1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)

1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1) X α α 1 : I X α 1 (s) = α(1 s) ( )α 1 1.1 X p X Ω(p) p loop α, β Ω(p) α β rel {0, 1} α β ( ) 1.2 α, β Ω(p) α β = α β well-defined ( ) 1.3 Π 1 (X, p) = { α α Ω(p)} ( ) α, β, γ Π 1 (X, p) ( α β ) γ = α β γ = (α β) γ = α (β γ) = α β γ = α ( β γ ) α Π 1 (X, p) α e p = α e p = α e p α = e p α = α e p Π 1 (X, p) α Π 1 (X, p) α α 1 = α α 1 = e p α 1 α = α 1 α = e p α 1 α Q.E.D

1.4 X α, β, γ α(1) = β(0), β(1) = γ(0) (α β) γ α(β γ) rel {0, 1} ( ) 1.5 X α : I X α(0) = x, α(1) = y e x α α rel {0, 1}, α e y α rel {0, 1} e z : I X, e z (s) = z (s I) e p : I X e p (s) = p e p Π 1 (X, p) ( ) 1.6 X α : I X α(0) = x, α(1) = y α α 1 e x rel {0, 1}, α 1 α e y rel {0, 1} α 1 : I X, α 1 (s) = α(1 s) (s I) α 1 = α 1 α ( )

1.7 f : X Y f : Π 1 (X, p) Π 1 (Y, q) (q = f(p)) f ( α ) = fα (1)f well-defined (2)f ( ) (1) α = β = fα = fβ α β rel {0, 1} F : I I X F (s, 0) = α(s) F (s, 1) = β(s) F (0, t) = α(0) = β(0) = p F (1, t) = α(1) = β(1) = p (s I, t I) G = ff : I I X G(s, 0) = (ff )(s, 0) = f(f (s, 0)) = fα(s) G(s, 1) = (ff )(s, 1) = f(f (s, 1)) = fβ(s) G(0, t) = (ff )(0, t) = f(f (0, t)) = f(p) = q G(1, t) = (ff )(1, t) = f(f (1, t)) = f(p) = q (s I, t I) fα fβ rel {0, 1} (2) α, β Π 1 (X, p) f ( α β ) = f ( α )f ( β ) = f ( α β ) = f(α β) = fα fβ = (fα) (fβ) f(α β) = (fα) (fβ) (α β)(s) = (f(α β))(s) = { α(2s) (0 s 1 ) 2 β(2s 1) ( 1 s 1) 2 { (fα)(2s) (0 s 1 ) 2 (fβ)(2s 1) ( 1 s 1) 2 } = ((fα) (fβ))(s) = Q.E.D

1.8 X, Y, Z f : X Y, g : Y Z (gf) = g f ( ) α Π 1 (X, p) (gf) ( α ) = (gf)α = g(fα) = g ( fα ) = g (f ( α )) = (g f )( α ) (gf) = g f Q.E.D 1.9 (1) 1 X : X X (1 X ) = 1 Π1 (X,p) (2) f : X Y, g : X Y f(p) = g(p) f g rel {p} = f = g (3) f : X Y f : Π 1 (X, p) Π 1 (Y, f(p)) (f 1 ) = (f ) 1 ( ) (1) α Π 1 (X, p) (1 X ) ( α ) = 1 X α = α (1 X ) = 1 Π1 (X,p) (2) f g rel {p} F : X I Y F (x, 0) = f(x) F (x, 1) = g(x) α Π 1 (X, p) F (p, t) = f(p) = g(p) (x X, t I) f ( α ) = fα g ( α ) = gα G : I I Y G(s, t) = F (α(s), t) G = F (α 1 I ) G G(s, 0) = F (α(s), 0) = f(α(s)) = (fα)(s) G(s, 1) = F (α(s), 1) = g(α(s)) = (gα)(s) G(0, t) = F (α(0), t) = F (p, t) = f(p) = f(α(0)) = (fα)(0) = (gα)(0) G(1, t) = F (α(0), t) = F (p, t) = f(p) = f(α(1)) = (fα)(1) = (gα)(1) (s I, t I) G : fα gα rel {0, 1} f ( α ) = fα = gα = g ( α ) (3) (f 1 f) = (1 X ) (f 1 ) f = 1 Π1 (X,p) f (f 1 ) = 1 Π1 (X,p) f (f ) 1 = (f 1 ) Q.E.D

1.10 X X p, q Π 1 (X, p) = Π 1 (X, q) ( ) X p q γ ϕ : Π 1 (X, p) Π 1 (X, q) ϕ( α ) = γ 1 α γ (1) ϕ well-difined α = β = γ 1 α γ = γ 1 β γ α β rel {0, 1} F : I I X F (s, 0) = α(s) F (s, 1) = β(s) F (0, t) = α(0) = β(0) = p F (1, t) = α(1) = β(1) = p (s I, t I) G : I I X G G(s, 0) = G(s, 1) = γ 1 (2s) = γ(1 2s) (0 s 1) 2 G(s, t) = F (4s 2, t) ( 1 2 s 3) 4 γ(4s 3) ( 3 s 1) 4 γ 1 (2s) = γ(1 2s) (0 s 1 2 ) F (4s 2, 0) ( 1 2 s 3 4 ) γ(4s 3) ( 3 4 s 1) γ 1 (2s) = γ(1 2s) (0 s 1 2 ) F (4s 2, 1) ( 1 2 s 3 4 ) γ(4s 3) ( 3 4 s 1) = {γ 1 (α γ)}(s) = {γ 1 (β γ)}(s) G(0, t) = γ 1 (0) = q = {γ 1 (α γ)}(0) = {γ 1 (β γ)}(0) G(1, t) = γ 1 (1) = q = {γ 1 (α γ)}(1) = {γ 1 (β γ)}(1) γ 1 α γ γ 1 β γ rel {0, 1} (2) ϕ (homomorphism) α, β Π 1 (X, p) ϕ( α β ) = ϕ( α β ) = γ 1 α β γ = γ 1 α γ γ 1 β γ = γ 1 α γ γ 1 β γ = ϕ( α )ϕ( β ) ϕ (3) ϕ (injective) ϕ( α ) = ϕ( β ) γ 1 α γ = γ 1 β γ γ 1 α γ γ 1 β γ rel {0, 1} γ γ 1 α γ γ 1 γ γ 1 β γ γ 1 rel {0, 1} α β rel {0, 1} (4) ϕ (surjective) β Π 1 (X, q) γ β γ 1 Π 1 (X, p) ϕ( γ β γ 1 ) = γ 1 γ β γ 1 γ = β Q.E.D

2 R n Convex Set R n C convex set C x, y (1 t)x + ty C (0 t 1) 2.1 C R n convex set A C f : C X, g : C X f(x) = g(x) f g rel {A} ( ) f, g f g F : X I C F (x, t) = (1 t)f(x) + tg(x) (x X, t I) 2.2 C R n convex set Π 1 (C, p) { e p } ( ) α Π 1 (X, p) α : I C e p : I C e p (s) = p (s I) α(0) = e p (0) = p α(1) = e p (1) = p α e p rel {0, 1} Q.E.D X Π 1 (X, p) = {e} X simply connected

3 S 1 X A X X x x N N (A\{x}) 3.1 (Bolzano-Weierstrass ) X compact X A ( ) A X x x A N x (A\{x}) = x N x x O x N x O x O x (A\{x}) N x (A\{x}) = O x (A\{x}) = O x A {x} O = {O x x X} X x X O x X X = x X O x O X open cover X compact x 1, x 2,, x k X X = O x1 O x2 O xk A = A X = A (O x1 O x2 O xk ) = (A O x1 ) (A O x2 ) (A O xk ) = {x 1 } {x 2 } {x k } A Q.E.D 3.2 (Lebergue ) (X, d) compact metric space X open cover O = {O λ λ Λ} δ X A diameter A < δ = A O λ0 λ 0 Λ diameter A = sup{d(x, y) x, y A} δ Leberque ( ) n = 1, 2, X A 1, A 2, diameter A n < 1 n λ Λ A n / O λ A n A n a n P = {a 1, a 2,, a n, } (1) P n 1 < n 2 < a n1 = a n2 = = a nk = = p p X = λ Λ O λ p O λ0 λ 0 Λ O λ0 ε S ε (p) O λ0 Archimedes 1 m 0 < ε m 0 n k0 m 0 n k0 x A nk0 d(x, p) = d(x, a nk0 ) diametera nk0 < 1 n k0 1 m 0 < ε

( ) x S ε (p) A nk0 S ε (p) O λ (2) P X compact Bolzano-Weirestrass P p p X = λ Λ O λ p O λ0 λ 0 Λ ε S ε (p) O λ Archimedes 1 m 0 < ε 2 m 0 ( ) P = {a 1, a 2, } P k = {a k, a k+1, }(k = 1, 2, ) p P p P k p P p P m0 S ε 2 (p) (P m 0 \{p}) a l S ε 2 (p) (P m 0 \{p}) ( l m 0 ) A l A l x d(x, p) d(x, a l ) + d(a l, p) d(x, a l ) diametera l < 1 l < 1 m 0 < ε 2 a l S ε 2 (p) d(a l, p) < ε 2 d(x, p) < ε 2 + ε 2 = ε x S ε (p) A l A l 0 λ0 Q.E.D ( ) = k = 1 = p P p N N (P k \{p}) p O N O ε S ε (p) O ε ε d(a 1, p), d(a 2, p),, d(a k 1, p) 0 < ε ε, ε d(a i, p)(i = 1, 2,, k 1) ( d(a i, p) 0 ) p P S ε (p) (P \{p}) S ε (p) a 1, a 2,, a k 1 p S ε (p) (P k \{p}) S ε (p) S ε (p) O N N (P k \{p}) p Q.E.D

4 S 1 R S 1 = {(x 1, x 2 ) R 2 x 2 1 + x 2 2 = 1} π : R S 1 π(x) = e 2πix (x X) π 4.1 Π 1 (S 1, p) = Z n γ n : I R γ n (s) = ns (1)γ n 0 n R (2)πγ n 1 S 1 loop (3)πγ n n S 1 n n S 1 n ( ) ( 4.2 4.3 ) 4.2 φ : Z Π 1 (S 1, 1) φ(n) = πγ n φ Z = Π 1 (S 1, 1) ( ) φ Π 1 (S 1, 1) α α : I S 1 path α(0) = α(1) = 1 path-lifting lemma α : I R α(0) = 0, π α = α π( α(1)) = α(1) = 1 α(1) π 1 ({1}) α(1) n α 0 n α γ n rel {0, 1} π α πγ n rel {0, 1} α πγ n φ rel {0, 1} φ(n) = πγ n = α φ φ(n) = e 1 = n = 0 φ(n) = πγ n = e 1 πγ n e 1 F : I I S 1 rel {0, 1} F (s, 0) = e 1 (s), F (s, 1) = πγ n (s), F (0, t) = F (1, t) = 1 homotopy-lifting lemma F : I I R π F = F, F (0, t) = 0 P = ({0} I) (I {0}) ({1} I) F (P ) = {1} π F (P ) = F (P ) = {1} F (P ) π 1 (π F (P )) = π 1 ({1}) F (P ) π 1 ({1}) = Z( ) P F (P ) F (P ) F (0, t) = 0 F (P ) = {0} γ(s) = F (s, 1) γ R γ(0) = F (0, 1) = 0, γ(1) = F (1, 1) = 0 πγ(s) = π F (s, 1) = F (s, 1) = πγ n (s) πγ = πγ n γ, γ n πγ n lifting path-lifting γ = γ n γ n (1) = n, γ(1) = 0 n = 0 4.3 φ φ Q.E.D

4.3 φ ( ) γ : I R 0 n R convex set γ γ n rel {0, 1} πγ πγ n rel {0, 1} m, n φ(m + n) = φ(m)φ(n) σ : I R σ(s) = γ n (s) + m (0 s 1) σ(0) = γ n (0) + m = m σ(1) = γ n (1) + m = m + n σ m m + n R γ n σ (γ n σ)(s) = { γn (2s) (0 s 1 2 ) σ(2s 1) ( 1 2 s 1) (γ n σ)(0) = γ n (0) = 0 (γ n σ)(1) = σ(1) = m + n γ n σ 0 m + n γ n σ γ m+n rel {0, 1} π(γ n σ) πγ m+n rel {0, 1} φ(m + n) = πγ m+n = π(γ m σ) = (πγ m ) πσ = πγ m πσ = φ(m) πσ (πσ)(s) = π(σ(s)) = e 2πiσ(s) = e 2πi{γn(s)+m} = e 2πiγn(s)+2πim = e 2πiγn(s) 1 = π(γ n (s)) = (πγ n )(s) (πσ)(s) = (πγ n )(s) φ(m + n) = φ(m) πσ = φ(m) πγ n = φ(m)φ(n) Q.E.D

4.4 (Path-Lifting Lemma) S 1 σ : I S 1 σ(0) = 1 R σ : I R σ(0) = 0, π σ = σ σ σ path-lifting ( ) U = S 1 \{ 1}, V = S 1 \{1} U, V S 1 U V = S 1 n Z A n = (n 1 2, n + 1 2 ), B n = (n, n + 1) (1)π 1 (U) = n Z A n (2)m n A m A n = (3)π An : A n U (4)π 1 (V ) = n Z B n (5)m n B m B n = (6)π Bn : B n V σ : I S 1 I = σ 1 (U) σ 1 (V ) σ 1 (U), σ 1 (V ) I I compact metric space Lebergue 0 = s 0 < s 1 < s 2 < < s r = 1 [s i, s i+1 ] σ 1 (U) [s i, s i+1 ] σ 1 (V ) σ([s i, s i+1 ]) U σ([s i, s i+1 ]) V σ(s 0 ) = σ(0) = 1 / V σ([s i, s i+1 ]) / V σ([s i, s i+1 ]) U π A0 : A 0 U f : U A 0 σ : [s 0, s 1 ] R σ(s) = f(σ(s)) = (fσ)(s) (s [s 0, s 1 ]) σ(0) = f(σ(0)) = f(1) = 0, π σ = π(fσ) = (πf)σ = σ on[s 0, s 1 ] σ σ σ : [s 0, s k ] R σ(0) = 0, π σ = σ σ([s i, s i+1 ]) U σ([s i, s i+1 ]) V σ([s i, s i+1 ]) U σ(s k ) = π( σ(s k )) U σ(s k ) π 1 (U) = n Z A n m n A m A n = σ(s k ) A n π An : A n U g σ : [s k, s k+1 ] R σ(s) = g(σ(s)) = (gσ)(s) (s [s k, s k+1 ]) π σ = πgσ = σ σ : [s 0, s k+1 ] R σ(0) = 0, π σ = σ σ([s i, s i+1 ]) V σ(s k ) = π( σ(s k )) V σ(s k ) π 1 (V ) = n Z B n m n B m B n = σ(s k ) B n π Bn : B n V h σ : [s k, s k+1 ] R σ(s) = h(σ(s)) = (hσ)(s) (s [s k, s k+1 ]) π σ = πhσ = σ σ : [s 0, s k+1 ] R σ(0) = 0, π σ = σ σ [s 0, s k ] [s 0, s k+1 ] Q.E.D

4.5 (Homotopy-Lifting Lemma) F : I I S 1 F (0, t) = F (1, t) = 1 (t I) F : I I R π F = F, F (0, t) = 0(t I) F F homotopt-lifting ( ) U = S 1 \{ 1}, V = S 1 \{1} {U, V } S 1 n Z A n = (n 1 2, n + 1 2 ), B n = (n, n + 1) (1)π 1 (U) = n Z A n (2)m n A m A n = (3)π An : A n U (4)π 1 (V ) = n Z B n (5)m n B m B n = (6)π Bn : B n V {F 1 (U), F 1 (V )} I I I I compact metric space Lebergue 0 = s 0 < s 1 < s 2 < < s m = 1, 0 = t 0 < t 1 < t 2 < < t n = 1 [s i, s i+1 ] [t j, t j+1 ] F 1 (U) [s i, s i+1 ] [t j, t j+1 ] F 1 (V ) F (0, t) = 1(0 t t 1 ) F (0, t) / V [0, s 1 ] [0, t 1 ] F 1 (U) {0} [0, t 1 ] [0, t 1 ] {0} [0, t 1 ] F (0, t) = 1 (π 1 F )({0} [0, t 1 ]) = π 1 ({1}) = Z F : [0, s 1 ] [0, t 1 ] R F (0, 0) = 0 π A0 : A 0 U f f : U A 0 F : [0, s 1 ] [0, t 1 ] R F (s, t) = f(f (s, t))(0 s s 1, 0 t t 1 ) (π F )(s, t) = π(f(f (s, t))) = (πf)(f (s, t)) = F (s, t) [0, s 1 ] [0, t 1 ] π F = F F ({0} [t 0, t 1 ]) π F ({0} [t 0, t 1 ]) = F ({0} [t 0, t 1 ]) = 1 F (0, 0) = 0 F ({0} [t 0, t 1 ]) = {0} F : [s k, s k+1 ] [t 0, t 1 ] R π F = F F (0, 0) = 0 [s k, s k+1 ] [t 0, t 1 ] F 1 (U) [s k, s k+1 ] [t 0, t 1 ] F 1 (V ) [s k, s k+1 ] [t 0, t 1 ] F 1 (U) F ([s k, s k+1 ] [t 0, t 1 ]) U {s k } [t 0, t 1 ] F ({s k } [t 0, t 1 ]) π F ({s k } [t 0, t 1 ]) = F ({s k } [t 0, t 1 ]) U F ({s k } [t 0, t 1 ]) π 1 (U) = n Z A n n Z A n p F ({s k } [t 0, t 1 ]) A p π An : A n U g : U A p F : [s k, s k+1 ] [t 0, t 1 ] A p F = gf π F (s, t) = π An F (s, t) = π An g(f (s, t)) = F (s, t) F : [s k, s k+1 ] [t 0, t 1 ] R π F = F

( ) [s k, s k+1 ] [t 0, t 1 ] F 1 (V ) F ([s k, s k+1 ] [t 0, t 1 ]) V F ({s k } [t 0, t 1 ]) π F ({s k } [t 0, t 1 ]) = F ({s k } [t 0, t 1 ]) V, F ({s k } [t 0, t 1 ]) π 1 (V ) = n Z B n F ({s k } [t 0, t 1 ]) B p p π Bn : B n V h : V B p F : [s k, s k+1 ] [t 0, t l ] R F = hf F F : ([s 0, s m ] [t 0, t l ]) ([s 0, s k ] [t l, t l+1 ]) R [s k, s k+1 ] [t l, t l+1 ] F 1 (U) [s k, s k+1 ] [t l, t l+1 ] F 1 (V ) [s k, s k+1 ] [t l, t l+1 ] F 1 (U) F ([s k, s k+1 ] [t l, t l+1 ]) U P = ({s k } [t l, t l+1 ]) ([s k, s k+1 ] {t l }) π F (({s k } [t l, t l+1 ]) ([s k, s k+1 ] {t l })) = F (P ) U F (P ) π 1 (U) = n Z A n P F (P ) F (P ) A p p π Ap : A p U g : U A p F : [s k, s k+1 ] [t l, t l+1 ] R F = gf F [s k, s k+1 ] [t l, t l+1 ] F 1 (V ) F : ([s 0, s m ] [t 0, t l ]) ([s 0, s k+1 ] [t l, t l+1 ]) R Q.E.D

5 S n (n 2) X Π 1 (X, p) = {e} X simply connected 5.1 n 2 Π 1 (X, p) = {e} ( ) S n x, y U = S n \{y}, V = S n \{x} U R V U, V simply connected {x}, {y} U, V U V U V = S n 5.2 simply connected Q.E.D 5.2 X simply connected U, V U V X simply connected ( ) p U V α Π 1 (X, p) α = e p X = U V α 1 (U) α 1 (V ) = I R I compact {α 1 (U), α 1 (V )} I open cover Lebergue 0 = s 0 < s 1 < s 2 < < s r = 1 [s i, s i+1 ] α 1 (U) α 1 (V ) α([s i, s i+1 ]) U V i = 0, 1,, n 1 α i : I X α i (s) = α((1 s)s i + s s i+1 ) (0 s 1) (1 s)s i + s s i+1 [s i, s i+1 ] α i α(s i ) α(s i+1 ) α i (I) U V α α 0 α n 1 (1) α(s i ) U γ i p α(s i ) U (2) α(s i ) V γ i p α(s i ) V (3) α(s i ) U V γ i p α(s i ) U V γ 0, γ n γ 0 (s) = γ n (s) = p(0 s 1) α α 0 α n 1 γ 0 α 0 γ 1 1 γ 1 α 1 γ 1 2 γ n 1 α n 1 γ 1 n γ i α i γ 1 i+1 p loop α i (I) U γ i α i γ 1 i+1 U loop α i (I) V γ i α i γ 1 i+1 V loop U V simply connected γ i α i γ 1 i+1 e p rel {0, 1} α e p e p e p rel {0, 1} α e p rel {0, 1} Q.E.D

6 6.1 X, Y p X q Y Π 1 (X Y, (p, q)) = Π 1 (X, p) Π 1 (Y, q) ( ) 1. Torus S 1 S 1 Π 1 (S 1 S 1 ) = Z Z 2. Π 1 (S m S n ) = {e}(m 2, n 2) 3. Π 1 (S 1 I) = Z ( ) φ : Π 1 (X Y, (p, q)) Π 1 (X, p) Π 1 (Y, q) φ( α ) = ( P X α, P Y α ) (1) φ well-defined α β rel {0, 1} = P X α P X β rel {0, 1}, P Y α P Y β rel {0, 1} (2) φ homomorphism φ( α α ) = ( (P X α) (P X β), (P Y α) (P Y β) ) = ( P X α P X β, P Y α P Y β ) = ( P X α, P X β )( P Y α, P Y β ) = φ( α )φ( α ) (3) φ injective φ( α ) = ( e p, e p ) = α = e (p.q) φ( α ) = ( P X α, P Y α ) = ( e p, e p ) F : P X α e p rel {0, 1} F : P X α e p rel {0, 1} H : I I X Y H(s, t) = (F (s, t), G(s, t)) H H(s, 0) = (F (s, 0), G(s, 0)) = (P X α(s), P y α(s)) = (P X (α(s)), P Y (α(s))) = α(s) H(s, 1) = (F (s, 1), G(s, 1)) = (e p (s), e q (s)) = (p.q) = e (p,q) (s) H(0, t) = (F (0, t), G(0, t)) = (p, q) H(1, t) = (F (1, t), G(1, t)) = (p, q) H : α e (p.q) α = e (p.q) (4) φ surjective

7 Homotopy Type X Y f : X Y (1) f (2) f (3) f 1 X, Y homotopy type (homotopy equivalent) f : X Y, g : Y X gf 1 X fg 1 Y f homotopy equivalent g homotopy inverse X Y X Y = Π 1 (X, p) = Π 1 (Y, q) 1 X Y = X Y 2 C R n convex set C { } 3 S n 1 R n \{0} (n 2) (S n 1 R n \{0} ) 7.1 ( ) X A X retract γ : X A γ(a) = a (a A) γ retraction 7.2 A X deformation retract A X ( ) i : A X, r : X A x A (ri)(x) = r(i(x)) = r(x) = x = 1 A (x) ri = 1 A F : X I X F (x, 0) = x = 1 X (x) F (x, 1) = r(x) = (ir)(x) F (a, t) = a(a A, t I, x X) 1 A ir A X Q.E.D 7.3 A X deformation retract i : Π 1 (A, p) Π 1 (X, p) p A ( ) i : A X i : Π 1 (A, p) Π 1 (X, p) homomorphism A X deformation retract ri = 1 A ir 1 X rel A ri = 1 A (ri) = (1 A ) r i = 1 Π1 (A,p) 1 Π1 (A,p) i ir 1 X rel A ir 1 X rel {p} (ir) = (1 X ) i r = 1 Π1 (A,p) 1 Π1 (A,p) i Q.E.D

3 S n 1 R n deformation retract 4 B n = {x R n x < 1} S n 1 B n \{0} deformation retract 7.4 f : X Y, g : X Y F : f g g = γ f γ(t) = F (p, t) (0 t 1) ( ) 7.5 X, Y X Y Π 1 (X, p) = Π 1 (Y, q) ( )