INTRODUCTION TO THE FUNDAMENTAL GROUPS 2014 年前期理科大 MUTSUO OKA 1. 基本概念 1.1. Loops and Fundamental group. Let X be an arcwise connected topological space. Consider the homotopy classes of loops starting at a ﬁxed point b0 ∈ X : Ω(X, b0 ). Recall that a loop is a continuous mapping σ : (I, ∂I) → (X, b0 ). Composition σ ∗ τ of two loops σ, τ is deﬁned as σ(2t), 0 ≤ t ≤ 1/2 (σ ∗ τ )(t) = τ (2t − 1), 1/2 ≤ t ≤ 1 Let c be the constant loop at b0 and σ −1 be the loop deﬁned by σ −1 (t) = σ(1 − t). Proposition 1. (1) (Well-deﬁned up-to homotopy) σ ≃ σ ′ , τ ≃ τ ′ =⇒ σ ∗ τ ≃ σ′ ∗ τ ′. (2) (Associativity) σ ∗ (τ ∗ ω) ≃ (σ ∗ τ ) ∗ ω. (3) (Unit element) σ ∗ c ≃ σ ≃ c ∗ σ. (4) (Inverse) σ ∗ σ −1 ≃ σ −1 ∗ σ ≃ c. In particular, Ω(X, b0 ) is a group. We denote this group by π1 (X, b0 ). Proposition 2. If (X, b0 ) ≃ (Y, y0 ), π1 (X, b0 ) ∼ = π1 (Y, y0 ). Example 3. π1 (S 1 , b0 ) ∼ = Z. π1 (R2 \ {o}) ∼ = Z. Proposition 4. b′0 ∈ X, ℓ : [0, 1] → X, ℓ(0) = b0 , ℓ(1) = b′0 . ψ : π1 (X, b0 ) → π1 (X, b′0 ), ω → ℓ−1 ωℓ は群の同型写像。 参考図書：田村一郎、岩波全書「トポロジー」小島 定吉： 「トポロジー入門」共立 1991 Mathematics Subject Classification. 14H30,14H45, 32S55. Key words and phrases. θ-Alexander polynomial, fundamental group. 1 2 M. OKA 1.2. Hurewicz theorem. Theorem 5. The canonical homomorphism π1 (X, b0 ) → H1 (X) is a surjective and the kernel is the commutator subgroup of π1 (X, b0 ). 1.3. van Kampen Lemma. Assume we have groups G1 , G2 , G12 and group homomorphisms ϕ1 : G12 → G1 and ϕ2 : G12 → G2 . Then amalgamated product G1 ∗ G2 of G12 G1 , G2 over G12 is the quotient group of the free product G1 ∗ G2 by the normal group generated by the elements {ϕ1 (g)ϕ2 (g)−1 | g ∈ G12 }. (1) If ϕ1 , ϕ2 are trivial homomorphisms, G1 ∗ G2 ∼ = G1 ∗ G2 . G12 (2) If ϕ2 is a surjective, G1 ∗ G2 ∼ = G1 /N ϕ1 (Kerϕ2 ) where N H is the normal Example 6. G12 subgroup generated by H. Suppose that X = X1 ∪ X2 and put X12 = X1 ∩ X2 . Assume that those subspaces X12 , X1 , X2 are arcwise connected and b0 ∈ X12 . Lemma 7. (van Kampen) The fundamental group π1 (X, b0 ) is isomorphic to π1 (X1 ) ∗ π1 (X2 ). π1 (X12 ,b0 ) Lemma 8. (Homotopy invariance) Suppose that f : (X, b0 ) → (Y, c0 ) is a homotopy equivalence. Then π1 (X, b0 ) ∼ = π1 (Y, c0 ). Example 9. 1. π1 (P1 \ {k + 1points}) ∼ = F (k) where F (k) is a free group of rank k. 2 2. Show that a 2-simplex ∆ and a unit disk D = {(x, y) | x2 +y 2 ≤ 1} are homotopic. 1.4. Covering. Let π : (E, e0 ) → (X, b0 ) be a covering. The covering transformation G(E/X) is the group of transformations φ : E → E such that π ◦ φ = π. The covering is called normal (or Galois) if G(E/X) acts transitively to π −1 (b0 ). Proposition 10. (1) Let π : (E, e0 ) → (X, b0 ) is a covering. Then π♯ : π1 (E, e0 ) → π1 (X, b0 ) is injective. (2) It is normal iﬀ π1 (E, e0 ) is a normal subgroup of π1 (X, b0 ). In this case, the following is exact. 1 → π1 (E, e0 ) → π1 (X, b0 ) → G(E/X) → 1 証明：(1)π♯ is injective: If π(σ) ≃ c, there is a homotopy h : σ ◦ π ≃ c. This can be ˜ : I × I → E so that h|t = 0 = σ. Then h|t = 1 = c. lifted to h 記号：E の道 ℓ の π による像を ℓ, B の道 ℓ に対し e ∈ π −1 (ℓ(0)) から始まる持ち上 げを ℓ˜e であらわす。 2) Suppose that π1 (E) is a normal subgroup. Take σ ∈ π1 (E, e0 ) and e ∈ π −1 (b0 ). Then the lift of π♯ σ from e is also closed: (正規なことは基点を変えても同じに注意。) INTRODUCTION TO THE FUNDAMENTAL GROUPS 3 (∵): Take a path ℓ from e0 to e. Then ℓσℓ−1 is conjugate to σ. So its lift is also closed by the normality. ℓσℓ−1 = ℓ˜ σe ℓ−1 σ˜e (1) これは σ ˜e (1) = e を意味する。 a) ∀σ ∈ π1 (X), deﬁne ϕσ : E → E: x ∈ E, take a path ℓ from ϕσ (x) to be the end point of the lift of (ℓ−1 σℓ)x : 0 to e0 to x and deﬁne ϕσ (x) = ℓ−1 σ ˜e0 (ℓ)σ˜e0 (1) (1) well-deﬁnedness: Suppose τ = ωσω −1 , ∃ω ∈ π1 (B, b0 ). Put ξ := στ −1 = σωσ −1 ω −1 ∈ π1 (E, e0 ) で上からのループ。 ((ℓ−1 σℓ)(ℓ−1 τ −1 ℓ))∼ = ℓ−1 στ −1 e0 ˜ℓe0 = ℓ−1 ξ˜x ℓ でループ。すなわち σ, τ に依存すない。 ｂ）Suppose that ℓ′ is another path from e0 to x. すると ω := ℓℓ′ −1 ∈ π1 (E, e0 ). −1 −1 −1 ℓ−1 σℓ(ℓ′ σℓ′ )−1 = ℓ−1 σ(ℓℓ′ )σ −1 ℓ′ = ℓ′ ω −1 σωσ −1 ℓ′ is closed. c) 逆に ψ ∈ G(E/X) をとる。ℓ を e0 から ψ(e0 ) への道を一つとれ。σ := π ◦ ℓ ∈ π1 (X, b0 ). ϕ を σ に対応する変換とすると、ϕ = ψ を示せ。 Proposition 11. Assume that a group G is acting to a space E freely and properly discontinuously. Let X be the quotient space E/G. Then the canonical mapping π : E → X is a normal covering. ”freely and properly discontinuously” とは任意の点 x ∈ E にたいし開近傍 U があっ て、{gU |g ∈ G} が交わらないときをいう。 Example 12. 1. Let S 3 = {(z1 , z2 ) ∈ C2 | ∥z∥ = 1}. Let ζ := exp(2πi/p). Consider the action Z/pZ × S 3 → S 3 , (ζ a , (z1 , z2 )) = (z1 ζ a , z2 ζ a ). The quotient space is called a Lens space L(p). We have π1 (L(P ) ∼ = Z/pZ. 1.5. Fibration. Assume that p : E → B is a locally trivial ﬁbration with a connected ﬁber F . Then we have the exact sequence: · · · → π2 (B) → π1 (F ) → π1 (E) → π1 (B) → 1 4 M. OKA Example 13. (Projective hypersurfaces are simply connected.) Let f (z) is a reduced homogeneous polynomial of n + 1) variables and let K = f −1 (0) ∩ S 2n+1 . Then by Milnor, K is (n − 2) -connected. Let V be the projective hypersurface in Pn . Thus if n ≥ 3, K is simply connected. From the above exact sequence, we see that π1 (V ) = 1 and π2 (V ) surjects to Z for dim V ≥ 2. 2. Simplicial space 2.1. 複体の基本群. 田村一郎、「トポロジー」、岩波全書 K を複体とし、ai で頂点を表す。 頂点列 ℓ = (ai0 , . . . , aiq ) : が 折れ線 ⇐⇒ (i) [aij , aij+1 ] が１単体かまたは (ii) aij = aij+1 . ℓ の終点と ℓ′ の始点が同じなら合成 ℓ · ℓ′ が自然に定義出来る。同様に逆折れ線 ℓ−1 も定義出来る。閉折れ線とは終点と始点が同じ時をいう。 基本変形： （I) (...ai ai ...) → (...ai ...) (II) (...ai aj ai ...) → (...ai ...) (III) (...ai aj ak ...) → (...ai ak ...) if (ai , aj , ak ): 2-simplex 有限回の基本変形で移りあうとき ℓ ≃ ℓ′ と書き、その同値類を [ℓ] とかく。 定義： π(K, a0 ): homotopy classes of closed path at a0 . Theorem 14. 自然な準同型 ρ : π(K, a0 ) → π1 (|K|, a0 ) is bijective. 単体近似定理を使うので復習。 ∪ 2.2. 単体近似. 定義： a ∈ VertexK, OK (a) := a<σ Intσ. ∩ 2. a0 , . . . , ar がある単体の頂点 ⇐⇒ ri=0 OK (ai ) ̸= ∅。 ∀a (vertex), f (OK (a)) ⊂ OK ′ (φ(a)). 3. φ : K → K ′ が単体写像から、その幾何的位相空間の連続写像が引き起される： φ¯ : |K| → |K ′ |. 定義：. f : |K| → |K ′ |, φ : K → K ′ が f の単体近似とは任意の頂点 a ∈ K に対し て、f (OK (a)) ⊂ OK ′ (φ(a)). Proposition 15. Let φ be a simplicial approximation of f . Then φ ≃ φ. ¯ の証明が意味を持つために次を証明せよ。 レポート問題 16. ∅. (1) 頂点 a1 , . . . , ak がある単体の σ の頂点 ⇐⇒ ∩ki=1 OK (ai ) ̸= INTRODUCTION TO THE FUNDAMENTAL GROUPS 5 (2) x ∈ Int(σ), σ = (a1 , . . . , ak ) とする. 命題の状況で x ∈ ∩ki=1 OK (ai ) =⇒ f (x) ∈ f (∩ki=1 OK (ai )) ⊂ ∩ki=1 f (OK (ai )) ⊂ ∩ki=1 OK ′ (φ(ai )) ⊂ Int(φ(a1 ), . . . , φ(ak )). Proposition 17. For f : |K| → |K ′ |, there exists n such that f (OK (Sdr (K), a)) ⊂ OK ′ (a′ ), ∃a′ for any r ≥ n. このとき単体写像 φ : K → K ′ があって、f は φ の単体 近似。 証明： a : vertex, ∃a′ , f (OSdn (K),a ) ⊂ OK ′ (a′ ). Deﬁne φ(a) = a′ . 前半の主張の証明は次の命題を示せ。 レポート問題 18. 単体 σ ∈ S に対し d(σ) をその直径。d(K) := max{d(σ)|σ ∈ S} で 定義する。K を m 次元複体とする時一回の重心細分で m d(Sd(K)) ≤ d(K). m+1 を示せ。 定理 14 の証明：全射：ω : (I, ∂I) → (|K|, a0 ) をとる。f = ω, L = I とする。 ∃n, ∀r ≥ n, f : |Sdr (L)| → |K| を単体近似とする単体写像が存在する： φ : Sdr L → K 単射：K の閉じた折線 ℓ′ = (a0 , ai1 , . . . , ajq , a0 ),ω ′ = ρ(ℓ) とする。ω ′ ≃ c とする。ℓ¯′ ≃ ¯ 0 ′ 2 by F : I × I → |K | なら ∃n, I × I を n の正方形に分けて正方形は対角線を加えて 2n2 三角形に分る。 f ([k/n, (k + 1)/n] × [j/n, (j + 1)/k]) ⊂ OK ′ (ajk ) と出来る。但し aj,n = a0 。F を単体写像 (k/n, j/n) → ajk で近似する。ψ : Sdn (I)2 → K ′. 2.3. 自由群、群の表示. 階数 n の自由群とは x1 , . . . , xn を生成元として全てのげんは w = xεi11 · · · xεikk , εj = ±1 とかけ j−1 j+2 ij = ij+1 , εj + εj+1 = 0 =⇒ w ∼ w′ = xεi11 · · · xij−1 xj+1 · · · xεikk ε ε この変換（簡約）を有限回おこなって長さ０の元になるとき自明と定めると自然に群とな る。簡約が出来無い表現を簡約かされた表現という。任意の元は唯ひとつの簡約かされた 表現をもつ。このとき G = F (n) と表す。R1 , . . . , Rs を F (n) の単語として N (R1 , . . . , Rs ) を R1 , . . . , Rs とその共役元で生成された正規部分ぐんとする。商群 G = F (n)/N を G =< x1 , . . . , xn | R1 , . . . , Rs > とかいて G の関係式表現、R1 , . . . , Rs を関係式という。 G =< x1 , . . . , xn | R1 = · · · = Rs = e > 6 M. OKA とかくことが多い。 例１。 < x1 | xk1 >∼ = Z/kZ 2.4. 自由積. G1 , G2 を群とする。G1 , G2 の自由積 G1 ∗ G2 とは xi1 xi2 · · · xik | xi ∈ G1 − {e1 } or xi ∈ G2 − {e2 }/ ∼ 上の表現で同じ Gj の元が続けてあれば xij , xij+1 ∈ Gs =⇒ w ∼ xi1 . . . xj−1 (xij xij+1 )xij+2 · · · xik と簡約する。特に xij xij+1 = e =⇒ w ∼ xi1 . . . xij−1 xij+2 · · · xik この同値るいの群を G1 ∗ G2 とかく。とくに このような簡約を出来るかぎり行って簡約が出来無い表現にしたものが唯１つある。 このような元の全体に積を自明にいれて群としたもの。 例２。 Zp ∗ Zq =< x, y | xp = y q = e > 2.5. 単体複体の基本関係式. 各頂点 ai に a0 から折線で継いで固定する。それを ℓ[ai ] と 起く。ℓ[a0 ] = (a0 ). 1-単体全体を {τ1 , . . . , τs }. τi =< ai , a′i > のとき閉じた道 ℓ(< τi >) := ℓ[ai ]τi ℓ[a′i ]−1 を対応させる。これらを基底にとる. x1 , . . . , xs . すなわち xi = ℓ(ai )τi ℓ(a′i )−1 . ℓ[ai ] = (a0 , . . . , ai ), ℓ[a′i ] = (a0 , . . . , a′i ) =⇒ xi = ℓ(< τi >) = (a0 , . . . , ai , ai , a′i , . . . , a0 ) これに変形（I）をして既約にしたものを (a0 , a1 , . . . , ap , a0 ) とすると、これらは基底で かけて ri = xi (xεi11 . . . xiqq )−1 , i = 1, . . . , s ε の関係式をえる。また２単体を σ1 , . . . , σt とするとたとえば σi = (ak1 ak2 ak3 ) なら ℓ(< ak1 ak2 >)ℓ(< ak2 ak3 >)ℓ(< ak3 ak1 ) = 1 から関係式 ri′ = xεa1 xεb2 xεc3 = 1 をえる。 Theorem 19. π(K, a0 ) ∼ = {x1 , . . . , xs | r1 , . . . , rs = r1′ = · · · = rt′ = e} INTRODUCTION TO THE FUNDAMENTAL GROUPS 7 例：K = ∂∆2 , 頂点 {a0 , a1 , a2 }, 辺：τ0 =< a0 , a1 >, τ1 =< a1 , a2 >, τ2 =< a2 , a0 >. ℓ[a1 ] = (a0 , a1 ), ℓ[a2 ] =< a0 , a1 , a2 > x0 = (a0 , a0 , a1 , a0 ), x1 = (a0 , a1 , a1 , a2 , a2 , a1 , a0 ), x2 = (a0 , a1 , a2 , a2 , a0 , a0 ) r0 = x0 = (a0 , a1 , a0 ) = x0 x−1 ⇐⇒ x0 = e 0 r1 = x1 = (a0 , a1 , a2 , a1 , a0 ) = x0 x1 x−1 ⇐⇒ x1 = e 1 r2 = x2 = (a0 , a1 , a2 , a0 ) = x0 x1 x2 ⇐⇒ r2 = ∅ すなわち π(K, a0 ) ∼ =< x0 , x1 , x2 | x0 = x1 = e >∼ = Z. 2.5.1. ファンカンペン 多面体. K = K1 ∪K2 , K0 = K1 ∩K2 ,K0 , K1 , K2 は連結。a0 ∈ K0 にとる。ιi : K0 → Ki . このとき Theorem 20. N を π(K1 , a0 ) ∗ π(K2 , a0 ) の正規部分群で {ι1 (α)ι2 (α)−1 | α ∈ π(K0 , a0 )} で生成されるものとする。このとき π(K, a0 ) ∼ = π(K1 , a0 ) ∗ π(K2 ∗ a0 )/N. 証明：K0 の頂点：a0 , . . . , as0 K1 の頂点：a0 , . . . , as0 , b1 , . . . , bs0 +s1 , K2 の頂点：a0 , . . . , as0 , b′s0 +1 , . . . , b′s0 +s2 , ℓ(ai ) は K0 ないで選ぶ。 1-simplex: K0 : τ1 , . . . , τt0 K1 : τ1 , . . . , τt0 , τt0 +1 , . . . , τt0 +t1 K2 : τ1 , . . . , τt0 , τt′0 +t1 +1 , . . . , τt′0 +t1 +t2 2-simplex: K0 : σ1 , . . . , σp0 K1 : σ1 , . . . , σp0 , σp0 +1 , . . . , σp0 +p1 K2 : σ1 , . . . , σp0 , σp′ 0 +p1 +1 , . . . , σp′ 0 +p1 +p2 そうすると π(K0 , a0 ) = ⟨x1 , . . . , xt0 |r1 , . . . , rt0 , r1′ , . . . , rp′ 0 ⟩ π(K1 , a0 ) = ⟨¯ x1 , . . . , x¯t0 , xt0 +1 . . . , xt0 +t1 |r1 , . . . , rt0 , . . . , rt0 +t1 , r1′ , . . . , rp′ 0 , . . . , rp′ 0 +p1 ⟩ π(K2 , a0 ) = ⟨¯ x′1 , . . . , x¯′t0 , xt0 +t1 +1 . . . , x¯t0 +t1 +t2 |r1 , . . . , rt0 , rt0 +t1 +1 . . . , rt0 +t1 +t2 , r1′ , . . . , rp′ 0 , rp′ 0 ,p1 +1 . . . , rp′ 0 +p1 ⟩ π(K, a0 ) = ⟨ˆ x1 , . . . , xˆt0 , xˆt0 +t1 +1 . . . , xˆt0 +t1 +t2 |r1 , . . . , . . . , rt0 +t1 +t2 , r1′ , . . . , . . . , rp′ 0 +p1 ⟩ 8 M. OKA 一方 ιi : xi → x¯i , or x¯′i N : x¯i (¯ x′i )−1 , i ≤ t0 従って、π(K, a0 ) ∼ = π(K1, a0 ) ∗ π(K2 , a0 )/N . 1 章の練習問題 (レポート） loop の合成に関して結合律 σ(τ ξ) ≃ στ )ξ を示せ。 loop の合成に関し σc ≃ cσ ≃ σ を示せ. loop の合成に関して σσ −1 ≃ c を示せ. 基底点の取替えに関して ψ : π1 (X, b0 ) → π1 (X, b′0 ) が群の同型写像であることを 示せ。 (5) 被覆写像 p : E → X に対し、e0 ∈ E, b0 ∈ X を p(e0 ) = b0 にとる。p♯ : π1 (E, e0 ) → π1 (X, b0 ) は単射であることを示せ。 (6) 被覆写像 p : E → X に対し、N := p♯ (π1 (E, e0 )) が正規部分群とする。g¯ ∈ π1 (X, b0 )/N に対し、g を持ち上げて g˜ : ([0, 1], 0) → (E, e0 )、e0 → g˜(1) を考え れば、Ψ : π1 (X, b0 )/N → G(E/X) の同型写像となることを示せ。 (7) トーラス T := I 2 / ∼ を考える。I 2 の単体分割 (1) (2) (3) (4) a0 a1 c3 b2 b1 c1 a0 a1 a2 a0 c4 b2 c2 b1 a2 a0 Figure 1. トーラスの単体分割 (8) (9) (10) (11) を考える。１単体、２単体を全てあげよ。π(T, a0 ) をけいさんせよ。 穴のあいた円 Sn := {(x, y) ∈ R2 | x2 + y 2 = 1/2 + 1/n, x + yi ̸= exp(nπi/10)} を考えて、X := R2 \ ∪∞ n=1 Sn とする。X は連結であることを示せ。X は弧状連 結でないことを示せ。（ひんと：O と (1, 0) は道でつなげないことを示せ。 R2 \ {(0, 0)} は S 1 = {(x, y)|x2 + y 2 = 1} に変位レトラクトすることを示せ。 弧状連結な位相空間 X, Y に対し、x0 ∈ X, y0 ∈ Y を固定する。π1 (X × Y ) ∼ = π1 (X, x0 ) × π1 (Y, y0 ) を示せ。 一点和 X ∨ Y を X ⨿ Y /x0 = y0 で定義する。S 1 ∨ S 1 の基本群を求めよ。 INTRODUCTION TO THE FUNDAMENTAL GROUPS 9 (12) T \ {(1/2, 1/2)} は S 1 ∨ S 1 とホモトーピックなることを示せ。 (13) Xn を R2 から n 点除いた空間とする。X2 ≃ S 1 ∨ S 1 を示せ。同様に Xn ≃ S 1 ∨ · · · ∨ S 1 （ｎ個） を示せ。 (14) π1 (R3 − {onepoint}) = 1 を示せ。 (15) X を 地球野表面に南極と北極をつないだ直線を合わせた空間とする： X := {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1} ∪ {(0, 0, z) | − 1 ≤ z ≤ 1}. この時 π1 (X) ≃ Z を示せ。 10 M. OKA 3. 射影曲線の補空間の基本群 The rest of lectures are mainly based on my lecture: Oka, Mutsuo A survey on Alexander polynomials of plane curves. Singularits Franco-Japonaises, 209―232, S´emin. Congr., 10, Soc. Math. France, Paris, 2005. 3.1. Introduction. For a given hypersurface V ⊂ Pn , the fundamental group π1 (Pn − V ) plays a crucial role when we study geometrical objects over Pn which are branched over V . By the hyperplane section theorem of Zariski [59], Hamm-Lˆe [19], Theorem 21. (Hamm-Le,[19]) The fundamental group π1 (Pn − V ) is isomorphic to the fundamental group π1 (P2 − C) where P2 is a generic projective subspace of dimension 2 and C = V ∩ P2 . A systematic study of the fundamental group was started by Zariski [58] and further developments have been made by many authors. See for example Zariski [58], Oka [33] ∼ [35], Libgober [24]. For a given plane curve, the fundamental group π1 (P2 − C) is a strong invariant but it is not easy to compute. Another invariant which is weaker but easier to compute is the Alexander polynomial ∆C (t). This is related to a certain inﬁnite cyclic covering space branched over C. Important contributions are done by Libgober, Randell, Artal, Loeser-Vaqui´e, and so on. See for example [22, 16, 48, 28, 52, 13, 10, 46, 31, 1, 17, 26, 49, 2, 51, 8] 3.2. Fundamental groups. 3.2.1. van Kampen Pencil method. Let C ⊂ P2 be a projective curve which is deﬁned by C = {[X, Y, Z] ∈ P2 | F (X, Y, Z) = 0} where F (X, Y, Z) is a reduced homogeneous polynomial F (X, Y, Z) of degree d. We assume that B0 ∈ / C. We consider Pencil of lines passing through B0 . For simplify, we assume that B0 = (0 : 1 : 0) and pencil lines are Lη : x = η (X = ηZ). Lτ is called a generic line of the pencil for C if Lτ and C meet transversally. If Lτ is not generic, either Lτ passes through a singular point of C or Lτ is tangent to C at some smooth point. We ﬁx two generic lines Lτ0 and L∞ = {Z = 0}. We denote the aﬃne line Lτ − {B0 } by Laτ . Note that Laτ ∼ = C. The complement a a 2 Lτ0 − Lτ0 ∩ C (resp. Lτ0 − Lτ0 ∩ C) is topologically S minus d points (resp. (d + 1) points). We take b0 = B0 as the base point in the case of π1 (P2 − C). In the aﬃne case π1 (C2 − C), we take the base point b0 on Lτ0 which is suﬃciently near to B0 but b0 ̸= B0 . Let us consider two free groups F1 = π1 (Lτ0 − Lτ0 ∩ C, b0 ) and of rank d − 1 and d respectively. F2 = π1 (Laτ0 − Laτ0 ∩ C, b0 ). INTRODUCTION TO THE FUNDAMENTAL GROUPS 11 We consider the set Σ := {τ ∈ P1 | Lτ is a non-generic line} ∪ {∞}. Thus there exists canonical action of π1 (P1 − Σ, τ0 ) on F1 and F2 . We call this action the monodromy action of π1 (P1 − Σ, τ0 ). For σ ∈ π1 (P1 − Σ, τ0 ) and g ∈ F1 or F2 , we denote the action of σ on g by g σ . The relations in the group Fν (R1 ) ⟨g −1 g σ = e | g ∈ Fν , σ ∈ π1 (P1 − Σ, τ0 )⟩, ν = 1, 2 are called the monodromy relations. The normal subgroup of Fν , ν = 1, 2 which are normally generated by the elements {g −1 g σ , | g ∈ Fν } are called the groups of the monodromy relations and we denote them by Nν for ν = 1, 2 respectively. The original van Kampen Theorem can be stated as follows. See also [7, 6]. Theorem 22. ([55]) The following canonical sequences are exact. 1 → N1 → π1 (Lτ0 − Lτ0 ∩ C, b0 ) → π1 (P2 − C, b0 ) → 1 1 → N2 → π1 (Laτ0 − Laτ0 ∩ C, b0 ) → π1 (C2 − C, b0 ) → 1 Here 1 is the trivial group. Thus the fundamental groups π1 (P2 − C, b0 ) and π1 (C2 − C, b0 ) are isomorphic to the quotient groups F1 /N1 and F2 /N2 respectively. For a group G, we denote the commutator subgroup of G by D(G). The relation of the fundamental groups π1 (P2 −C, b0 ) and π1 (C2 −C, b0 ) are described by the following. Let ι : C2 − C → P2 − C be the inclusion map. Lemma 23. ([32]) Assume that L∞ is generic. (1) We have the following central extension. γ ι♯ 1 → Z−→π1 (C2 − C, b0 )−→π1 (P2 − C, b0 ) → 1 A generator of the kernel Ker ι♯ of ι♯ is given by a lasso ω for L∞ . (2) Furthermore, their commutator subgroups coincide i.e., D(π1 (C2 −C)) = D(π1 (P2 − C)). In particular, π1 (P2 − C) is abelian oﬀ π1 (C2 − C) is abelian. Proof. A loop ω is called a lasso for an irreducible curve D if ω is homotopic to a path written as ℓ ◦ τ ◦ ℓ−1 where τ is the boundary positively oriented circle of a normal small disk of D at a smooth point and ℓ is a path connecting the base point. For the assertion (1), see [32]. We only prove the second assertion. Assume that C has r irreducible components of degree d1 , . . . , dr . The restriction of the homomorphism ι♯ gives a surjective morphism ι♯ : D(π1 (C2 − C)) → D(π1 (P2 − C)). The generator τ of the kernel Z is a lasso going around L∞ and it goes to d1 [C1 ] + · · · + dr [Cr ]. Thus ker ι♯ ∩ D(π1 (C2 − C)) = {1}. 12 M. OKA 3.3. Examples of monodromy relations. (I) Tangent relation. Assume that C and L0 intersect at a simple point P with intersection multiplicity p. Such a point is called a flex point of order p − 2 if p ≥ 3 ([58]). This corresponds to the case q = 1. Then the monodromy relation gives ξ0 = ξ1 = · · · = ξp−1 and thus π1 (C2 − C) ∼ = Z. ξ2 ξ1 ξ0 Figure 2. Local Generators As a corollary, Zariski proves that the fundamental group π1 (P2 − C) is abelian if C has a ﬂex of order ≥ d − 3. In fact, if C has a ﬂex of order at least d − 3, the monodromy relation is given by ξ0 = · · · = ξd−2 . On the other hand, we have one more relation ξd−1 . . . ξ0 = e. In particular, considering the smooth curve deﬁned by C0 = {X d − Y d = Z d }, we get that π1 (P2 − C) is abelian for a smooth plane curve C, as C can be joined to C0 by a path in the space of smooth curves of degree d. (II) Nodal relation. Assume that C has an ordinary double point (i.e., a node) at the origin and assume that C is deﬁned by x2 − y 2 = 0 near the origin. This is the case when p = q = 2. Then as the monodromy relation, we get the commuting relation: ξ1 ξ2 = ξ2 ξ1 . Assume that C has only nodes as singularities. The commutativity of π1 (P2 − C) was ﬁrst asserted by Zariski [58] and is proved by Fulton-Deligne [14, 18]. See also [30, 46, 45]. (III) Cuspidal relation. Assume that C has a cusp at the origin which is locally deﬁned by y 2 − x3 = 0 (p = 2, q = 3). Then monodromy relation is: ξ1 ξ2 ξ1 = ξ2 ξ1 ξ2 . This relation is known as the generating relation of the braid group B3 (Artin [4]). Similarly in the case p = 3, q = 2, we get the relation ξ1 = ξ3 , ξ1 ξ2 ξ1 = ξ2 ξ1 ξ2 . 3.4. First Homology. Let X be a path-connected topological space. By the theorem of Hurewicz, H1 (X, Z) is isomorphic to the the quotient group of π1 (X) by the commutator subgroup (see [50]). Now assume that C is a projective curve with r irreducible components C1 , . . . , Cr of degree d1 , . . . , dr respectively. By Lefschetz duality, we have the following. INTRODUCTION TO THE FUNDAMENTAL GROUPS 13 Proposition 24. H1 (P2 −C, Z) is isomorphic to Zr−1 ×(Z/d0 Z) where d0 = gcd(d1 , . . . , dr ). In particular, if C is irreducible (r = 1), the fundamental group is a cyclic group of order d1 . 3.5. Generic cuspidal relation. We consider a model curve Cp,q which is deﬁned by y p − xq = 0 and we study π1 (C2 − Cp,q ). For this purpose, we consider the pencil lines x = t, t ∈ C. We consider the local monodromy relations for σ, which is represented by the loop x = ε(2πit), 0 ≤ t ≤ 1. We take local generators ξ0 , ξ1 , . . . , ξp−1 of π1 (Lε , b0 )) as in Figure 2. Every loops are counter-clockwise oriented. It is easy to see that each point of Cp,q ∩ Lε are rotated by the angle 2π × q/p. Let q = mp + q ′ , 0 ≤ q ′ < p. Then the monodromy relations are: ω m ξ ′ ω −m , 0 ≤ j < p − q ′ j+q (R1 ) ξj (= ξjσ ) = ω m+1 ξj+q′ −p ω −(m+1) , p − q ′ ≤ j ≤ p − 1 (R2 ) ω = ξp−1 · · · ξ0 . 3.5.1. Typical groups. For the convenience, we introduce two groups G(p, q) and G(p, q, r). G(p, q) := ⟨ξ1 , . . . , ξp , ω | R1 , R2 ⟩, G(p, q, r) := ⟨ξ1 , . . . , ξp , ω | R1 , R2 , R3 ⟩ where R3 is the vanishing relation of the big circle ∂DR = {|y| = R}: (R3 ) ω r = e. Now the above computation gives the following. Lemma 25. We have π1 (C2 − Cp,q , b0 ) ∼ = G(p, q) and if p ≥ q, π1 (P2 − Cp,q , b0 ) ∼ = G(p, q, 1). Note that and G(p, q, 1) = Z/pZ. The groups of G(p, q) and G(p, q, r) are studied in [34, 15]. For instance, we have Theorem 26. ([34]) (i) Let s = gcd(p, q), p1 = p/s, q1 = q/s. Then ω q1 is the center of G(p, q). (ii) Put a = gcd(q1 , r). Then ω a is in the center of G(p, q, r) and has order r/a and the quotient group G(p, q, r)/ < ω a > is isomorphic to Zp/s ∗ Za ∗ F (s − 1). Corollary 27. ([34]) Assume that r = q. Then G(p, q, q) = Zp1 ∗ Zq1 ∗ F (s − 1). In particular, if gcd(p, q) = 1, G(p, q, q) ∼ = Zp ∗ Zq . Let us recall some useful relations which follow from the above model. 14 M. OKA 3.6. Relation with Milnor Fibration. Let F (X, Y, Z) be a reduced homogeneous polynomial of degree d which deﬁnes C ⊂ P2 . We consider the Milnor ﬁbration of F [27] F : C3 − F −1 (0) → C∗ and let M = F −1 (1) be the Milnor ﬁber. By the theorem of Kato-Matsumoto [21], M is path-connected. We consider the following diagram where the vertical map is the restriction of the Hopf ﬁbration. C∗ i M ι j ↘ → C3 − F −1 (0) −→ C∗ p q ↘ F P2 − C Proposition 28. ([32])(I) The following conditions are equivalent. (i) π1 (P2 − C) is abelian. (ii) π1 (C3 − F −1 (0)) is abelian. (iii) π1 (M ) is abelian and the ﬁrst monodromy of the Milnor ﬁbration h∗ : H1 (M ) → H1 (M ) is trivial. (II) Assume that C is irreducible. Then π1 (M ) is isomorphic to the commutator subgroup of π1 (P2 − C). In particular, π1 (P2 − C) is abelian if and only if M is simply connected. 3.7. Degenerations and fundamental groups. Let C be a reduced plane curve. The total Milnor number µ(C) is deﬁned by the sum of the local Milnor numbers µ(C, P ) at singular points P ∈ C. We consider an analytic family of reduced projective curves Ct = {Ft (X, Y, Z) = 0}, t ∈ U where U is a connected open set with 0 ∈ C and Ft (X, Y, Z) is a homogeneous polynomial of degree d for any t. We assume that Ct , t ̸= 0 have the same conﬁguration of singularities so that they are topologically equivalent but C0 obtain a bigger total Milnor number, i.e., µtot (Ct ) < µtot (C0 ). We call such a family a degeneration of Ct at t = 0 and we denote this, for brevity, as Ct → C0 . Then we have the following property about the fundamental groups. Theorem 29. There is a canonical surjective homomorphism for t ̸= 0: φ : π1 (P2 − C0 ) → π1 (P2 − Ct ). In particular, if π1 (P2 − C0 ) is abelian, so is π1 (P2 − Ct ). Proof. Take a generic line L which cuts C0 transversely. Let N be a neighborhood of C0 so that ι : P2 − N → P2 − C0 is a homotopy equivalence. For instance, N can be a regular neighborhood of C0 with respect to a triangulation of (P2 , C0 ). Take suﬃciently small t ̸= 0 so that Ct ⊂ N . Then taking a common base point at the base point of the pencil, we deﬁne φ as the composition: ι−1 # π1 (P − C0 , b0 )−→π1 (P2 − N, b0 ) → π1 (P2 − Ct , b0 ) 2 INTRODUCTION TO THE FUNDAMENTAL GROUPS 15 We can assume that Ct and L intersect transversely for any t ≤ ε and L − L ∩ N → L − L ∩ Ct is a homotopy equivalence for 0 ≤ t ≤ ε. Then the surjectivity of φ follows from the following commutative diagram β′ α′ π1 (L − L∩ C0 , b0 ) ←− π1 (L − L ∩ N, b0 ) −→ π1 (L − L∩ Ct , b0 ) a c b π1 (P2 − C0 , b0 ) α ←− π1 (P2 − N, b0 ) β −→ π1 (P2 − Ct , b0 ) where the vertical homomorphisms a, c are surjective by Theorem 22 and α, α′ , β ′ are canonically bijective. Thus β is also surjective. Thus deﬁne φ : π1 (P2 − C0 , b0 ) → π1 (P2 − Ct , b0 ) by the composition α−1 ◦ β. The second assertion is immediate from the ﬁrst assertion. This completes the proof. Applying Theorem 29 to the degeneration Ct ∪ L → C0 ∪ L, we get Corollary 30. There is a a surjective homomorphism: π1 (C2 − C0 ) → π1 (C2 − Ct ). Corollary 31. Let Ct , t ∈ C be a degeneration family. Assume that we have a presentation π1 (P2 − C0 ) ∼ = ⟨g1 , . . . , gd | R1 , . . . , Rs ⟩ Then π1 (P2 − Ct ), t ̸= 0 can be presented by adding a ﬁnite number of other relations. Corollary 32. (Sandwich isomorphism) Assume that we have two degeneration families Ct → C0 and Ds → D0 such that D1 = C0 . Assume that the composition π1 (P2 − D0 ) → π1 (P2 − Ds ) = π1 (P2 − C0 ) → π1 (P2 − Ct ) is an isomorphism. Then we have isomorphisms π1 (P2 − D0 ) ∼ = π1 (P2 − Dt ) and π1 (P2 − C0 ) ∼ = π1 (P2 − Ct ). Example 33. Assume that C is a reduced curve of degree d with n nodes as singularities () with n < d2 . By a result of J. Harris [20], there is a degeneration Ct of C = C1 so that C0 obtains more nodes and C0 has no other singularities. (This was asserted by Severi but his proof had a gap.) Repeating this type of degenerations, one can deform a given () nodal curve C to a reduced curve C0 with d2 nodes, which is a union of d generic lines. On the other hand, π1 (P2 − C0 ) is abelian by Corollary 36 below. Thus we have Theorem 34. ([58, 14, 20, 30, 46]) Let C be a nodal curve. Then π1 (P2 −C) is abelian. 3.8. Product formula. Assume that C1 and C2 are reduced curves of degree d1 and d2 respectively which intersect transversely and let C := C1 ∪ C2 . We take a generic line L∞ for C and we consider the the corresponding aﬃne space C2 = P2 − L∞ . Theorem 35. (Oka-Sakamoto [44]) Let φk : C2 −C → C2 −Ci , k = 1, 2 be the inclusion maps. Then the homomorphism φ1# × φ2# : π1 (C2 − C) → π1 (C2 − C1 ) × π1 (C2 − C2 ) is isomorphic. 16 M. OKA Corollary 36. Assume that C1 , . . . , Cr are the irreducible components of C and π1 (P2 − Cj ) is abelian for each j and they intersect transversely so that Ci ∩ Cj ∩ Ck = ∅ for any distinct three i, j, k. Then π1 (P2 − C) is abelian. 3.9. Zariski Pair. Let C, C ′ be a pair of plane curves. The pair is called a Zariski pair if (1) deg C = deg C ′ . (2) There exist regular neighborhoods N and N ′ of C and C ′ and a homeomorphism ψ : (N, C) → (N ′ , C ′ ) in which singular points of C are bijectively mapped to singular points of C ′ . (3) The pair (P2 , C) and (P2 , C ′ are not homeomorphic. First example is found by Zariski in [58]. It is a pair of irreducible sextics with 6 cusps (C, C ′ where C is a so-called of torus type and its deﬁning polynomial is written as f (x, y) = f2 (x, y)3 + f3 (x, y)2 with deg fi = iand π1 (P2 − C) ∼ = Z2 ∗ Z3 . On the other hand, there are no conic which contains 6 cusps of C ′ and Zariski proved that π1 (P2 − C ′ is not isomorphic to Z2 ∗ Z3 . In [35], we gave a ﬁrst explicit example of such septic C ′ and proved that π1 (P2 − C ′ ) ∼ = Z6 . In [12], A. Degtyarev showed that the moduli of such sextics is irreducible which implies that any non-torus sextics with 6 cusps has Z6 as the fundamental group. There are many Zariski pairs are found after Zariski. Main tool for distinguish the topology of the complements of the curve is the fundamental group π1 (P2 − C) or Alexander polynomial ∆C (t) which is deﬁned later [39, 40, 41, 3, 11, 12]. There are also a method of the existence or non-existence of a dihedral cover by Tokunaga [53, 54]. 3.10. Covering transformation. Assume that C is a reduced curve deﬁned by f (x, y) = 0 in the aﬃne space C2 := P2 − L∞ . The line at inﬁnity L∞ is assumed to be generic so that we can write f (x, y) = d ∏ (y − αi x) + (lower terms), α1 , . . . , αd ∈ C∗ , αj ̸= αk , j ̸= k. i=1 Take positive integers n ≥ m ≥ 1. We assume that the origin O is not on C and the coordinate axes x = 0 and y = 0 intersect C transversely and C ∩ {x = 0} and C ∩ {y = 0} has no point on L∞ . Consider the doubly branched cyclic covering Φm,n : C2 → C2 , (x, y) → (xm , y n ). Put fm,n (x, y) := f (xm , y n ) and put Cm,n = {fm,n (x, y) = 0} = Φ−1 m,n (C). The topology of the complement of Cm,n (C) depends only on C and m, n. We will call Cm,n (C) as a generic (m, n)-fold covering transform of C. INTRODUCTION TO THE FUNDAMENTAL GROUPS 17 If n > m, Cm,n (C) has one singularity at ρ∞ = [1; 0; 0] and the local equation at ρ∞ takes the following form: d ∏ (ζ n − αi ξ n−m ) + (higher terms), ζ = Y /X, ξ = Z/X i=1 In the case m = n, we have no singularity at inﬁnity. We denote the canonical homomorphism (Φm,n )♯ : π1 (C2 − Cm,n (C)) → π1 (C2 − C) by ϕm,n for simplicity. Theorem 37. ([36]) Assume that n ≥ m ≥ 1 and let Cm,n (C) be as above. Then the canonical homomorphism ϕm,n : π1 (C2 − Cm,n (C)) → π1 (C2 − C) is an isomorphism and it induces a central extension of groups ι ϕm,n 1 → Z/nZ−→π1 (P2 − Cm,n (C))−→π1 (P2 − C) → 1 The kernel of ϕm,n is generated by an element ω ′ in the center and ϕm,n (ω ′ ) is homotopic to a lasso ω for L∞ in the target space. The restriction of ϕm,n gives an isomorphism of the respective commutator groups ϕm,n ♯ : D(π1 (P2 − Cm,n (C))) → D(π1 (P2 − C)). We have also the exact sequence for the ﬁrst homology groups: Φm,n 1 → Z/nZ → H1 (P2 − Cm,n (C)) −→ H1 (P2 − C) → 1 Corollary 38. (1) π1 (P2 − Cm,n (C)) is abelian if and only if π1 (P2 − C) is abelian. (2) Assume that C is irreducible. Put F (x, y, z) = z d f (x/z, y/z), Fm,n (x, y, z) = z dn fm,n (x/z, y/z) Let Mm,n and M be the Minor ﬁbers of Fm,n and F respectively. Then we have an isomorphism of the respective fundamental groups: π1 (Mm,n ) ∼ = π1 (M ). For a group G, we consider the following condition : Z(G) ∩ D(G) = {e} where Z(G) is the center of G. This is equivalent to the injectivity of the composition: Z(G) → G → H1 (G). When this condition is satisﬁed, we say that G satisﬁes homological injectivity condition of the center (or (H.I.C)-condition in short). A pair of reduced plane curves of a same degree {C, C ′ } is called a Zariski pair if there is an bijection α : Σ(C) → Σ(C ′ ) of their singular points so that the two germs (C, P ), (C ′ , α(P )) are topologically equivalent for each P ∈ Σ(C) and the fundamental group of the complement π1 (P2 − C) and π1 (P2 − C ′ ) are not isomorphic. (This deﬁnition is slightly stronger than that in [36].) Corollary 39. ([36]) Let {C, C ′ } be a Zariski pair and assume that π1 (P2 −C ′ ) satisﬁes (H.I.C)-condition. Then for any n ≥ m ≥ 1, {Cm,n (C), Cm,n (C ′ )} is a Zariski pair. See also Shimada [49]. 18 M. OKA 3.11. Non-generic covering. Let C be a given aﬃne curve deﬁned by f (x, y) = 0. The covering ψn : C2 → C2 , (x, y) → (x, y n ) is more interesting for the purpose of constructing a new curve with more singularities. Suppose that the line y = 0 is not ∏ generic with respect to C and assume that f (x, 0) = sj=1 (x − αj )νi . We assume that C is non-singular on these points (αj , 0) and assume that {Pk | k = 1, . . . , t} are the singular points of C. Then the pull-back Cn : f (x, y n ) = 0 contains the following singularities. • On C2 − {y = 0}, each singular point Pk produce n isomorphic singuarities on Cn . • On y = 0, the point (αj , 0) ∈ C produce a singular point which is locally written as v n − uνj = 0 if νj ≥ 2. Consider the following curve C deﬁned by f (x, y) = 0. ( ) ( ) f (x, y) = x2 (x − 1)2 x2 + 2 x −x2 (x − 1)2 (y − 1)+1/3 x2 − 1 (y − 1)2 −1/27 (y − 1)3 Its graph is as Figure 3. Figure 3. Graph of C √ The curve as two ﬂex points at (± 2/3, 0). Let C2 be the double cover f (x, y 2 ) = 0. It has six cusps and they are not on any single conic. INTRODUCTION TO THE FUNDAMENTAL GROUPS Figure 4. Graph of C2 19 20 M. OKA 4. Lecture 3 4.1. Alexander polynomial. Let X be a topological space which has a homotopy type of a ﬁnite CW-complex and assume that we have a surjective homomorphism: ϕ : π1 (X) → Z. Let t be a generator of Z and put Λ = C[t, t−1 ]. Note that Λ is a principal ideal domain. Consider an inﬁnite cyclic covering p : X → X such that p# (π1 (X)) = Ker ϕ. Then H1 (X, C) has a structure of Λ-module where t acts as the canonical covering transformation. Thus we have an identiﬁcation: H1 (X, C) ∼ = Λ/λ1 ⊕ · · · ⊕ Λ/λn as Λ-modules. We normalize the denominators so that λi is a polynomial in t with λi (0) ̸= 0 for each i = 1, . . . , n. The Alexander polynomial ∆(t) is deﬁned by the ∏ product ni=1 λi (t). The classical one is the case X = S 3 − K where K is a knot. As H1 (S 3 − K) = Z, we have a canonical surjective homomorphism ϕ : π1 (S 3 − K) → H1 (S 3 − K, Z) induced by the Hurewicz homomorphism. The corresponding Alexander polynomial is called the Alexander polynomial of the knot K. In our situation, we consider a plane curve C deﬁned by a homogeneous polynomial F (X, Y, Z) of degree d. Unless otherwise stated, we always assume that the line at inﬁnity L∞ is generic for C and we identify the complement P2 − L∞ with C2 . Let ϕ : π1 (C2 − C) → Z be the canonical homomorphism induced by the composition π1 (C2 − C)−→H1 (C2 − C, Z) ∼ = Zr −→Z ξ s ∑ where ξ is the Hurewicz homomorphism and s is deﬁned by s(a1 , . . . , ar ) = ri=1 ai and r is the number of irreducible components of C. We call s the summation homomorphism. ˜ → C2 − C be the inﬁnite cyclic covering corresponding to Ker ϕ. The correLet X sponding Alexander polynomial is called the generic Alexander polynomial of C and we denote it by ∆C (t) or simply ∆(t) if no ambiguity is likely. It does not depend on the choice of the generic line at inﬁnity L∞ . Let M = F −1 (1) ⊂ C3 be the Milnor ﬁber of F . The monodromy map h : M → M is deﬁned by the coordinatewise multiplication of exp(2πi/d). Randell showed in [48] the following important theorem. Theorem 40. The Alexander polynomial ∆(t) is equal to the characteristic polynomial of the monodromy h∗ : H1 (M ) → H1 (M ). Thus the degree of ∆(t) is equal to the ﬁrst Betti number b1 (M ). Lemma 41. Assume that C has r irreducible components. Then the multiplicity of the factor (t − 1) in ∆(t) is r − 1. Proof. As hd = idM , the monodromy map h∗ : H1 (M ) → H1 (M ) has a period d. This implies that h∗ can be diagonalized. Assume that ρ is the multiplicity of (t − 1) INTRODUCTION TO THE FUNDAMENTAL GROUPS 21 in ∆(t). Consider the Wang sequence: h −id ∗ H1 (M ) −→ H1 (M ) → H1 (E) → H0 (M ) → 0 where E := S 5 − V ∩ S 5 and V = F −1 (0). Then we get b1 (E) = ρ + 1. On the other hand, by Alexander duality, we have H1 (E) ∼ = H 3 (S 5 , V ∩ S 5 ) and b1 (E) = r. Thus we conclude that ρ = r − 1. The following Lemma describes the relation between the generic Alexander polynomial and local singularities. Lemma 42. (Libgober [22]) Let P1 , . . . , Pk be the singular points of C and let ∆i (t) be the characteristic polynomial of the Milnor ﬁbration of the germ (C, Pi ). Then the ∏ generic Alexander polynomial ∆(t) divides the product ki=1 ∆i (t) Lemma 43. (Libgober [22]) Let d be the degree of C. Then the Alexander polynomial ∆(t) divides the Alexander polynomial at inﬁnity ∆∞ (t) which is given by (td −1)d−2 (t− 1). In particular, the roots of Alexander polynomial are d-th roots of unity. The last assertion also follows from Theorem 40 and the periodicity of the monodromy. 4.2. Fox calculus. Suppose that ϕ : π1 (X) → Z is a given surjective homomorphism. Assume that π1 (X) has a ﬁnite presentation as π1 (X) ∼ = ⟨x1 , . . . , xn | R1 , . . . , Rm ⟩ where Ri is a word of x1 , . . . , xn . Thus we have a surjective homomorphism ψ : F (n) → π1 (X) where F (n) is a free group of rank n, generated by x1 , . . . , xn . Consider the group ring of F (n) with C-coeﬃcients C[F (n)]. The Fox diﬀerential ∂ : C[F (n)] → C[F (n)] ∂xj is C-linear map which is characterized by the property ∂ xi = δi,j , ∂xj ∂ ∂u ∂v (uv) = +u , u, v ∈ C[F (n)] ∂xj ∂xj ∂xj The composition ϕ ◦ ψ : F (n) → Z gives a ring homomorphism γ : C[F (n)] → C[t, t−1 ]. The Alexander matrix A is m × n matrix with coeﬃcients in C[t, t−1 ] and its (i, j)i component is given by γ( ∂R ). Then it is known that the Alexander polynomial ∆(t) ∂xj is given by the greatest common divisor of (n − 1)-minors of A ([9]). The following formula will be useful. ∂ k ∂ ∂ −k ∂ k ω = (1 + ω + · · · + ω k−1 ) ω, ω = −ω −k ω ∂xj ∂xj ∂xj ∂xj 22 M. OKA Example 44. We gives several examples. 1. Consider the trivial case π1 (X) = Z and ϕ is the canonical isomorphism. Then π1 (X) ∼ = ⟨x1 ⟩ (no relation) and ∆(t) = 1. More generally assume that π1 (X) = Zr with ϕ(n1 , . . . , nr ) = n1 + · · · + nr . Then π1 (X) = ⟨x1 , . . . , xr | Ri,j = xi xj xi −1 xj −1 , 1 ≤ i < j ≤ r⟩ As we have ( γ ∂ Ri,j ∂xℓ 1 − t ) = ℓ=i t−1 ℓ=j 0 ℓ ̸= i, j , we have ∆(t) = (t − 1)r−1 . Definition 45. We say that Alexander polynomial of a curve C is trivial if ∆(t) = (t − 1)r−1 where r is the number of the irreducible components of C. 2. Let C = {y 2 − x3 = 0} and X = C2 − C. Then π1 (X) = ⟨x1 , x2 | x1 x2 x1 = x2 x1 x2 ⟩. is known as the braid group B(3) of three strings. The Alexander polynomial is given by ∆(t) = t2 − t + 1. 3. Let us consider the curve C := {y 2 − x5 = 0} ⊂ C2 and put X = C2 − C. Then by §2.2, −2 π1 (X) ∼ = G(2, 5) = ⟨x0 , x1 | x0 (x1 x0 )2 x−1 1 (x1 x0 ) ⟩ In this case, we get ∆(t) = t4 − t3 + t2 − t + 1 = (t10 − 1)(t − 1) . (t2 − 1)(t5 − 1) 4.3. Degeneration and Alexander polynomial. We consider a degeneration Ct → C0 . By Corollary 31 and Fox calculus, we have Theorem 46. Assume that we have a degeneration family of reduced curves {Cs | s ∈ U } at s = 0. Let ∆s (t) be the Alexander polynomial of Cs . Then ∆s (t)|∆0 (t) for s ̸= 0. Corollary 47. (Sandwich principle) Suppose that we have two families of degeneration Cs → C0 and Dτ → D0 such that D1 = C0 and assume that ∆D0 (t) and ∆Cs (t), s ̸= 0 coincide. Then we have also ∆Cs (t) = ∆C0 (t), s ̸= 0. INTRODUCTION TO THE FUNDAMENTAL GROUPS 23 4.4. Algebric geometrical computation of Alexander polynomials. Let C be a given plane curve of degree d deﬁned by f (x, y) = 0 and let Σ(C) be the singular locus of C and let P ∈ Σ(C) be a singular point. Consider an embedded resolution of C, π : U˜ → U where U is an open neighbourhood of P in P2 and let E1 , . . . , Es be the exceptional divisors. Let us choose (u, v) be a local coordinate system centered at P and let ki and mi be the order of zero of the canonical two form π ∗ (du ∧ dv) and π ∗ f respectively along the divisor Ei . We consider an ideal of OP generated by the function germ ϕ such that the pull-back π ∗ ϕ vanishes of order at least −ki + [kmi /d] along Ei and we denote this ideal by JP,k,d . Namely ∑ JP,k,d = {ϕ ∈ OP , (π ∗ ϕ) ≥ (−ki + [kmi /d])Ei } i Let us consider the canonical homomorphisms induced by the restrictions: ⊕ σk,P : OP → OP /JP,k,d , σk : H 0 (P2 , O(k − 3)) → OP /JP,k,d P ∈Σ(C) where the right side of σk is the sum over singular points of C. We deﬁne two invariants: ∑ ρ(P, k) ρ(P, k) = dimC OP /JP,k,d , ρ(k) = P ∈Σ(C) Let ℓk be the dimension of the cokernel σk . Then the formula of Libgober [23] and Loeser-Vaqui´e [26], combined with a result of Esnault and Artal [17, 1], can be stated as follows. Lemma 48. The polynomial ∆(t) is written as the product ˜ ∆(t) = d−1 ∏ ∆k (t)ℓk , k = 1, . . . , d − 1 k=1 where ∆k = (t − exp( 2kπi −2kπi ))(t − exp( )). d d Note that for the case of sextics d = 6, the above polynomials take the form: ∆5 (t) = ∆1 (t) = t2 − t + 1, ∆4 (t) = ∆2 (t) = t2 + t + 1, ∆3 (t) = (t + 1)2 . 4.5. Triviality of the Alexander Polynomials. We have seen that the Alexander polynomial is trivial if C is irreducible and π1 (P2 − C) is abelian. However this is not a necessary condition, as we will see in the following. Let F (X, Y, Z) be the deﬁning homogeneous polynomial of C and let M = F −1 (1) ⊂ C3 the Milnor ﬁber of F . Theorem 49. Assume that C is an irreducible curve. The Alexander polynomial ∆(t) of C is trivial if and only if the ﬁrst homology group of the Milnor ﬁber H1 (M ) is at most a ﬁnite group. 24 M. OKA Proof. By Theorem 40, the ﬁrst Betti number of M is equal to the degree of ∆(t). Corollary 50. Assume that π1 (P2 − C) is a ﬁnite group. Then the Alexander polynomial is trivial. Proof. This is immediate from Theorem 28 as D(π1 (P2 − C)) = π1 (M ) and it is a ﬁnite group under the assumption. 4.6. Examples. 1. (Zariski’s three cuspidal quartic, [58]) Let Z4 be a quartic curve with three A2 -singularities. The corresponding moduli space is irreducible. Then the fundamental groups are given by [58, 35] as π1 (C2 − Z4 ) ∼ = ⟨ρ, ξ | ρ ξ ρ = ξ ρ ξ, ρ2 = ξ 2 ⟩ π1 (P2 − Z4 ) ∼ = ⟨ρ, ξ | ρ ξ ρ = ξ ρ ξ, ρ2 ξ 2 = e⟩ Then by an easy calculation, ∆(t) = 1. This also follows from Theorem 49 as π1 (P2 −Z4 ) is a ﬁnite group of order 12 by Zariski [58]. By Theorem 37, the generic covering transform Cn,n (Z4 ) has also a trivial Alexander polynomial for any n. 2. (Libgober’s criterion) Assume that for any singularity P of C, the characteristic polynomial of (C, P ) does not have any root which is a d-th root of unity. Then by Lemma 42 and Lemma 43, the Alexander polynomial is trivial. For example, an irreducible curve C with only A2 or A1 as singularity has a trivial Alexander polynomial if the degree d is not divisible by 6. 3. Curves of torus type. A curve C of degree d is called of (p, q)-torus type if its deﬁning polynomial F (X, Y, Z) is written as F (X, Y, Z) = F1 (X, Y, Z)p + F2 (X, Y, Z)q where Fi is a homogeneous polynomial of degree di , i = 1, 2 so that d = pd1 = qd2 . Assume that (1) two curves F1 = 0 and F2 = 0 intersect transversely at d1 d2 distinct points, and (2) the singularities of C are only on the intersection F1 = F2 = 0. We say C is a generic curve of (p, q)-torus type if the above conditions are satisﬁed. Let M be the space {(F1 , F2 ) | degree F1 = d1 , degreeF2 = d2 } and let M′ be the subspace for which the conditions (1) and (2) are satisﬁed. Then by an easy argument, M is an ( ) ( ) aﬃne space of dimension d12+2 + d22+2 and M′ is a Zariski open subset of M. Thus the topology of the complement of C does not depend on a generic choice of C ∈ M′ . Fundamental groups π1 (P2 − C) and π1 (C2 − C) for a generic curve of (p, q)-torus type are computed as follows. Put s = gcd(p, q), p1 = p/s, q1 = q/s. As d1 p = d2 q, we can write d1 = q1 m and d2 = p1 m. Theorem 51. (a) ([33, 34]) Then we have π1 (C2 − C) ∼ = G(p, q) and π1 (P2 − C) ∼ = G(p, q, mq1 ) INTRODUCTION TO THE FUNDAMENTAL GROUPS 25 (b) The Alexander polynomial is the same as the characteristic polynomial of the Pham-Brieskorn singularity Bp,q which is given by (1) ∆(t) = (tp1 q1 s − 1)s (t − 1) (tp − 1) (tq − 1) Proof. The assertion for the fundamental group is proved in [33, 34]. For the assertion about the Alexander polynomial, we can use Fox calculus for small p and q. For example, assume that p = 2, q = 3 and d1 = 3, d2 = 2. Thus C is a sextic with six (2,3)-cusps. As π1 (C2 − C) = ⟨x0 , x1 | x0 x1 x0 = x1 x0 x1 ⟩, the assertion follows from 2 of Example 42. Assume that p = 4, q = 6 and d1 = 3, d2 = 2. Then C has two irreducible component of degree 6: C : f3 (x, y)4 − f2 (x, y)6 = (f3 (x, y)2 − f2 (x, y)3 ) (f3 (x, y)2 + f2 (x, y)3 ) where deg fk (x, y) = k. By Lemma 25, we have π1 (C2 − C) = ⟨ξ0 , . . . , ξ3 | Rj , j = 0, 1, 2⟩ where using ω := ξ3 . . . ξ0 the relations are given as R1 : ξ0 ωξ2−1 ω −1 , R2 : ξ1 ωξ3 ω −1 , R3 : ξ2 ω 2 ξ0 ω −2 Thus by an easy calculation, the Alexander matrix is given as 1 + t4 − t3 (t − 1) t2 − (−t + t3 + 1) t (t − 1) t3 1 + t3 − t2 (t − 1) t t−1 t − t4 − 1 − (−t + t4 + 1) t3 (t + t5 − t4 − 1) t2 1 + t2 + t6 − t5 − t t + t5 − t4 − 1 and the Alexander polynomial is given by (t12 − 1)2 (t − 1) ∆(t) = (t − 1) (t + 1) (t + t + 1) (t − t + 1) (t − t + 1) = 4 (t − 1)(t6 − 1) 2 2 2 4 2 2 To prove the assertion for general p, q, we use the following result of Nemethi [29]. Lemma 52. Let u := (f, g) : (Cn+1 , O) → (C2 , O) and P : (C2 , O) → (C, O) be germs of analytic mappings and assume that u deﬁnes an isolated complete intersection variety at O. Let D be the discriminant locus of u and V (P ) := {P = 0}. Consider the composition f = P ◦ u : (Cn+1 , O) → (C, O). Let Mf and MP be the respective Minor ﬁbers of f and P . Assume that D ∩ V (P ) = {O} in a neighbourhood of O. Then the characteristic polynomial of the Minor ﬁbration of f on H1 (Mf ) is equal to the characteristic polynomial of the Milnor ﬁbration of P on H1 (MP ), provided n ≥ 2. 26 M. OKA Proof of the equality (1). Let F1 , F2 be homogeneous polynomials of degree q1 m, p1 m respectively and let F = F1p + F2q . We assume that F1 , F2 are generic so that the singularities of F = 0 are only the intersection F1 = F2 = 0 which are p1 q1 m2 distinct points. Then by Lemma 52, the characteristic polynomial of the monodromy h∗ : H1 (MF ) → H1 (MF ) of the Milnor ﬁbration of F is equal to that of P (x, y) := y p + xq . Thus the assertion follows from [5, 27] and Theorem 40. 4.7. Sextics of torus type. Let us consider a sextic of torus type C : f2 (x, y)3 + f3 (x, y)2 = 0, degree fj = j, j = 2, 3, as an example. Assume that C is reduced and irreducible. A sextic of torus type is called tame if the singularities are on the intersection of the conic f2 (x, y) = 0 and the cubic f3 (x, y) = 0. A generic sextic of torus type is tame but the converse is not true. Then the possibility of Alexander polynomials for sextics of torus type is determined as follows. Theorem 53. ([42, 38]) Assume that Cis an irreducible sextic of torus type. The Alexander polynomial of C is one of the following. (t2 − t + 1), (t2 − t + 1)2 , (t2 − t + 1)3 Moreover for tame sextics of torus type, the Alexander polynomial is given by t2 − t + 1 and the fundamental group of the complement in P2 is isomorphic to Z2 ∗ Z3 except the case when the conﬁguration is [C3,9 , 3A2 ]. In the exceptional case, the Alexander polynomial is given by (t2 − t + 1)2 . 4.8. Weakness of Alexander polynomial. Let C1 and C2 be curves which intersect transversely. We take a generic line at inﬁnity for C1 ∪ C2 . Theorem 35 says that π1 (C2 − C1 ∪ C2 ) ∼ = π1 (C2 − C1 ) × π1 (C2 − C2 ) which tell us that the fundamental group of the union of two curves keeps informations about each curves C1 , C2 . On the other hand, the Alexander polynomial of C1 ∪ C2 keeps little information about each curves C1 , C2 . In fact, we have Theorem 54. Assume that C1 and C2 intersect transversely and let C = C1 ∪C2 . Then the generic Alexander polynomial ∆(t) of C is given by given by (t − 1)r−1 where r is the number of irreducible components of C. Proof. Assume that π1 (C2 − Cj ), j = 1, 2 is presented as π1 (C2 − C1 ) = ⟨g1 , . . . , gs1 | R1 , . . . , Rp1 ⟩, π1 (C2 − C2 ) = ⟨h1 , . . . , hs2 | S1 , . . . , Sp2 ⟩ Then by Theorem 35, we have π1 (C2 −C) = ⟨g1 , . . . , gs1 , h1 , . . . , hs2 | R1 , . . . , Rp1 , S1 , . . . , Sp2 , Ti,j , 1 ≤ i ≤ s1 , 1 ≤ j ≤ s2 ⟩ INTRODUCTION TO THE FUNDAMENTAL GROUPS 27 where Ti,j is the commutativity relation gi hj gi−1 h−1 j . Let γ : C[g1 , . . . , gs1 , h1 , . . . , hs2 ] → −1 C[t, t ] be the ring homomorphism deﬁned before (§3.1). Put gs1 +j = hj for brevity. Then the submatrix of the Alexander matrix corresponding to ( ) ∂Ti,j γ( ) , {i = 1, . . . , s1 , j = s2 } or {i = s1 , j = 1, . . . , s2 −1}, and 1 ≤ k ≤ s1 +s2 −1 ∂gk is given by (1 − t) × A where ( ) Es1 0 A= K −Es2 −1 and Eℓ is the ℓ × ℓ-identity matrix and K is a (s2 − 1) × s1 matrix with only the last column is non-zero. Thus the determinant of this matrix gives ±(t − 1)s1 +s2 −1 and the Alexander polynomial must be a factor of (t − 1)s1 +s2 −1 . As the monodromy of the Milnor ﬁbration of the deﬁning homogeneous polynomial F (X, Y, Z) of C is periodic, this implies that h∗ : H1 (M ) → H1 (M ) is the identity map. Thus ∆(t) = (t − 1)b1 where b1 is the ﬁrst Betti number of M . On the other hand, b1 = r − 1 by Lemma 41. 5. Dual curves This lecture is based on: Oka, Mutsuo Geometry of cuspidal sextics and their dual curves. Singularities― Sapporo 1998, 245―277, Adv. Stud. Pure Math., 29,Kinokuniya, Tokyo, 2000. References [1] E. Artal. Sur les couples des Zariski. J. 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