Diploma Thesis - Universidad de Buenos Aires

K

K

K
K0
K1

R
K0(R) K1(R)
R

K
K0
(R, +, 0, ·, 1) =: R
R P
M f −→ P → 0
R r
f ◦ r ≡ IdP
r
M
f
P → 0
P R
Q
I
P ⊕ Q ∼ = R (I)
⇒ P {xi}i∈I
R P
R (I)
P ∼ = Im(r ◦ f) (r ◦ f)
M Q IdM − (r ◦ f)
r
f
P
M (r◦f)⊕(IdM −r◦f)
−−−−−−−−−−−−→ Im(r ◦ f) ⊕ Im(IdM − r ◦ f) ∼ = P ⊕ Q
⇐ P ⊕Q ∼ = R (I)
M f −→ P → 0
Q Id
−→ Q f
P ⊕ Q
˜r
R (I)
M ⊕ Q
P ⊕ Q
f⊕IdQ
0

M
π◦˜r
M
f
R (I)
i
P
f
(π · ˜r · i) ◦ f = IdP
R
R n n ∈ N
R
Proj(R)
S
G(S) j : S → G(S)
φ : S → H H
S
∀f ∃! ¯ f :
G(S)
¯f
f
H
j
S
j < j(S) >⊆ G(S)
G(S)
G(S)
(x1, x2) ∈ S × S
(x1, x2) ∼ (y1, y2) ⇐⇒ ∃t ∈ S : t+x1+y2 = t+y1+x2

j x ↦→ (x + p, p) ∈ S × S/ ∼
p ∈ S
¯f(x1, x2) = f(x1) − f(x2)
K0 R
R
K0(R) := G(Proj(R))
E F
[E] = [F ] ∈ K0(R)
Q E ⊕ Q ∼ = F ⊕ Q ∈ Proj(R) Q
[E] = [F ] ⇐⇒ (∃k) E ⊕ R k ∼ = F ⊕ R k
Proj(R)
R n n
Gln(R) n × n
R
n
1
Gln(R) i
Gln+1(R)
A ↦→
A 0
0 1
Gl(R)
R × = Gl1(R) → · · · → Gln(R) → Gln+1(R) → · · · −→ Gl(R)

Mn(R)
M(R)
Mn(R)
i
Mn+1(R)
A
A ↦→
0
0
0
M(R) Idemn(R) ⊆
Idem(R)
f ∈ Idem(R) g ∈ Idem(R)
Im(f) ∼ = Im(g) α ∈ Gl(R) αfα −1 = g
⇐
u ∈ GlN(R) R
⇒ R δ : R n f −→ R m g
R n f
R m
d ∈ R n×m δ −1 =: ɛ e ∈ R m×n
d · e = f
e · d = g
d = f · d = d · g
e = g · e = e · f
(1 − f) f
N = n + m
1 − f d
e 1 − g
2
1n 0
=
0 1m
f g
1 − f
e
d
1 − g
f
·
0
0 1 − f
·
0 e
d
1 − g
=
=
1 − f d
e 1 − g
0 d
·
0 0
=
0 0
0 g
M(R)
Idem(R) Gl(R)

Idem(R) → Proj(R) e ↦→ Im(e)
Idem(R)/Gl(R) ∼ = Proj(R)
e ∈ Idemn(R) f ∈ Idemm(R)
P Q
e 0
e ⊕ f =
0 f
P ⊕ Q
K
Proj(K) = N0
K0(K) ∼ = Z
R
R
M
M ∼
= R/In
n≤N
In ⊂ R
R N → M
R
A = C ∞ (X) X
X
A
C ∞ X
E
(E p −→ X) ↦→ Γ(X, E) = {s : X → E/p ◦ s = IdX} ∈ Proj(A)
E C ∞ (X)
R
A
F → X X E ⊕ F ∼ =

X × R N
Γ(X, E) ⊕ Γ(X, F ) ∼ = Γ(X, E ⊕ F ) ∼ = Γ(X, X × R N ) ∼ = A N
X
C ∞
A x ∈ X
P A
Q P ⊕ Q ∼ = A N P ⊆ A N
f : X → R N
{(x, v) ∈ X × R N /∃f ∈ P f(x) = v}
f1, . . . , fk ∈ P
f1(x), . . . , fk(x)
g1 · · · gN−k ∈ Q Q x
k × k
{fi(x)}i N − k × N − k
{gj(x)}j
f1, . . . , fk
Z → R K0
K0(Z) ∼ = Z i∗
−→ K0(R)
K0 R
˜K0(R) := K0(R)/i∗(Z)
˜ K0(R) = 0
K0
Ri i = 1, 2
Mn(R) Idemn(R) Gln(R)
Idem(R1 × R2) ∼ = Idem(R2) × Idem(R2)
Gl(R1 × R2) ∼ = Gl(R1) × Gl(R2)
K0(R1 × R2) ∼ = K0(R1) × K0(R2)

f : R1 → R2
K0(R1) f∗
−→ K0(R2)
K0
K0
K0
I ⊆ R
R I R × R
D(R, I) = {(x, y) ∈ R × R : x − y ∈ I}
K0
P1 : D(R, I) → R K0 I R
K0(R, I) := Ker{(P1)∗ : K0(D(R, I)) → K0(R)}
π : R → R/I a ∈ R a ∈ Mn(R)
π ā ∈ Mn(R/I)
K0(R, I) (P2)∗
−−−→ K0(R) π∗
−→ K0(R/I)
[e] − [f] ∈ K0(R, I) ⊆ K0(R × R)
e = (e1, e2), f = (f1, f2) ∈ Idem(D(R, I))
K0(R × R) = K0(R) ⊕ K0(R)
[e] = ([e1], [e2])
[f] = ([f1], [f2])
[e] − [f] = ([e1] − [f1], [e2] − [f2]) ∈ K0(R × R)

K0(R, I)
(P1)∗([e] − [f]) = [e1] − [f1] = 0 ∈ K0(R)
I
[ē1] − [ ¯ f1] = 0
ē1 − ē2 = 0
¯f1 − ¯ f2 = 0
[ē2] = [ ¯ f2] ∈ K0(R/I)
(P2)∗(K(R, I)) ⊆ Ker(π∗)
[e] − [f] ∈ K0(R)
0 = π∗([e] − [f]) = [ē] − [ ¯ f]
K0
ē ¯ f
ē⊕1n ¯ f ⊕1m
ē ¯ f
∃¯g ∈ GLn(R/I) / ē = ¯g ¯ f ¯g −1 ∈ Idem(R/I)
R
R/I R
R
Ā ∈ Gln(R/I) =⇒ Ā + Ā−1 ∈ π∗(Gl2n(R))
ē + 0n ē ¯ f
¯g ⊕ ¯g −1 ¯g R
I R π R/I
Ā ∈ Gln(R/I) =⇒ Ā ⊕ Ā−1 ∈ π∗(Gl2n(R))
Ā Ā −1
=
1 Ā
1
1
−( Ā)−1 1
1 Ā
1
1
−1
Ā A Ā−1 N
R

K1
K1
Gln(R)
K1(R) := Gl(R)/[Gl(R), Gl(R)]
eij 1
(i, j)
En(R) Gln(R)
(Idn + α · eij) j = i
En(R) :=< (Idn + α · eij) / α ∈ R, j = i >⊂ Gln(R)
E(R) Gl(R)
E(R) :=
En(R) ⊂ Gl(R)
E(R)
n
⎛
⎜
⎝
1 0
∗ 1
⎞
⎟
⎠ ∈ E(R)
eij(a)
⎛
a ∈
⎞
R i > j
1 ∗
⎜
⎝
⎟
⎠ ∈ E(R)
0 1
0 −1
1 0
∈ E(R)
0
1
−1
0
1
=
0
−1
1
1 0
1 1
1 −1
0 1

A ∈ Gln(R)
A
0
0
A−1
∈ E2n(R)
[Gl(R), Gl(R)] = [E(R), E(R)] = E(R)
E(R) [E(R), E(R)]
eij(α) = [eik(α), ekj(1)]
i, j, k α ∈ R
E(R) ⊆ [E(R), E(R)] ⊆ [Gl(R), Gl(R)]
A, B ∈ Gln(R)
ABA −1 B −1 0
0 1
=
AB 0
0 B −1 A −1
A −1 0
0 A
B −1 0
0 B
E2n(R)
K1(R)
−1 A 0 AB 0 B 0
=
0 B 0 1 0 B
R R × R ×
ab
R × = Gl1(R) → Gl(R)
R ×
ab
K1
K1(R)
˜K1(R) := K1(R)/R ×
ab = Gl(R)/ < [Gl(R), Gl(R)], R× >
R ˜ K1(R) = 1

A ∈ Gl(R) A ∈ Gln(R) n ∈ N
K1
⎛
⎞
⎜
⎝
1p
λ
0
λ −1
0 1q
⎟
⎠ ∈ E2n(R)
1p 1q p + q = 2n − 2
λ ∈ R ×
ab
λ 0
0 1n−1
1 ∈ ˜ K1(R)
R I := R − R ×
R R I
K1(R) ∼ = R ×
ab
R
K1(R)
det : K1(R) → R ×
Gln(R)
R × =
R ×
ab
K1(R)
K1(R)
K1(R) = R × ⊕ ˜ K1(R)
K1(R)
R 2
1

−1 K1(R)
˜ K1(R)
0 1
1 0
=
−1 0
0 1
0 −1
1 0
≡ 1 ∈ ˜ K1(R)
Mk(Mn(R)) = Mnk(R)
Gl(R) ∼ = Gl(Mn(R))
1
MN(R) N ∈ N
n
K1(R) ∼ = K1(Mn(R))
G Z[G]
Z[G]
±g g ∈ G
{±g : g ∈ G} ⊆ Z[G] ×
K1(Z[G])
{±¯g : g ∈ G} ⊆ Z[G] × a b ⊂ K1(Z[G])
G
W h(G) = K1(Z(G))/{±¯g, g ∈ G}
X G = π1(X, x0)
X x0
W h(π1(X, x0)) = W h(X)
x1
W h(π1(X, x0)) ∼ = W h(π1(X, x1))
x0 x1
γ ∈ π1(X, x0) Gl(R)
γ

K1(R)
W h(X)
W h(X) :=
W h(π1(X, p))
p∈π0(X)

R (C•, ∂•) R
Z −1
∂• ◦ ∂• ≡ 0
R Cn
∂n
· · · → Cn+1
∂n+1 ∂n
−−−→ Cn −→ Cn−1 → · · ·
∂n+1 ◦ ∂n ≡ 0
(C•, c•) f•
−→ (D•, d•) R
0
fn
{Cn −→ Dn : n ∈ Z} dn ◦ fn = fn−1 ◦ cn
. . .
Cn+1
. . .
fn+1
cn+1
Cn
fn
dn+1 Dn+1
Dn
cn
Cn−1
fn−1
dn
Dn−1
. . .
. . .
cn : Cn → Cn+1
Bn(C) := Im(∂n+1) ⊆ Ker(∂n) =: Zn(C)
Bn(C) n n Zn(C)
n
Hn(C•) = Zn(C)/Bn(C)
f
f∗
Hn(C•) f∗
−→ Hn(D•)
Hn
R

Zn(C) Hn(C)
Bn(C)
Hn(C) = 0, ∀n ∈ Z
R
M ′
∂ ′
′ N
0
Ker(∂ ′ )
Coker(∂ ′ )
f
f
f
f
M
∂
N
Ker(∂)
S
g
g
M ′′
∂ ′′
′′ N
g
Coker(∂) g
0
′′ Ker(∂ )
′′ Coker(∂ )
f g
S
S = π ′ ◦ (f −1 ) ◦ ∂ ◦ (g −1 )
f g
π ′
∂ ′
0 → C ′ •
f• g•
−→ C• −→ C ′′
• → 0
Hn+1(C ′′ ) ∂ −→ Hn(C)
f∗ g∗
· · · → Hn+1(C ′′ ) ∂∗
−→ Hn(C ′ ) f∗
−→ Hn(C) g∗
−→ Hn(C ′′ ) → · · ·

R
k
0
C ′ k
0
∂ ′
k
C ′ k−1
fk
Ck
∂k
fk−1
Ck−1
gk
gk−1
C ′′
k
∂ ′′
k
C ′′
k−1
0
0
k k−1
0
Ker(∂ ′ k )
Coker(∂ ′ k )
f
f
Ker(∂k)
Sk
g
Coker(∂k) g
Ker(∂ ′′
k )
Coker(∂ ′′
k )
0
0
Coker(∂ ′ k )
∂ ′
k−1
fk−1
Coker(∂k)
∂k−1
gk−1
Coker(∂ ′′
k )
∂ ′′
k−1
Ker(∂ ′ k−2 ) fk−2
Ker(∂k−2) gk−2
Ker(∂ ′′
k−2 )
Ker( Coker(∂k) ∂k−1
−−−→ Ker(∂k−2)) = Hk−1(C)
Coker( Coker(∂k) ∂k−1
−−−→ Ker(∂k−2)) = Hk−2(C)
· · · → Hk−1(C ′ ) f∗
−→ Hk−1(C) g∗
−→ Hk−1(C ′′ ) ∂∗
−→ Hk−2(C ′ ) → · · ·
R
0

(C•, ∂)
Σ k C• k ∈ Z
(−1) k
(Σ k C•)n = Cn−k
∂ Σ k C• = (−1) k .∂C•
∂n
R
( ˘ C•)n := HomR(C−n, R)
−1 (−n+1) (∂−n+1) ∗ =: ˘ ∂n
˘Cn := HomR(C−n, R) (∂−n+1)∗
−−−−−−→ HomR(C−n+1, R) =: ˘ Cn−1
n
n
Σ n ( ˘ C•) =: ˘ C•−n
Σ n ( ˘ C•) ∼ =
˘
(Σ −n C) •
(C, c) (D, d) R S
(C ⊗D, ∂)
R ⊗Z S ∂
(C ⊗D)n =
∂n =
k∈Z
k∈Z
Ck ⊗Z Dn−k
ck ⊗ 1Dn−k + (−1)k 1Ck
⊗ dn−k
∂• ◦ ∂• = 0
Ck ⊗ Dn−k
∂n−1 ◦ ∂n = ∂n−1 ◦ {ck ⊗ 1Dn−k } + ∂n−1 ◦ {−1k 1Ck ⊗ dn−k}
= (ck−1 ⊗ 1Dn−k ) ◦ (ck ⊗ 1Dn−k )
+ (ck ⊗ 1Dn−1−k ) ◦ ( (−1)k · 1Ck
⊗ dn−k)
+ ( −1k−1 · 1Ck−1 ⊗ dn−k ) ◦ (ck ⊗ 1Dn−k )
+ ( −1k−1 · 1Ck−1 ⊗ dn−k ) ◦ ( (−1) k 1Ck ⊗ dn−k)
c• d•

R (C, c)
(D, d) (Hom(C, D), ∂)
Hom(C, D)n :=
HomR(Ck−n, Dk)
k∈Z
f ∈ HomR(Ck−n, Dk)
∂n(f) = dk ◦ f − (−1) n f ◦ (ck−n+1)
D R
0 C
C D R D
D ⊗R HomR(C, R) −→ HomR(C, D)
x ⊗ φ ↦→ (x ⊗ φ) (y) = φ(y) · x
D
D ⊕ Q ∼ = R n {(d1, q1), . . . , (dn, qn)}
f ∈ HomR(C, D) φk C R k = 1, . . . , n
k dk ⊗ φk
f
(C, c) (D, d) R D
Hom
C D
Hom(C, D) = (D• ⊗R ˘ C•)
D•⊗ ˘ C• =
Dk ⊗ ( ˘ C•)n−k =
HomR(Ck−n, Dk) := Hom(C, D)n
k∈Z
k∈Z
∂n(f) = dk ◦ f + (−1) k 1D (−1) k−n+1 (ck−n+1) ∗ ◦ f
= (dk ⊗ 1C ′ + (−1)k ⊗ (c ′ )n−k)(f)
Hom
0 z ∈ (Hom(C, D))0 =
k HomR(Ck, Dk)
∂(z) = d ◦ z − z ◦ c = 0
f −g = ∂(h)

f ∼ = g f ∼ =h g
h ∈
k∈Z Hom(Ck+1, Dk)
[C•, D•] C• D•
H0(Hom(C, D)) = [C•, D•]
Hom
Hom(C, D)n =
Hom(Ck−n, Dk) =
k∈Z
k∈Z
Hom((Σ n C)k, Dk) = Hom(Σ n C, D)0
Hom Hom
Hn(Hom(C•, D•)) = [Σ n C•, D•]
f g [f] = [g] ∈
[C•, D•]
[C•, D•] → HomR(Hn(C), Hn(D))
[f] ↦→ {Hn(C•) f∗
−→ Hn(D•)}
fn − gn = ∂(h) = h ◦ cn + dn+1 ◦ h
h•
f g
cn
hn
Cn
dn+1
Dn+1
Dn
f−g
hn−1
Cn−1
∂(h) ∗ = (d ◦ f − f ◦ c)∗ = 0
Hn(D)
f ∼ = 0
(C, ∂) Sop(C) :=
{n ∈ Z/Cn = 0} Z

R
R
R
C IdC ≡ 0
Hn(C) = Id∗(Hn(C)) = 0 C
R
. . . → Cn
∂n
∂1
−→ . . . → C1 −→ C0 → 0
H0(C) = 0 C0 C0
r0
∂1 ◦ r0 = 1C0 r0 ◦ ∂1 ∈ Idem(EndR(C1))
Ker(∂1) ⊕ r0◦∂1(C1) = C1
Ker(∂1) ∂2
Ker(∂1) H1(C) = 0
rn ∂n+1
δ
δn = rn ⊕ 0 sobre Ker(∂n) ⊕ rn−1◦∂n(Cn)
(∂n+1 ◦δn) Ker(∂n) (δn−1 ◦∂n)
IdCn = δn−1 ◦ ∂n + ∂n+1 ◦ δn
R
f• : C• → D•
f : C• → D•
(f∗)n ∀n ∈ Z

f• : C• → D•
g• : D• → C•
+1 h• : C• → C•+1 h ′ • : D• → D•+1
g ◦ f − IdC = ∂ ◦ h + h ◦ ∂
f ◦ g − IdD = ∂ ◦ h ′ + h ′ ◦ ∂
f
Z
p 2
Zp 0
Zp Z p 2 p
Z p 2 p
. . . 0 → Zp → Z p 2
µp
−→ Z p 2 → · · · → Z p 2 . . .
δ x ∈ Z p 2 n
x = (µp ◦ δn + δn−1 ◦ µp)(x)
= p · (δn + δn−1)(x)
x = p 2 · (δn + δn−1) 2 (x) = 0 ∀x

f•
C• −→ D•
△(f)n = Cn−1 ⊕ Dn ∂n = (−cn−1, fn−1 + dn)
−cn−1
∂n(x, y) =
0
x
·
y
fn−1 dn
f•
C• −→ D•
0 → (D, d)•
i
p
−→ △(f)• −→ σ 1 C• → 0
. . . → Hn+1(△(f)) → Hn(C) ∂∗
−→ Hn(D) → Hn(△(f)) → . . .
∂∗ f
[x] ∈ Hn(C)
x
(p∗ −1 )(x) = (x, y)
∂n
∂n(x, y) = (−cn−1(x), fn−1(x) + dn(y) ) = ( 0 , fn−1(x) + dn(y) )
∂n−1([x]) = [fn−1(x) + dn(y)] = [fn−1(x)] = fn−1([x]) ∈ Hn(D)
f △(f)
f △(f)
∂∗ = f∗
⇐ f δ △(f)
g : D → C h h ′ g ◦ f ∼ =h 1C f ◦ g ∼ =h ′ 1D
δ ◦ ∂ + ∂ ◦ δ = 1C ⊕ 1D

δ(x, 0) = (h(x), s(x))
δ(0, y) = (g(y), −h ′ (y))
(x, 0) = (δ ◦ ∂ + ∂ ◦ δ)(x, 0)
= δ(−c(x), f(x)) + ∂(h(x), . . .)
= (−h ◦ c(x) + g ◦ f − c ◦ h(x), . . .)
h (g ◦f)
C
(0, y) = δ(0, d(y)) + ∂(g(y), −h ′ (y))
= (g ◦ d(y), −h ′ ◦ g(y)) + (−c ◦ g(y), f ◦ g(y) − d ◦ h ′ (y))
= (g ◦ d(y) − c ◦ g(y) , − h ′ ◦ g(y) + f ◦ g(y) − d ◦ h ′ (y))
g
h ′ (f ◦g)
D
⇒
δ(x, y) = ((h(x)+g(y)+g◦h ′ ◦f(x)+g◦f ◦h(x), −h ′ (y)+h ′ ◦f ◦h(x)−(h ′ ) 2 ◦f(x))
C D R
f : C → D R
C D
f
f
C 1 ∼ = C 2 D 1 ∼ = D 2
C i ⊗ D i
D 1 = D 2 = D C i
D i
( ¯ C, c) C 1 φ −→ C 2
s
1 ¯ C = s ◦ c + c ◦ s

(s ⊗ 1D)
(φ ⊗ 1D)
D
C 1 ⊗ D φ⊗IdD
−−−−→ C 2 ⊗ D
n
(C 1 ⊗ D)n−1 ⊕ (C 2 ⊗ D)n =
k∈Z (C1 n−1−k ⊗ Dk) ⊕ (C2 n−k
=
k∈Z (C1 n−1−k ⊕ C2 n−k ) ⊗ Dk)
∂ ¯ C⊗D = c ⊗ 1D + −1 grad( ¯ C) · 1 ¯C ⊗ d
⊗ Dk)
∂m(φ ⊗ 1d) = −(c 1 ⊗ 1D + −1 grad(C1 ) · 1C 1 ⊗ d) . . .
⊕ (φ ⊗ 1D) + (c 2 ⊗ 1D + −1 grad(C2 ) · 1C 2 ⊗ d)
= (−c 1 ⊕ (φ + c 2 )) ⊗ 1D + (−1 grad(C1 )−1 · 1C 1 ⊕ −1 grad(C2 ) · 1C 2) ⊗ d
= c ⊗ 1D + −1 grad( ¯ C) · 1 ¯C ⊗ d
grad( ¯ C) = grad(C 1 ) + 1 = grad(C 2 )
R
K0(R)
R C
χ(C)
K0(R) Cn
χ(C) =
(−1) n [Cn] ∈ K0(R)
k∈Z

χ
C
χ(C ⊕ D) = χ(C) + χ(D)
χ(Σ n C) = (−1) n χ(C)
χ(C•, ∂) = χ(C•, 0)
0 → C• → D• → E• → 0
χ(D) = χ(C) + χ(E)
0 → Cn → Dn → En → 0 En
Dn ∼ = Cn ⊕ En
(−1) n [Dn] =
(−1) n [Cn] + [En] ∈ K0(R)
n
n
C R
Hn(C)
χ(C) =
(−1) n [Hn(C)]
k∈Z
Z0 = C0 B0
n = 0
0 → Bn → Zn
p
−→ Hn → 0
Z1
n = 1
0 → Zn → Cn
∂n
−→ Bn−1 → 0
Bn
Zn+1
K0(R)
[Cn] = [Zn] + [Bn−1]

[Hn] + [Bn] = [Zn]
[Cn] = [Hn] + ( [Bn−1] + [Bn] )
χ(C) =
(−1) n [Hn]
k∈Z
C R
χ(C) = 0
f : C → D
R
C• R
R C ′
C
χ(C) := χ(C)
χ
R (C, ∂)
Hn(C)
n ∈ N
⇐
Cn Zn R
Hn(C) = Zn/Bn
⇒
C0 = Z0 H0 = Z0/B0 =< ¯z1, . . . , ¯zk > H0
k
R k → H0 H0
R
z1, . . . , zk ∈ C0
R k φ −→ C0
0
Ker(φ)
k R
φ0
H0
0

q R
Rk ¯φ0
H0
0
C ′ • φ C•
0 → C
φ : C ′ • → C•
C ′ 0 = R k
C ′ 1 = R q
φ0 := φ φ1
φ1 : C ′ 1 → C1
φ1(ej) = uj uj ∈ C1
∂(uj) = φ0 ◦ ∂(ej) ∀j ≤ q
φ0
C
0 (φ0)∗
· · ·
· · ·
C2
0
∂
C1
φ1
′ C 1
∂
C0
φ0
∂
′ C 0
k ≤ N − 1
Hk+1(C)
n = nk+1
R n → Ck+1
0
0
n ′
Rn′ ∂
n R
Hk+1
R n′ ∂ −→ R n
k + 1 k + 2 C ′
0

φ
k + 1 k + 2
· · ·
· · ·
Ck+3
0
Ck+2
φk+2
C ′ k+2
∂
Ck+1
φk+1
∂
C ′ k+1
· · ·
· · ·
C0
φ0
′ C 0
N C ′ N+1
N − 1 N
φ △(φ) =: △ k = N + 1
Hk(△) = 0
∂ ′′ N+1
HN+1(△) △N+1 = C ′ N
HN+1(△) = Ker(∂ ′′ N+1 ) C′ N
∂ ′ C ′ • φN
C ′ N
C ′ • C•
C R
R
K0
˜χ(C) := [χ(C)] = 0 ∈ ˜ K0(R)
C R ˜ K0(R)
˜χ(C) =
(−1) n [Cn] = 0 ∈ ˜ K0(R)
k∈Z
C
. . . → Cn
∂n
∂1
−→ . . . → C1 −→ C0 → 0
0
0

Qn Cn Cn ⊕ Qn =: Fn ∼ = R m
Qn ⊗ Σ n ∆(Id : R → R)
. . . → 0 → Qn
Id
−→ Qn → 0 → . . .
n n−1
0 → Fn → . . . F0 → Q0 → 0
Fi = Ci ⊕
Qi ⊕ Qi+1
[Qn] = −[Cn] ∈ ˜ K0(R)
[Qj] = −([Cj] + [Qj+1]) j = 0, . . . n − 1
Q0
[Q0] = −[C0] + ([C1] − [Q2]) + . . . = −
(−1) n [Cn] = 0 ∈ ˜ K0(R)
m ∈ N Rm ⊕ Q0 ∼ = Rk
k∈Z
. . . → 0 → R m Id
−→ R m → 0 → . . .
0 −1
R
R P
· · · → 0 → P → 0 → · · ·
· · · → 0 → F s −→ F → 0 → · · ·
Q ⊕ P F
s
F = P ⊕ Q ⊕ P ⊕ Q ⊕ · · ·
s(p0, q1, p1, q2, . . . ) = (p1, q1, p2, q2, . . . )
s Ker(s) = P

R S
C D
C ⊗Z D
R⊗Z S
R ⊗ S
K0
K0(R) × K0(S) → K0(R ⊗ S)
([P ], [Q]) ↦→ [P ⊗ Q]
χ(C ⊗ D) =
j (−1)j [
k (Cj−k ⊗ Dk)] ∈ K0(R ⊗ZS)
=
j, k (−1)j · [Cj−k ⊗ Dk]
=
j, k (−1)j · [Cj−k] · [Dk]
=
k (
j (−1)j · [Cj−k]) · [Dk]
χ(C ⊗ D) = χ(C) · χ(D)
D Z χ(D) K0(Z) = Z
K0
˜χ(C ⊗Z D) = ˜χ(C) · χ(D) ∈ ˜ K0(R)
f
R
(C•, ∂•) δ

(∂ + δ)
Cpar =
n
Cimpar =
n
C2n
C2n+1
χ(C) = 0
rg(Cpar) = rg(Cimpar)
Cpar
(∂•+δ•)par
−−−−−−−−→ Cimpar
R
(∂• + δ•) 2 := (∂• + δ•)impar ◦ (∂• + δ•)par
= ∂ 2 + δ ◦ ∂ + ∂ ◦ δ + (δ) 2
= 1C + (δ) 2
(∂• + δ•) 2 = 1 ∈ K1(R)
δ
δ 2 +2
1C + (δ) 2
K1(R)
[(∂ + δ)par] = [(∂ + δ)impar] −1 ∈ K1(R)
R
˜τ(C) = [(∂ + δ)par] ∈ K1(R)
δ
C
f +g = 1C g◦f = 0
f + g = 1C = (f + g) 2 = f 2 + f g + g f + g 2 = f ◦ (f + g) + g 2 = f + g 2
=⇒ g = g 2 , f = f 2
1Cn = (∂n+1 ◦ δn) + (δn−1 ◦ ∂n)

Bn = Im(∂n+1) Cn
pn = ∂n+1 ◦ δn
˜ Bn−1 = Im(δn−1 ◦ ∂n)
Cn = Bn ⊕ ˜ Bn−1 ∂( ˜ Bk) = Bk
Cn
(pn⊕∂n)
−−−−−→ Bn ⊕ Bn−1
Bn Cn
in ⊕ δn−1
C•
Bi
C• R
¯ C•
C B ′ i τ(C) = τ( ¯ C)
0 → Ck → · · · → C2
∂2 ∂1
−→ C1 −→ C0 → 0
Ci B0 = C0
Bi
K0
0 = [C1] = [B0] + [B1] = [B1] ∈ ˜ K0(R)
F B0 R
F ⊕ B1 ∼ = R N
· · · → 0 → F IdF
−−→ F → 0 → · · ·
R ¯ C•
· · ·
C2 ⊕ F ∂2⊕IdF
C1 ⊕ F
δ⊕IdF
B ′ 1
∂1⊕0
C0 · · ·
B ′ 1 = Ker( ¯ ∂1) = Im( ¯ ∂2) = B1 ⊕ F ∼ = R N
Ci

F
(∂ + δ)par
K1
τ( ¯ C•) = [ (∂ + δ)par ]
= [(∂ + δ)par ⊕ IdF ]
= [(∂ + δ)par] · [IdF ]
= [(∂ + δ)par]
τ( ¯ C•) = τ(C•) ∈ K1(R)
Bi R
R [pn ⊕ ∂n] ∈ K1(R)
p ′ n
Ker(pn − p ′ n) ⊇ Bn
∂n pn − p ′ n = u ◦ ∂n
pn
1 u
0 1
∂n
=
1 u
0 1
p ′ n
K1(R)
∂n
[pk ⊕ ∂k] = [p ′ k ⊕ ∂k] ∈ K1(R)
K1(R)
[pk ⊕ ∂k] (−1)k
∈ K1(R)
k
Bi pi ⊕ ∂i−1
pi⊕∂i K1
R
τ(C) =
[pn ⊕ ∂n] (−1)n
∈ K1(R)
n

L
k (pk⊕∂k)
−−−−−−−→
(Bk ⊕ Bk−1)
Cpar
k
(Bk ⊕ Bk−1)
k
L
−1
k (pk⊕∂k)
−−−−−−−−−→ Cimpar
(pk ⊕ ∂k) −1 = ik ⊕ δk−1
(Bk ⊕ Bk−1)
k
L
k (δk⊕ik−1)
−−−−−−−−−→ Cimpar
K1(R)
pk = ∂k+1 ◦ δk
k
(δk ◦ ∂k+1 ◦ δk) ⊕ ∂k) = ((δ ◦ ∂ ◦ δ) ⊕ ∂) par
¯ δ := δ ◦ ∂ ◦ δ C• δ
( ¯ δ ⊕ ∂)par K1
τ(C) = [( ¯ δ + ∂)par] ∈ K1(R) ¯ δ
( ¯ δ + ∂)impar ◦ ( ¯ δ ⊕ ∂)par = ¯ δ2 + ∂ ◦ ¯ δ ⊕ ¯ δ ◦ ∂ + ∂2 = ¯ δ2 + IdB• ⊕ IdB•−1 ˜
= ( ¯ δ2 ⊕ 0) + IdC•
( δ ¯ + ∂)impar · ( δ¯ ⊕ ∂)par = 1 ∈ K1(R)
pn δ
R
u•
A• −→ B•
τ(A) · τ(B) −1 =
[uk] k
∈ K1(R)
k

u• δ
u δ ˜ δ B
˜δ• = u•+1 ◦ δ• ◦ u• −1
· · · A•+1
u
δ
∂ A
· · · B•+1 ∂ B
1Bk = uk · 1Ak
A• · · ·
u
B• · · ·
· u−1
k
= uk(δ · ∂A + ∂A · δ)u −1
k
= (uk δ u −1
k−1 ) · (uk−1 ∂A u −1
k
) + · · ·
· · · (uk ∂ A uk+1 −1 ) · (uk+1 δ u −1
k )
= ˜ δk−1 · ∂ B k + ∂B k+1 · ˜ δk
B
τ(B) = [(∂ B + ˜ δ)par] =
1Bk = ˜ δ · ∂ B + ∂ B · ˜ δ
τ(A) · τ(B) −1 =
k
[u2k+1] · [u2k] −1 · [(∂ A + δ)par]
k
[u2k+1] −1 · [u2k]
0 → (C ′ ; β ′ ) ι −→ (C; β) π −→ (C ′′ ; β ′′ ) → 0
ι(β ′ ) ⊂ β
β” = π(β \ ι(β ′ ))
R
0
′ C •
j
C•
p
′′ C •
τ(C ′ ) · τ(C) −1 · τ(C ′′ ) = 1 ∈ K1(R)
0

τ(C ′ •) · τ(C ′′ •) = τ(C ′ ⊕ C ′′ )•
β
β, β ′′
C•
Id⊕π
−−−→ (C ′ ⊕ C ′′ )•
[(j ⊕ p)k] = 1 ∈ K1(R)
τ(C ′ ⊕ C ′′ )• = τ(C•)
f : C• → D•
R
f
τ(f) := τ(∆(f))
R C• D• E•
f∗ g∗ h∗
0
0
′ C •
f∗
C•
′ D •
g∗
D•
h∗
E ′ •
E•
τ(f∗) · τ(g∗) −1 · τ(h∗) = 1 ∈ K1(R)
f, g : C → D R
f ∼ = g
0
0
τ(f∗) = τ(g∗)
f∗ : C• → D• g∗ : D• → E•
τ(g∗ ◦ f∗) = τ(g∗) · τ(f∗)

R
f
g h
F• G• f∗ g∗
C•−1 ⊕ D•
h
1 0
h 1
1
h
0
1
−∂C
·
f∗
0
−∂D
−∂C
=
g∗
0
∂D
1
·
h
0
1
f∗ g∗
h∗ : Σ−1△(g∗) → △(f∗)
0
−IdD
0
: Dk ⊕ Ek+1 → Ck−1 ⊕ Dk
0
△(h∗)k = Dk−1 ⊕ Ek ⊕ Ck−1 ⊕ Dk
0 → △(f∗) → △(h∗) → △(g∗) → 0
0 → △(g∗ ◦ f∗) ī −→ △(h∗) → △(IdD) → 0
D
⎛
⎜
[ ī ] = ⎜
⎝
f
0
IdC
0
IdE
0
⎞
⎟
⎠
0 0
: C•−1 ⊕ E• −→ D•−1 ⊕ E• ⊕ C•−1 ⊕ D•

△(h∗)
τ(h∗) = τ(f∗) · τ(g∗)
τ(h∗) = τ(g∗ ◦ f∗) · τ(IdD)
τ(g∗ ◦ f∗) = τ(f∗) · τ(g∗)

f : (X, A) → (Y, B)
f : X → Y A B
f |A: A → B
f, g : (X, A) → (Y, B)
H
H : X × I → Y
H | X×{0}≡ f
H | X×{1}≡ g
H | A×{0}≡ f |A
H | A×{1}≡ g |A
[(X, A), (Y, B)]
(X, A) (Y, B)
[(X, A), (Y, B)] := {γ : (X, A) → (Y, B)}/ ∼
X Y
F G
X
F
G
Y
G ◦ F ∼ =h IdX
F ◦ G ∼ =h ′ IdY
X Y
F G
I = [0, 1] I n
n n ≥ 1 ∂I n I 0
∂I 0 = ∅ I n = ∅ n
n X
x0 ∈ X
πn(X, x0) := [(I n , ∂I n ), (X, x0)]

f, g ∈ πn(X, x0)
f(2 · s1, ..., sn) s1 ≤ 1/2
f · g(s1, ..., sn) =
g(2 · s1 − 1, ..., sn) s1 ≥ 1/2
πn n ≥ 2
f · g =: f + g
k
f : (X k , X k−1 , X k−2 , . . . ) → (Y k , Y k−1 , Y k−2 , . . . )
f |X q: Xq → Y q k
n πn(X, A, x0) x0 ∈
A ⊆ X
πn(X, A, x0) := [(I n , ∂I n , J n−1 ), (X, A, x0)]
J n−1 := ∂I n \ I n−1◦ I n−1 ∂I n
x0 ∈ B ⊂ A ⊂ X
· · · → πn(A, B, x0) i∗
−→ πn(X, B, x0) j∗
−→ πn(X, A, x0) ∂ −→ πn−1(A, B, x0) → · · ·
i∗ j∗
∂ ∂I n (X, A) (Y, B) f : (X, A) →
(Y, B)
f∗ : πn(X, A, a0) → πn(Y, B, f(a0))
γ ∈ πn(X, A, a0) ↦→ f ◦ γ ∈ πn(Y, B, f(a0))
f, g : (X, A) → (Y, B)
f∗, g∗
f : X → Y k
f∗ : πn(X, x0) → πn(Y, f(x0))
n k n = k
k k ∈ N0

f : X → Y
f
f f∗ : S•(X) → S•(Y )
g f
f ◦g g◦f f∗
Id∗ = (f ◦ g)∗ = f∗ ◦ g∗
Sn(X) Z
n < 0
f ∈ πn(X, A, x0) = [(I n , ∂I n , J n−1 ), (X, A, x0)]
f∗ : Hn(I n , ∂I n ) → Hn(X, A)
α ∈ Hn(I n , ∂I n ) ∼ = Z
h
h : πn(X, A, x0) → Hn(X, A)
[f] ↦→ f∗(α)
f∗
α
(X, A)
f∗ : Hn(I n , ∂I n ) → Hn(X, A) n > 1
(X, A)
· · ·
· · ·
πn(X, x0)
Hn(X)
j∗
πn(X, A, x0)
p∗
Hn(X, A)
∂
πn−1(A, x0)
∂
Hn−1(A)
· · ·
· · ·

(X, A) n πq(X, A, x0)
q ≤ n x0 ∈ A
(X, A) (n − 1)
n ≥ 2 A = ∅
i n
h : πn(X, A, x0) → Hn(X, A)
Hi(X, A) = 0
Y ⊆ X
Y X
f : X → Y
f∗ : Hn(X) → Hn(Y ) n ∈ N
X f
Y
˜ f
˜X
˜f
Y˜
˜ f π
f X
f ◦ PX : ( ˜ X, p) → (Y, f(p))
p ∈ P −1
X (p)
PX
f
˜X
Y˜
X
f
Y
PY

f(p) ∈ P −1
Y (f(p)) ⊂ ˜ Y
γ ∈ π1(X) ˜ X
˜ Y
f ◦ γ ˜ Y
γ · p = p ′
γ · f(p) = f(p) ′
f(p ′ ) = f(p) ′
f
f ◦ PX : ( ˜ X, p ′ ) → (Y, f(p))
f(p) ′
∈ ˜ Y f ′
f ′ (p ′ ) = f(p) ′
f ≡ f ′
f(γ · p) = f(p ′ ) = f(p) ′
π
= γ · f(p) = γ · f(p)
g f
g ◦ f ∼ =H IdX f ◦ g ∼ =H ′ IdY
H p ∈ X ˜p ∈ P −1 (p)
(˜p, 0) ∈ ˜ X × I
P ×Id
(p, 0) ∈ X × I
H
H
X˜ ∋ f(p) ˜
X ∋ f(p)
˜ X × I
H0 ≡ Id ˜ X
H1 ≡ (g ◦ f)
H0 H0 ≡ IdX Id ˜ X H1 H1 = (g ◦ f)
g ◦ f ≡ (g ◦ f)
˜ X ˜ Y

F : X → Y
π Z[π]
S•( ˜ X) F∗
−→ S•( ˜ Y )
F
˜ F Z
˜ F π
Z[π]
X
X0 ⊆ X1 ⊆ X2 ⊂ · · ·
α∈An Sn−1
α∈An Dn
∪φα
∪Φα
Xn−1
Xn
˜ X
Xn ( ˜ X)n := P −1 (Xn)
X
˜ X X | π |
π
π ˜ X → X
α∈An
α∈An
π × Sn−1
π × Dn
∪ ¯ φα
∪ ¯ Φα
Xn−1
˜
Xn
˜
˜ X Z
X π
Z[π]
n |An |=: an
C•( ˜ X; Z[π]) ∼ = Z[π] an

X
Z
Hn(Xn, Xn−1)
α∈An
Hn(D n , S n−1 ) ⊕α(Φα,φα)
−−−−−−−→ Hn(Xn, Xn−1)
Hn(Xn, Xn−1)
−1
α∈An
Hn(π × (D n , S n−1 ))
⊕α( Φα, ˜ ˜ φα)
−−−−−−−→ Hn( ˜ Xn, ˜ Xn−1)
Hn(D n , S n−1 )
π
π α ∈ An
Z[π] Z[π]
Hn( ˜ Xn, ˜ Xn−1)
±g ∈ π
π
X W
f g
Id
X
f
g
X
W
X ∼ =h g ◦ f
X
X W
W
X
i
r
W
r ◦ i = IdX

S
S•(W ) r∗
−→ S•(X) → 0
S•(X)
W
X
X
R C R
D R C
f : C → D g : D → C g ◦ f ∼ = 1C C
X W
W
(S•( ˜ W ), f∗, g∗, ¯ h) S•( ˜ X) Z[π]
R
R
C R
χ(C) ∈ K0(R)
X
S•(X)
Z[π]
χ(X) := χ(S•(X)) ∈ K0(Z[π])
˜χ(X) := ˜χ(S•(X)) ∈ ˜ K0(Z[π])

X π
X
S•( ˜ X) Z[π]
X S•( ˜ X)
Z[π1(X)]
S•( ˜ X) X
X Y S•( ˜ X)
Y
Z[π]
Z[π] S•( ˜ X)
X Y X
Y
Z[π]
S•( ˜ X)
X Y
f : X → Y
˜ f
˜X
˜f
X
f
˜ f∗
Z[π]
Y˜
Y

τ( ¯ f∗) ∈ K1(Z[π])
f
τ(f) ∈ W h(π)
τ(f) = 1
K1(Z[π]) W h(π)
±g ∈ π
i
A f
X g
A B X f i
Y Yn j(Bn) g(Xn)
Z
0 → C•(A)
B
Y
j
f∗⊕ i∗
−−−−→ C•(B) ⊕ C•(X) j∗⊕g∗
−−−−→ C•(Y ) → 0
j ⊔ g Y
Y ∼ = (B ⊔ X)/(f ⊔ i)A
ĩ
Ã
˜f
˜X ˜g
A γ ∈ π1(A, a0)
˜ f f∗(γ) ∈ π1(B, f(a0)) B
j : B → Y Z[π1(B)] → Z[π1(Y )]
Z[π1(Y )] Z[π1(B)]
˜B
Y˜
C( ˜ B) Z[π1(Y )] = C( ˜ B) ⊗ Z[π1(B)] Z[π1(Y )]
˜j

Z[π1(B)]
Z[π1(Y )] C( ˜ X) Z[π1(Y )] C( Ã) Z[π1(Y )]
g g ◦ i = j ◦ f
Z[π(Y )]
0 → C•( Ã) Z[π(Y )] → C•( ˜ B) Z[π(Y )] ⊕ C•( ˜ X) Z[π(Y )] → C•( ˜ Y ) → 0
fi : Xi → Yi
fj ◦ ij = kjfj j = 1, 2 f : X → Y
l0 = l1k1 = l2k2 f
i2
X0
X2
i1
j2
X1
j1
k2
Y0
X Y2
k1
fi : Xi → Yi i = 0, 1, 2
f : X → Y
τ(f) = l1∗τ(f1) · l2∗τ(f2) · l0∗τ(f0) −1
l0 = l1 ◦ k1 = l2 ◦ k2
l2
Y1
Y
l1
∈ W h(Y )
A B f, g : A → B
f ∼ = g
τ(f) = τ(g)
f : X → Y g : Y → Z
τ(g ◦ f) = τ(g) · g∗τ(f) ∈ W h(Z)
f : A ′ → A g : B ′ → B
a0 ∈ A b0 ∈ B i : A → A × B, i(a) =
(a, b0) j : B → A × B j(b) = (a0, b)
τ(f × g) = (i∗τ(f)) χ(Y ) · (j∗τ(g)) χ(X)

W h(Y ) j∗
−→ W h(X × Y )
x0 ∈ X i∗
A B χ(A) χ(B)
Z
0
0
C•( ˜ X0)
( ˜ f0)∗
C•( ˜ Y0)
C•( ˜ X1) ⊕ C•( ˜ X2)
( ˜ f1)∗⊕(f2)∗
C•( ˜ Y1) ⊕ C•( ˜ Y2)
C•( ˜ X)
˜f∗
C•( ˜ Y )
Z[π(Y )]
π =
π(X)
0 → △(f0∗) Z[π] → △(f1∗) Z[π] ⊕ △(f2∗) Z[π] → △(f∗) → 0
Z[π]
˜ f∗ ˜ f
(li)∗ i = 0, 1, 2
τ(△(fi∗) Z[π]) = (li)∗τ(fi)
R
τ(f) = (l1)∗τ(f1) · (l2)∗τ(f2) · ((l0)∗τ(f0)) −1 ∈ W h(Y )
g∗ W h(Y ) W h(Z)
f × g = (f × IdY ) ◦ (IdX ′ × g)
τ(f × g) = τ(f × IdY ) · (f ×IdY )∗τ(IdX ′ × g)
0
0

τ(f × IdY ) = i∗(τ(f) χ(Y ) ) ∈ W h(X × Y )
Y
Y
X × Y = X ⊔ · · · ⊔ X
τ(f × IdY ) = τ(f) ⊕ · · · ⊕ τ(f) ∈ W h(X × Y ) :=
W h(X)
Sk−1 φ
Y
Dk Φ
φ Y
f : X ′ → X
X ′ × Sk−1 (Id,φ)
X ′ × D k
X ′ × Y
(Id,i)
′ ′ X × Y
(Id,Φ)
i
′ Y
y∈Y
X × Sk−1 (Id,φ)
X × Dk (Id,Φ)
X × Y
(Id,i)
′ X × Y
τ(f ×IdY ′) = (Id, i)∗τ(f ×IdY )·(Id, Φ)∗τ(f ×Id D k)·(Id, i◦φ)∗τ(f ×Id S k−1) −1
τ(f × IdY ) = i∗(τ(f) χ(Y ) )
τ(f × Id D k) = i∗(τ(f) χ(Dk ) )
τ(f × Id S k−1) = i∗(τ(f) χ(Sk−1 ) )
X
Y ′
i∗ : W h(X) → W h(X × Y ′ )
τ(f × IdY ′) = i∗τ(f) χ(Y )+χ(Dk )−χ(S k−1 ) ∈ W h(Y ′ )
Y ′
χ
τ(f × IdY ′) = i∗τ(f) χ(Y ′ ) ∈ W h(Y ′ )

f : X → Y
τ(f) = 1 ∈ W h(π)
π1(X) ∼ = π1(Y ) =: π
(X, Y ) X
Y
e k e k+1
S k−1
D
k
Y
Y ∪Sk−1 Dk
S k
D
k+1
g
Y ∪Sk−1 Dk g (D k ) ◦
g −1 ((D k ) ◦ ) S k e k+1 g
X Y
Y X
Y X
X

C•( ˜ X, ˜ Y )
W h(X)
f : X → Y
⇐
C•(X, Y ) k k + 1
C•( ˜ X, ˜ Y )
· · · 0 → Z[π] ∂ −→ Z[π] → 0 → · · ·
e k+1
e k k Y
e k ∂ 1 ∈ Z[π] (k + 1) ±g
W h(π)
⇒
(X, Y )
Y π
w ∈ W h(π) X Y
w
w ∈ W h(π)
M = (mij)i,j ∈ Glq(Z[π])
n ≥ 2 y ∈ Y q
n
Y ′ := Y ∨
q n y ∈ Y
πn(Y ′ , y)
Z[π]
i≤q
S n i

y ∈ Y ′
[S n i ] ∈ πn(Y ′ , y) Y
µi =
1≤j≤q
mij[S n i ] ∈ πn(Y ′ , y)
fi : S n → Y ′ i = 1, . . . , q
q n + 1 X
n ≥ 2
π := π1(Y, y) ∼ = π1(X, y) ( ˜ X, ˜ Y )
Cj( ˜ X, ˜ Y ) =
(Z[π]) q k = n, n + 1
0 k = n, n + 1
∂
· · · → 0 → (Z[π]) q ∂ −→ (Z[π]) q → 0 → · · ·
M ∈ Glq(Z[π])
˜y ∈ P −1 (y)
P
H•( ˜ Y , ˜ X) = 0
(Y, X) 1
τ(X, Y ) = w (−1)n+1
W h(π)
n
C ∞ n
M0 M1
M0 ⊔ M1 W
∂W = ∂0W ⊔ ∂1W

∂jW ∼ = Mj j = 0, 1
(W, M0, M1)
(W, M0, f0, M0, f0) j = 0, 1
fj
Mj
fj
∂jW
W
∂0W ∼ = M + 0 ∂1W ∼ = M − 1
M1 W
M0 × I
(W, M0, M1)
(W ′ , M1, M2) M1 (W ∪M1 W ′ , M0, M2)
M0 M2
n
Ωn
[M0] + [M1] = [M0 ⊔ M1]
n
D n Mi
W = (M0 ⊔ M1 × I) ∪ (D n ×j→Mj×1) D n × I
W M0 ⊔ M1
M0#M1 := (M0 ⊔ M1) ∪ S n−1 ×{j} (S n−1 × I)
M
S n
Ωn
M0 (W, M0, f0, M1, f1)
(W ′ , M0, f ′ 0, M ′ 1, f ′ 1)
F : W → W ′ F ◦ f0 ≡ f ′ 0
F (M1) ∼ = M ′ 1
M0
∂

M0 M1
(W, M0, f0, M1, f1)
Mj → W j = 0, 1
(W, M0)
τ(W, M0) ∈ W h(π1(M0))
M0 n ≥ 5
π := π1(M0)
∂
τ(W, M0) = 1 ∈ W h(π)
M0
∀x ∈ W h(π) ∃ (W, M0, M1) / τ(W, M0) = x
M0
M0
π = ∗
M
5
n M
S n
n = 1
n = 2
n ≥ 6
n = 5 n = 3, 4
n = 3

M n ≥ 6 Sn
Z k = 0, n
Hk(M) =
0 k = 0, n
1
n − 1 πn(M, p) = Hn(M, p)
p ∈ M
n
f : S n → M
g : M → S n f
n
M D n 0 D n 1 g
˜g : M\(D n 0 ∪ D n 1 ) −→ S n−1 × I
W := M\(D n 0 ∪ D n 1 )
S n−1 ˜g
W ∂D n 0 ∼ = S n−1
F : (W, ∂D n 0 , ∂D n 1 ) −→ (∂D n 0 × I, ∂D n 0 × 0, ∂D n 0 × 1)
D n 0
D n 1 F1 : ∂D n 1 → ∂D n 0 × 1
D n 1 −→ D n
(r, θ) ↦→ (r, r · F1(1, θ))
r θ ∈ S n−1
r = 0
M = W ∪ D n 0 ∪ D n 1 −→ ∼ (∂D n 0 × I) ∪ D n 0 ∪ F1(∂D n 1 ) D n ∼ = S n

X n
T : I n → X
n T (s1, . . . , sn)
n
Qn(X)
n Dn(X) n
Sn(X)
Sn(X) = Qn(X)/Dn(X)
Sn(X)
n
n T : I n → X
(n − 1) T
1 ≤ i ≤ n
AiT (s1, . . . , sn−1) = T (s1, . . . , si−1, 0, si, . . . , sn−1)
BiT (s1, . . . , sn−1) := T (s1, . . . , si−1, 1, si, . . . , sn−1)
∂n : Qn(X) → Qn−1(X)
∂n(T ) :=
n
(−1) i [AiT − BiT ]
i=1
(n−1)
∂n : Qn(X) → Qn−1(X)
∂n∂n−1 ≡ 0
∂n(Dn(X)) ⊆ Dn−1(X)
1 ≤ i < j ≤ n
AiAj(T ) = Aj−1Ai(T )
BiBj(T ) = Bj−1Bi(T )
AiBj(T ) = Aj−1Bi(T )
BiAj(T ) = Bj−1Ai(T )
T i
AiT = Bi

X
∂n : Sn(X) → Sn−1(X)
(S•(X), ∂) Z
X
(S•(X), ∂)
Hn(X) := Hn(S•(X))
(X, A)
X A
Sn(X, A) := Sn(X)/Sn(A)
∂ A ∂(Sn(A)) ⊆
Sn−1(A) (S•(X, A), ∂)
∂
0 → S•(A) i −→ S•(X) p −→ S•(X, A) → 0
· · · ∂ −→ Hn(A) i∗
−→ Hn(X) p∗
−→ Hn(X, A) ∂ −→ Hn−1(A) → · · ·
Hn(X) Hn(X, A)
X
A Hn(X, A) ∂ −→ Hn−1(A)
(X, A) (Y, B) f : (X, A) → (Y, B)
f∗ : Sn(X, A) → Sn(Y, B)
f
f f∗
f ∼ = g : X → Y
f∗, g∗ : Sn(X) → Sn(Y )
U V X
X
· · · Hn(U ∩ V) iU ⊕−iV
−−−−−−→ Hn(U) ⊕ Hn(V) jU +jV
−−−−→ Hn(X) δ −→ Hn−1(U ∩ V) → · · ·

iU iV
U ∩V U V jU jV
X
δ : Hn(X) → Hn−1(U ∩ V)
z ∈ Hn(X)
U V
z = u + v
0 = ∂z = ∂u + ∂v
δ : z ↦→ ∂u = −∂v ∈ Hn−1(U ∩ V)
z
z = u ′ + v ′ u ′ ⊂ U v ′ ⊂ V
0 = u − u ′ + v − v ′
u − u ′ ⊂ U ∩ V
∂(u − u ′ ) = 0 ∈ Hn−1(U ∩ V)
X Y
a : Z → X b : Z → Y E
a
Z
X
b p.o.
Y
E
E = (X ⊔ Y )/a(z) ∼ b(z)
E
X Y X Y
n D n S n−1

n
X
α∈An Sn−1
∪φα
α∈An Dn
∪ ¯ φα
Y X n
¯ φ : D n → X n
S n−1 → D n
φα ¯ φα
an =|An | n
X
X
Y
∅ := X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ Xn ⊂ · · ·
Xn Xn−1 n X
n Xn n
n
D n
D n◦ X
X
A
X 0 := A ⊔ D D
(X, A)
A X A
X/A
X = X/∅ ∼ = (X, A)
A ⊂ X
X (X, A)
A
n Xn X
(X, Xn)
(X, A) X/A

Hn(X, A) ∼ = ˜ Hn(X/A)
(X, A)
· · · → H•(A) i∗
−→ H•(X) j −→ ˜ H•(X/A) ∂ −→ H•−1(A) → · · ·
X
Hk(Xn, Xn−1) =
∀k > n Hk(Xn) = 0
0 k = n
Z an k = n
∀k < n Hk(Xn) ∼ = Hk(X)
Xn/Xn−1 n
Xn/Xn−1 ∼
=
α∈An
S n α
Hk(Xn, Xn−1) ∼ = ˜ Hk(
α∈An
U
V
· · · Hk(U∩V) iU ⊕−iV
−−−−−−→ Hk(U)⊕Hk(V) jU +jV
−−−−→ Hk(
S n )
α∈An
S n ) δ −→ Hk−1(U∩V) → · · ·
Hk(
U
α∈An
˜Hk(U ∩ V) = 0
S n ) ∼ = Hk(U) ⊕ Hk(V)
Hk(U) ∼ = Hk(S n )

Hk(S n ) = Z k = n
V
˜Hn(
S n ) ∼ = Z an
α∈An
an =|An |
Xn−1 ⊆ Xn
0 → Hn(Xn)) jn
−→ Hn(Xn, Xn−1) ∂n
−→ Hn−1(Xn−1) i∗
−→ Hn−1(X) → 0
X Cn(X) :=
Hn(Xn, Xn−1) dn = jn−1 ◦ ∂n
· · ·
Hn(Xn, Xn−1)
dn
Hn−1(Xn−1, Xn−2)
∂n
jn−1
Hn−1(Xn−1)
dn ◦ dn+1 = (jn−1 ◦ ∂n) ◦ (jn−2 ◦ ∂n−1)
= jn−1 ◦ (∂n ◦ jn−2) ◦ ∂n−1
= jn−1 ◦ (0) ◦ ∂n−1
dn ◦ dn+1 = 0
· · ·
C•(X)
∂n+1
· · · Hn+1(Xn+1, Xn)
Hn(Xn)
jn
dn+1
0
Hn(Xn, Xn−1)
∂n
i∗
Hn(X) → 0
dn
0 → Hn−1(Xn−1) Hn−1(Xn−1, Xn−2)
jn−1
Hn(X) ∼ = Hn(Xn)
Im(∂n+1)

jn
Hn(X) ∼ = jn(Hn(X))
Im(jn) = Ker(∂n) = Ker(dn)
Hn(X) ∼ = jn(Hn(Xn)/Im(∂n+1) ∼ = Ker(∂n)
=: Hn(C•(X))
Im(dn+1)
X
X x ∈ U V
x ∈ V ⊆ U
X
x0 ∈ X U
π1(U, x0) i∗
π1(X, x0)
i∗ ≡ 0
X E
P
E P
X
X {Ui}i P
Ui P −1 (Ui)
X
˜ X P
˜ X
˜X P
X

X
f : X → B p : E → B
x0 ∈ X e0 ∈ E f(x0) = p(e0)
f∗(π1(X, x0)) ⊆ p∗(π1(E, e0))
˜ f : X → E p ◦ ˜ f = f
E
˜f
X
B
f
p

n
M
n
∀x ∈ M ∃ U ⊂ M
R n φ : U −→ Ũ ⊆ Rn
U φ
φ : U −→ φ(U) ψ : V −→ ψ(V )
φ ◦ ψ −1 : ψ(U ∩ V ) → φ(U ∩ V )
R n
C k n
M n C k
k
k = 0 k = ∞
M
R n + := {(x1 · · · xn) : xn ≥ 0} n
C k f : M m → N n
C k φ : U ⊂ M −→ φ(U) ⊂ R m
ψ : V ⊂ N −→ ψ(V ) ⊂ R n
ψfφ −1 : φ(f −1 (V ) ∩ U) ⊆ R m −→ ψ(f(U) ∩ V ) ⊆ R n
k
R n + R n
R n
f : M m → N n
x ∈ M
D(ψfφ −1 ) φ(x) ∈ HomR(R m , R n )
f : M → N
f

M N (N) ≥ 2·(M)+1 M
f : M → N
M N
n M n
R 2n+1
p : E → X p −1 (x) p
x ∈ X X E
p : E → X
K n X
C k E E p −→ X
X E
X {(Ui, φi)}
p −1 (U)
p
φi
∼
U × Kn π1
U
Ui ∩Uj
n ψ◦φ−1
Ui∩Uj × K −−−−→ Ui∩Uj × K n
(x, ¯v) ↦→ (x, gij x (¯v))
gij
gij : Ui∩Uj → Gln(K)
C q
0 ≤ q ≤ k ≤ ∞ X
C k E
C k
E φ −→ F

E
F
X
φx : Ex → Fx
φx : Ex → Fx
M n C k k ≥ 1
T M M n {Ui ⊂ M φi
−→ Ũi ⊂
Rn }i∈I T M
Ui × R n / ∼
(x, u) ∼ (y, v) x = y v = D(φjφ −1
i ) φi(x)(u)
T M → M
{Ui × R n }i φi × Id
x ↦→ D(φjφ −1
i ) φi(x) ∈ Gln(R)
p : E → X k
X p ′ : E ′ → X
N ∈ N
E ⊕ E ′ ∼ = X × R N
E
R N
E R N
T E ⊕ NE ∼ = E × R N
X ⊂ E T X
TXE E X
T R k ∼ = R k Ex ∼ = T Ex
E TXE
X
TXE ∼ = T M ⊕ E

E (x, v) x ∈ X
v ∈ Ex TXE ((x, 0, u, v))
x ∈ X u ∈ T X v ∈ T Ex 0
E X
X
E
E ⊕ (T M ⊕ NXE) ∼ = TXE ⊕ NXE ∼ = X × R N

K
CW
CW