Algebraic Topology II Notes Week 12 1 Cohomology Theory (Continued)

Algebraic Topology II Notes Week 12

1 Cohomology Theory (Continued)

1.1 More Applications of Poincar´eDuality

2n f 2n Proposition 1.1. Any homotopy equivalence CP → CP preserves orientation (n ≥ 1).

In other words,

2n 2n f∗ : H4n(CP ) → H4n(CP ) 2n 2n [CP ] 7→ [CP ]

2n Proof. Say α generates H2(CP ). Then, as f induces an isomorphism on H2, we have: 2n 2n f ∗(α) = ±α. We also know that α2n generates H4n(CP ), i.e. hα2n, [CP ]i = 1, or

2n 4n 2n ∩[CP ] 2n H (CP ) −−−−→ H0(CP ) α2n 7−→ 1

2n 2n If f∗([CP ]) = −[CP ], then 2n 2n 2n 2n 1 = hα , [CP ]i = hα , −f∗([CP ])i ∗ 2n 2n ∗ 2n 2n = −hf (α ), [CP ]i = −h(f (α)) , [CP ]i (1) 2n 2n 2n 2n = −h(±α) ), [CP ]i = −hα , [CP ]i = −1, which is a contradiction! Proposition 1.2. Let M n be a closed, connected, oriented n-manifold and let f : Sn → M be a continuous map of non-zero degree, i.e., the morphism

n f∗ : Hn(S ; Z) → Hn(M; Z) ∼ n is non-trivial. Show that H∗(M; Q) = H∗(S ; Q).

Proof. Assume that ∃ 1 ≤ i ≤ n − 1 such that Hi(M; Q) = 0. Then by Universal Coeﬃcient Theorem, i ∼ H (M; Q) = HomQ(Hi(M; Q), Q) 6= 0 If 0 6= α ∈ Hi(M; Q) is the generator, then by Poincar´eDuality, ∃ β ∈ Hn−i(M; Q) such that α ∪ β generates Hn(M; Q) = Q. Especially, α ∪ β 6= 0. On one hand, f ∗ : Hi(M; Q) → Hi(Sn; Q) = 0 is a zero map. Consequently, f ∗(α ∪ β) = f ∗(α) ∪ f ∗(β) = 0 ∪ 0 = 0.

On the other hand, f ∗(α∪β) = (deg f)·generator of Hn(Sn; Q) 6= 0, a contradiction.

1 Deﬁnition 1.3 (Manifold with Boundary). M is a n-manifold with boundary if any x ∈ M n n has a neighbourhood Ux homeomorphic to R or R+(xn ≥ 0).

∼ n ∼ • if Ux = R , Hn(M,M − x) = Hn(Ux,Ux − x) = Z ∼ n • if Ux = R+,

n n Hn(M,M − x) = Hn(Ux,Ux − x) = Hn(R+, R+ − {0}) = 0

And the boundary of M is deﬁned to be

∂M = {x ∈ M | Hn(M,M − x) = 0}.

n n−1 n n−1 Example 1.4. ∂(D ) = S , ∂(R+) = R . Remark 1. ∂M is a manifold of dimenison n − 1 with no boundary.

Deﬁnition 1.5. (M, ∂M) is orientable, if M \ ∂M is orientable as a manifold with no boundary.

Proposition 1.6. If (M, ∂M) is compact, orientable n-manifold with boundary, then ∃! µM ∈ Hn(M, ∂M) inducing local orientations µx ∈ Hn(M,M − x) at all x ∈ M \ ∂M. Note 1. In the long exact sequence for the pair (M, ∂M), we have

∂ Hn(M, ∂M) −→ Hn−1(∂M)

[M] = µM 7−→ [∂M]

if ∂M is oriented.

Theorem 1.7 (Poincar´eDuality). If (M, ∂M) is a connected, oriented n-manifold with boundary, then

i ∩µM i ∩µM Hc(M) −−−→ Hn−i(M, ∂M) or Hc(M, ∂M) −−−→ Hn−i(M). =∼ =∼

i def i where Hc(M, ∂M) === limKcompact H (M, (M \ K) ∪ ∂M) is the cohomology with compact −→ ⊂M\∂M support.

Proposition 1.8. If M n = ∂V n+1 is a connected manifold with V compact, then the Euler characteristic χ(M) is even.

An immediate corollary is

2n 2n 2n Corollary 1.9. RP , CP , HP cannot be boundaries of compact manifolds.

In order to prove Proposition 1.8, we need another proposition:

2 Proposition 1.10. Assume V 2n+1 is an oriented, compact manifold with connected boundary 2n n ∂V = M . If R is a ﬁeld (especially Z/2Z if M is non-orientable), then dimR H (M; R) is even.

Proof of Proposition 1.10. Let’s consider the long exact sequence for the pair (V,M):

i∗ δ Hn(V ; R) - Hn(M; R) - Hn+1(V,M; R)

∼= ∩[M] ∼= ∩[V ]

? ? i∗ - Hn(M; R) Hn(V ; R)

∗ where i , i∗ are induced by the inclusion i : M = ∂V ,→ V . P.D. ∗ ∼ ∼ By exactness, Im i = Ker δ = Ker i∗, so

∗ dim Im i = dim Ker i∗ = dim Hn(M; R) − dim Im i∗.

∗ ∗ Since i , i∗ are Hom-dual, dim Im i = dim Im i∗. n Therefore, dim H (M; R) = dim Hn(M; R) = 2 dim Im i∗ is even.

Proof of Proposition 1.8. If n = dim M is odd, then χ(M) = 0 is even. If n = 2m is even, then work with Z/2Z-coeﬃcients,

2m X i χ(M) = (−1) dimZ/2Z Hi(M; Z/2Z) i=0 m−1 (1) X i m = 2 (−1) dimZ/2Z Hi(M; Z/2Z) + (−1) dimZ/2Z Hm(M; Z/2Z) i=0

≡ dimZ/2Z Hm(M; Z/2Z) mod 2 (2) ≡ 0 mod 2

Equation (1) is due to Poincar´eDuality, dimZ/2Z Hn−i(M; Z/2Z) = dimZ/2Z Hi(M; Z/2Z), 0 ≤ i ≤ m − 1. Congruence (2) is by Proposition 1.10.

Note 2. • Im i∗ ⊂ Hn(M 2n) is self-annihilating with respect to cup product ∪, i.e. if α, β ∈ Im i∗, then α ∪ β = 0.

∗ 1 n 2n • by Proposition 1.10, dim Im i = 2 dim H (M ).

Proof. For any α = i∗(α), β = i∗(β) ∈ B, where α, β ∈ H2n(V ), we have

δ(α ∪ β) = δ(i∗(α) ∪ i∗(β)) = δi∗(α ∪ β) = 0

3 Hence, α ∪ β ∈ Ker (δ : H4n(M) → H4n+1(V )) = 0 by the following commutative diagram

δ H4n(M) - H4n+1(V ) ∼= P.D. ∼= P.D. ? ? - H0(M) H0(V ) with the bottom arrow an injection.

1.2 Signature

Let M n be a closed, oriented, n-manifold.

• σ(M) = 0 if 4 6| dim M = n.

• if dim M = 4k, then σ(M) is deﬁned to be the signature of the non-degenerate cup product pairing

2k 2k (, ): H (M; R) × H (M; R) → R (α, β) 7→ (α, β)[M]

σ(M):=(the number of positive eignvalues)-(the number of negative eignvalues).

0 1 Example 1.11. σ(S2 × S2) = σ = 0, σ( 2n) = 1, σ( 2# 2) = 2. 1 0 CP CP CP

Remark 2. Signature σ is a cobordism invariant, i.e. if ∂W = M t−N, then σ(M) = σ(N).

Theorem 1.12. If in the above notations M 4k = ∂V 4k+1 is connected with V compact and orientable, then σ(M) = 0.

2 2 2 2 Remark 3. • σ(CP #−CP ) = 0. In fact, CP #−CP is the boundary of a connected, oriented 5-manifold;

2 2 2 2 • σ(CP #CP ) = 2 6= 0, so CP #CP is NOT the boundary of a connected, oriented 5-manifold, but as we will see later, it IS the boundary of a non-orientable 5-manifold.

4 Proof. Let A = H2n(M) and we’ll always work over R-coeﬃcients. We have a non-degenerate and symmetric pairing ϕ : A × A → R.

Let A+ be the subspace on which the pairing is positive-deﬁnite, A− be the subspace on which the pairing is negative-deﬁnite.

Let r = dim A+, 2l = dim A (by Proposition 1.10). Then, automatically, dim A− = 2l − r and σ(M) = r − (2l − r) = 2r − 2l. In order to prove that σ(M) = 0, we want to show that r = l! Let B ⊂ A be the self-annihilating l-dimensional subspace given by Proposition 1.8 and Note 2.

Therefore, A+ ∩ B = {0},A− ∩ B = {0}. Hence,

dim A+ + dim B ≤ dim A = 2l, i.e. r + l ≤ 2l i.e. r ≤ l

dim A− + dim B ≤ dim A = 2l, i.e. 2l − r + l ≤ 2l i.e. r ≥ l In conclusion, r = l and σ(M) = 0.

1.3 Connected Sums

Deﬁnition 1.13 (Connected Sum). Let M n,N n be closed, connected, oriented n-manifolds, then their connected sum

is deﬁned to be n n M#N := (M \ D1 ) ∪f (N \ D2 ) n n−1 n n−1 where f : ∂D1 = S → ∂D2 = S is an orientation-reversing homeomorphism. Remark 4. M#N is still a closed, connected, oriented n-manifold, H0(M#N) = Z, n k ∼ k k H (M#N) = Z and H (M#N) = H (M) ⊕ H (N), 0 < k < n. Remark 5. Cup product α ∪ β = 0 for any α ∈ Hk(M) and β ∈ Hl(N) with k, l > 0. 2 2 Example 1.14. S2 × S2 and CP #CP have the same cohomology groups, 0 2 4 H = Z,H = Z ⊕ Z = Zα ⊕ Zβ, H = Z, but diﬀerent cohomogy rings, since

2 2 2 2 α ∪ β 6= 0 in S × S ; while α ∪ β = 0 in CP #CP .

5 2 2 2 2 Example 1.15. σ(CP #CP ) = 2, so CP #CP cannot be the boundary of a compact 2 2 oriented 5-manifold. However, CP #CP = ∂W 5, where W 5 is a compact non-orientable 5-manifold. W can be constructed as follows: 2 2 1. Start with (CP × I)#(RP × S3). 2 2. Run an orientation reversing path γ from one CP to the other, by traveling along an 2 orientation reversing path in RP . 3. Enlarge the path to a tube and remove its interior. What is left is a 5-dimensional 2 2 non-orientable manifold with ∂W = CP #CP .

2 Homotopy Theory

Let X be a topological space, x0 ∈ X. Earlier we deﬁned the fundamental group

π1(X, x0) = {f :(I, ∂I) → (X, x0)}/ ∼ 1 = {f :(S , s0) → (X, x0)}/ ∼ where ∼ refers to the homotopy of maps.

And also some properties of π1:

• functorial, i.e. f : X → Y induces f∗ : π1(X, x0) → π1(Y, f(x0)); • homotopy invariant;

• group structure;

• related to H1.

2.1 Higher Homotopy Groups

Deﬁnition 2.1 (Higher homotopy groups).

n n πn(X, x0) = {f :(I , ∂I ) → (X, x0)}/ ∼ n = {f :(S , s0) → (X, x0)}/ ∼

6 n n n n where I = [0, 1] , ∂I = {(x1, . . . , xn) ∈ I | ∃ i, s.t. xi ∈ {0, 1}} and ∼ refers to homotopy of maps.

When n = 0, I0 =point, ∂I0 = ∅. So

π0 = set of connected components of X

Deﬁne for n ≥ 1, f, g ∈ πn(X, x0),

1 f(2s1, s2, . . . , sn) 0 ≤ s1 ≤ 2 (f + g)(s1, . . . , sn) = 1 g(2s1 − 1, s2, . . . , sn) 2 ≤ s1 ≤ 1

Lemma 2.2. If n ≥ 2, πn is an abelian group.

Goal: Prove Whitehead’s Theorem.

Theorem 2.3 (Whitehead’s Theorem). If a map f : X → Y between connected CW com- plexes induces isomorphisms on πn for all n, then f is a homotopy equivalence. If moreover, f is the inclusion of a subcomplex X,→ Y , then X is a deformation retract of Y .