s y, t if and only if x, s y, t or s t 0 or s t 1. = × [ ] ∼ Example( ) ∼ ( 6.15.) Let A (X ) =0,(2 3) =ΣX= and let= B= X 1 3, 1 ΣX, then A B CX and A B X 1 2, 2 3 X. Apply Mayer-Vietoris to get = × [ ~ ]∼ ⊂ = × [ ~ ] ∼ ⊂
≃ ≃ ≃ ∗ ∩ = × [ ~Hn~Σ]X≃ Hn−1 X . ( ) ≅ ̃ ( ) 7 The degree of a map of spheres
n n n Let f S S be a continuous map. Recall Hn S Z for n 0.
Definition∶ → 7.1. f∗ in dimension n is a homomorphism̃ ( ) ≃ Z Z,≥ i.e. f∗ in dimension n is multi- plication by an integer deg f , the degree of f. → ( ) 15 n n Note 7.2. (i) deg idS 1 as idS ∗ idHn (ii) deg f g deg f deg g as f g ∗ f∗ g∗ ( ) = ( ) = (iii) f g deg f deg g as f∗ g∗ ( ○ ) = ( ) ⋅ ( ) ( ○ ) = ○ (iv) f deg f 0 as f∗ 0. ≃ Ô⇒ ( ) = ( ) = Example 7.3. ≃ ∗ Ô⇒ ( ) = = n n 1 (1) f is induced by the reflection in R i 0 R + , then 1 0 deg f 1: n 2 Hn S ∆1 ∆2 , f interchanges× ∆{ 1} and⊂ ∆2. So ( ) = − f∗ ∆1 ∆2 ∆2 ∆1 ∆1 ∆2 . ( ) = ⟨ − ⟩ n n n+1 (2) f is( the− antipodal) = ( map− S) = −(S ,−x ) x, then deg f 1 : −1 0 ⋯ 0 1 0 ⋯ 0 1 ⋯ 0 0 f is induced by I n+1 n+1, I 0 1 0 0 −1 0 ⋮ ⋱ ⋮ . R →R ↦ − ⋮ ⋱ ⋮ (⋮) = ⋱(− ⋮ ) 0 1 0 0 0 ⋯ 1 0 0 ⋯ 1 0 ⋯ 0 −1 Each of the n 1− matrices∶ → on the right− = hand side is a reflection⋯ homotopic to the one in (1) hence, using (ii) and (iii), deg f 1 n+1. + n n 1 Application 7.4. A continuous( function) = (− )v S R + defines a tangent vector field on Sn if v x x for all x Sn. Sn has a continuous nowhere zero tangent vector field if∶ and→ only if n is odd. ( ) ⊥ ∈ n n 1 Proof. Suppose v S R + is continuous with v x x for all x. Suppose further v x 0 for all x. Consider F Sn 0, 1 Sn given by F x, t cos πt x sin πt v(x) . ∶ → ( ) ⊥ Sv(x)S ( ) ≠ Note F x, 0 x and∶ F x,× 1[ ] →x. Hence F defines( ) a= homotopy( ) + ( ) v(x) from the identity map to the antipodal map. Hence they must have Sv(x)S the same( degree) = (by (iii))( and) = 1 − 1 n+1. So n must be odd.
Conversely, if n is odd, define = (− ) x x n n 1 v S R + − x1, . . . , x2k x2, x1,..., x2k, x2k 1 . ∶ Ð→ − 2 Corollary 7.5( (Hairy ball) z→ theorem)(− . You− cannot comb) a hairy ball (S ). Definition 7.6. Let f x y and U x, V y be open neighbourhoods of x and y respectively such that ( ) = f ∋U, U ∋x V,V y ( ) i.e. the only point in U mapping to∶y( is x.∖ ) Ð→ ( ∖ ) ∗ From the relative long exact sequence and excision we have
n n n (fSx)∗ n Z Hn S Hn S ,S x Hn U, U x Hn V,V y Hn S Z. ̃ ̃ Then f x≅ ∗ is( multiplication) ≅ ( by∖ an) integer≅ ( deg ∖f x),ÐÐÐ→ the local degree( ∖of)f≅at x(. ) ≅ ( S ) ( S )
16 1 Proposition 7.7. Assume f − y x1, . . . , xk and for each i 1, . . . , k there exist disjoint open sets Ui xi each satisfying condition ( ). Then ( ) = { } = ∋ k ∗ x2 deg f deg f xi . x1 y i=1 ( ) = Q ( S ) If f −1 y then deg f 0.
Proof.( The) = ∅ proposition( follows) = from the following commutative diagram:
f n ∗ n Hn S / Hn S
q ̃ ( ) ̃ ( ≅ ) n n f∗ Hn S ,S x1, . . . , xk / Hn V,V y j4 jjjj ≅ jjj ( ∖ { jj})j f ( ∖ ) jjjj ∑i( Sxi )∗ k H U ,U x i=1 n i i i 2 2 n Example 7.8. Let f S > C ( ∖ S) be define by f z z . Let y 1, then f −1 y e2πki~n k 1, . . . n . Since f is orientation preserving and is a homeomorphism near ∶ = ∪ {∞} → ( ) = = each root of unity, deg f e2πki~n 1. Hence deg f n. ( ) = { ∶ = } • For p an arbitrary( polynomialS ) = of degree n,( p) =f (via a homotopy that moves all zeros to 0). Hence deg p n. ≃ • deg f 0 n. ( ) = ( S ) = 8 Cellular homology
Definition 8.1 (CW-complex). Inductive construction: (1) Start with X0, a disjoint set (X−1 ).
n n n−1 n−1 (2) Let Dα be an n-disk and φα ∂Dα= ∅S X be a continuous map. For a collection n n n−1 n . n−1 n n Dα, φα construct X from X by X . X α Dα x φα x for all x ∂Dα. ∶ ≃ → n n (3) X{ n X} is given the weak topology (A =X is open∐ if and∼ only( ) if A X is∈ open for all n). = ⋃ ⊂ ∩ ˚n n n Dα eα is called an n-cell. Each cell has a characteristic map eα X.
Example= 8.2. ↪ X0 pt X1 = X2 D2 φ S1 S1 of degree 3 = φ = ∶ →
17 n 1 Example 8.3. Real projective space - the space of lines in R + . X0 pt X1 P 1 = R 2 1 2 2 1 1 n n X X D RP φ S S of degree 2 RP S v v = φ= = = ∶ → = ∼ − n n 1 n n 1 n 1 X X − D φ S − RP − , x x ⋮ φ
= ∶ → n ↦1 {± } Example 8.4. Complex projective space - the space of complex lines in C + . 0 0 X pt CP X1 X0 = = 2 0 2 1 1 X CP D CP φ S pt = φ n 2n 1 1 3 2 CP S + v λv, λ S X = X = ∶ → 4 1 4 2 3 3 1 2 X CP D CP φ S S S S = ∼ ∈ = φ = = ∶ → ≃ X2k+1 X2k ⋮
Let X be a cell complex with m skeleton Xm,=m 0. Then Xm,Xm−1 is good and the quotient space Xm Xm−1 Sm is a wedge of m-spheres, one for each m-cell. ≥ ( ) CW . m m−1 Definition 8.5. Define C≃m⋁ X . Hm X ,X free abelian group generated by the m-cells. Define the boundary map ( ) = ( ) = δ q d CCW X H Xm,Xm−1 H X H X ,X CCW X . m m m−1 m−1 m−1 m−1 m−2 m−1
∶ ( ) = ( ) Ð→ ( ) Ð→ ( m ) =m ( )m−1 The boundary homomorphism δ is induced by the attaching maps, δ Dα ∂Dα φα S . m m−1 Therefore d also has the following description. For an m-cell eα and an m 1 cell eβ define [ ] = [ ≡ ( )]
. m 1 m 1 m 1 m 2 m 1 − m 1 dαβ . deg S − / X − / X − X − S − / S − φα q ∂Dm Dm−1 ∂ = α ≃ ⋁ β m . m−1 and d eα β dαβeβ . This is a finite sum as the image of a compact set is compact.
Lemma( 8.6.) = ∑d d 0 Proof. Consider○ the= following three long exact sequences in relative homology.
δ m+1 m / m / m+1 / Hm+1 X ,X Hm X Hm X ( ) jj jjjjj jjjjj ( jjj)jj ( ) ( ) ∗∗ jj δ / H Xm / H Xm,Xm−1 / H Xm−1 m q∗ m m−1 jj jjjjj ( ) jjjjj ( jj)jj ( ) / H Xm−1 / H Xm−1,Xm−2 m−1 q∗ m−1 ( ) ( ) 18 d d q∗δ q∗δ q∗ δq∗ δ 0 as δq∗ 0 by the exactness of the middle row. Definition○ = ( )( 8.7 )(Cellular= ( ) homology)= . =
HCW X . H CCW X , d n n ● n Remark 8.8. A ∆-complex X● is naturally( ) a= cell-complex ( ) with X the union of all k simplices, k n. Furthermore, with this identification there is an isomorphism of chain complexes
CCW X , d C∆ X , ∂ . ≤ ● ● ● ● m m m m 1 Example 8.9. S φ D , φ ∂D ( S) − ≅ ( )
m 0 m 2= ∗ ∪ 0 Z ∶ = 0 → Z∗ 0 d=0 0 m ≥ 1 0 Ð→ 0 Ð→ ⋯ Ð→Z Ð→Z Ð→0 φ S , deg φ 0 = Ð→ Ð→ ⋯ Ð→ ÐÐ→ Ð→ ∶ → ∗ = m CW n Z n 0, m so Hn S Cn S ⎧0 n 0, m. ⎪ = ( ) = ( ) = ⎨ m In particular we deduce that⎪ the≠ cell complex above for S has the minimal number of cells as we must have at least as many⎩ cells as generators for the homology.
CW Theorem 8.10. For any cell complex X, Hm X Hm X , and hence also for any ∆- ∆ complex, Hm X● Hm X● . ( ) = ( ) m 1 m m 1 m Proof. We note( that) = Hk( X) + ,X is zero for k m 1 as X + X is a wedge of m 1- spheres. Using induction and the l.e.s. for triples of space Xm+l,Xm+1,Xm , one shows that m 1 m Hk X Hk X + for all( k m, and) Hk X 0≠ for all+ k m. ~ + ( ) m+1 ( H)m= X ( Hm )X ≤ (from relative( l.e.s.)) = > δ m m+1 m m ( ) ≅ Hm(X )im Hm+1 X ,X Hm X (from first row in ( ))
m m m−1 m−1 q Hm X ≅ H( m X) ,X (is an injection) Ð→ as H( m X) 0. So im δm+1 ∗∗im dm+1 m CW (as d q δ). Now ker dm ker δm im q∗ Hm X so Hm X ker dm im dm+1 ∶ m( ) Ð→ ( ) ( ) = ≅ Hm X im δm 1 Hm X . = ○ + = ≅ ≅ ( ) ( ) = ≅ Example( ) 8.11. Real≅ projective( ) plane (cf. Example 8.3). n n RP S x y iff x y space of lines in n+1. = ∼ ∼ R = ± P 0 S0 R = P 1 S1 S1 R = ∼ = ∗ P 2 R = ∼ ≅ 3 t RP = SO 3 A M3×3 R det A 1, AA I . n RP has a cell structure≅ with( one) = { cell∈ in each( ) ∶ dimension= m = 0}, . . . , n and m-skeleton m m−1 m m m−1 m−1 m−1 RP RP φ D where φ ∂D S RP S is the natural quotient map. = φ q = ⋃ m−1 ∶ m−=1 →m−1 = m−1 ∼m−2 m−1 d dm deg S S RP RP RP S
= = ÐÐ→ ∼ = ÐÐ→ ≅ 19 q φ , q φ are homeomorphisms onto their images which are related to each other via the ∆˚+ ∆˚− antipodal map. ○ S ○ S d dm deg q φ deg id deg antipodal map 1 1 m Ô⇒ = = ( ○ ) = ( ) + ( ) 0 2 2 0 0 CW n 0 = Z Z+ (− ) Z Z 0 m odd C RP 0 2 2 0 0 ● ⎧0 0 0 m even ⎪ Ð→ ÐÐ→Z ÐÐ→ ⋯ ÐÐ→Z ÐÐ→Z ÐÐ→ Ô⇒ ( ) = ⎨ Z⎪ m 0 ⎩⎪ Ð→ ÐÐ→ ÐÐ→ ⋯ ÐÐ→ ÐÐ→ ÐÐ→ n ⎧Z 2 m odd and m n Hm RP ⎪ = ⎪Z m n odd ⎪ < Ô⇒ ( ) = ⎨0 otherwise. ⎪ = ⎪ ⎩⎪ 9 Cohomology and the Universal Coefficient Theorem
9.1 Cohomology
Definition 9.1. Let C●, ∂● be a chain complex of free Z-modules (or F-vector spaces). Define n . • n-cochains: C (. Hom) Cn, Z , group of group homomorphisms (or the dual vector space) n n n+1 • coboundary map:= ∂ C( C) , φ φ ∂ n n • n-cocycle: Z ker ∂∶ → ↦ ○ n n−1 • n-coboundary:=B ( im) ∂ n . ● ● n n • n-th cohomology group:= H C, ∂ . Hn C , ∂ Z B . n n n m We note that Hom Z , Z Z , and( if) a= matrix( A represents) = ~ an element in Hom Z , Z then the dual map is represented by its transpose. ( ) ≃ ( 3 Example 9.2. The cellular chain complex for RP is 0 2 0 0 Z Z Z Z 0. As Hom , the dual complex is Z Z Z Ð→ ÐÐ→ ÐÐ→ ÐÐ→ Ð→ 0 2 0 ( ) ≅ 0 Z Z Z Z 0. Hence ←Ð ←ÐÐ ←ÐÐ ←ÐÐ ←Ð Z n 0 ⎧0 n 1 n 3 ⎪ = H RP ⎪Z 2 n 2 ⎪ = ⎪Z n 3 ( ) = ⎨ ~ = ⎪0 n 3. ⎪ = ⎪ ⎪ Induced Homomorphism. f X Y induces⎩ >
f Cn X Cn Y and hence ♯∶ Ð→ f ♯ Cn Y Cn X , φ φ f ∶ ( ) Ð→ ( ) ♯ f ∗ Hn Y Hn X ∶ ( ) Ð→ ( ) z→ ○ ∶ ( ) Ð→ ( ) 20 ● ● ● ● Lemma 9.3. If f●, g● C●, ∂● C●, ∂● are chain homotopic then so are f , g C , ∂ C●, ∂● and f ● g● Hn C , ∂ Hn C , ∂ . ∶ ( ● )● → ̃ ̃ ● ● ∶ ̃ ̃ → ( ) = ∶ ̃ ̃ → ● ( ● ●) ● ● ● Proof. If f● g● ∂●h● h●∂● then f g h ∂ ∂ h . − = ̃ + − = ̃ + Homotopy Invariance. If f g X Y then f ∗ g∗ Hn Y Hn X .
i● j● Lemma 9.4. If 0 A● C● ≃ B∶● →0 is an exact= sequence∶ ( of) → free chain( ) complexes then so i● j● is 0 A● C● B● 0. → Ð→ Ð→ →
Proof.← Consider←Ð ←Ð 0 A←n Cn Bn 0. As Bn is free there is a splitting s Bn Cn and i⊕s Cn An Bn. Hence → → → → ∶ →
ÐÐ→ ⊕ n n n ≅ C Hom Cn, Z Hom An, Z Hom Bn, Z A B φ φ i , φ s = ( ) ≅ ( ) ⊕ ( ) ≅ ⊕ and ker i∗ im j∗. ↦ (( ○ ) ( ○ ))
= i∗ Excision. If Z¯ A˚ X then Hn X,A Hn X Z,A Z .
⊂ ⊂ ( ) Ð→ ( ∖ ∖ ) ≅ Long Exact Relative Cohomology Sequence. For any X,A there is a long exact se- quence δ1 ( ) H1 X,A H0 A H0 X H0 X,A 0.
⋯ ←Ð ( ) ←ÐÐ ( ) ←Ð ( ) ←Ð ( ) ←Ð Mayer-Vietoris. For A˚ B˚ X there is a long exact sequence
δ1 H1 X ∪ =H0 A B H0 A H0 B H0 X 0.
⋯ ←Ð ( ) ←ÐÐ ( ∩ ) ←Ð ( ) ⊕ ( ) ←Ð ( ) ←Ð 9.2 Universal Coefficient Theorem
∗ Relating H to H∗.
Let C●, ∂● be a chain complex of free abelian groups (or vector spaces). Then Cn, Bn and Zn are all free. Hence the short exact sequence ( ) ∂n 0 Zn Cn Bn−1 0 (9.1) gives rise to a short exact sequenceÐ→ of dualÐ→ chainÐÐ→ complexesÐ→
∂n 0 o Zn o Cn o n−1 o 0 O O B O n n n−1 ∂ =0 ∂ ∂ =0 ∂n−1 0 o Zn−1 o Cn−1 o Bn−2 o 0
21 and a long exact sequence in homology
n n n n−1 n−1 B Z H C● B Z so there are short exact⋯ ←Ð sequences←Ð ←Ð ( ) ←Ð ←Ð ←Ð ⋯
0 o ker in o Hn C o coker in o 0 ( ) 6 ● S _ k n n (∗ ) ∗ ∗ and ker i φ Z φ Bn 0 φ Zn Bn Hn C● . n n If we are working= { ∈ over∶ S a field≅ } and= assuming∈ that =H( C(● is)) finite dimensional, coker i 0 as every f Bn−1 can be extended to f Zn−1. ( ) = Theorem∈ 9.5. If C●, ∂● is a chaiñ∈ complex of vector spaces with finite dimensional cohomol- n ∗ ogy, then H C● Hn C● . ( ) n ∗ Corollary 9.6.( )For= ( any( space)) X of finite type and any field F, H X; F Hn X; F .
Let C●, ∂● now be a chain complex of free, finitely generated abelian( groups.) ≅ ( Then( ))
d1 ( ) ⋱ dk 1 in−1 ⎛ ⋱ ⎞ Bn−1 Zn−1 ⎜ 1 ⎟ ⎜ 0 ⋯ ⋯ ⋯ ⋯ 0 ⎟ = ⎜ ⋮ ⋮ ⎟ ∶ Ð→ ⎜ 0 ⋯ ⋯ ⋯ ⋯ 0 ⎟ ⎜ ⎟ l ⎝ ⎠ and hence Hn−1 Z Z d1Z Z dkZ . Also
d 0 0 = ⊕ ⊕ ⋯ ⊕ t 1 ⋯ d1 ⋱ ⋮ ⋮ in−1 ⋱ dk ⋮ ⋮ 1 1 ⋮ ⋮ 0 ⋯ 0 ⎛ ⋱ ⋮ ⋮ ⎞ ⎜ 1 0 ⋯ 0 ⎟ = = ⎜ ⎟ and ⎝ ⎠ n−1 n−1 n−1 coker i B im i Z d1Z Z dkZ Tor Hn−1
Theorem 9.7. If C●, ∂●= is a chain complex≅ of finitely ⊕ ⋯ ⊕ generated free≅ abelian( ) groups then
n ∗ ( ) H C● Hn C● Tor Hn−1 C● Corollary 9.8. For a cell complex( ) of≅ finite( ( type,)) ⊕ i.e. a( complex( with)) finitely many cells in each dimension, n H X Hn X Tor Hn X Tor Hn−1 X . Furthermore, H C ∗ is free and so the short exact sequence ( ) splits. n ● ( ) ≅ ( ) ( ( )) ⊕ ( ( ))
Remark 9.9. A( map( )) of chain complexes C● C● induces a map∗ of exact sequences in ( ). However, the splitting in Theorem 9.7 does not correspond, i.e. the splitting may not be natural. → ̃ ∗ Example 9.10.
3 3 3 3 H0 RP Z H1 RP Z 2Z H2 RP 0 H3 RP Z H0 P 3 H1 P 3 0 H2 P 3 2 H3 P 3 R = Z R = R = Z Z R = Z = = = =
22