five lemma

II

:

2006

11

2

(

)

1. five lemma

.

! #"%$'&)(+*,)-.0/2143

.

−−−→ A −−−→ ^{f 1} B −−−→ ^{f 2} C −−−→ ^{f 3} D −−−→ ^{f 4} E −−−→

α





y ^β





y ^γ





y ^δ





y ^ε



 y

−−−→ A ⁰ −−−→

f ₁ ⁰ B ⁰ −−−→

f ₂ ⁰ C ⁰ −−−→

f ₃ ⁰ D ⁰ −−−→

f ₄ ⁰ E ⁰ −−−→

5768+9%:<;>=4?@

3A'B

,

^C

68+9%:

α, β, δ, ε

:'DFE7G+HI?@

3AKJ#3

.

^L)MA0N

, γ

^O

D)EP?4@

34LIA<QRS

.

2. f

C ^∞

^T2UWVWX

N

YZ+[]\^[]_I`<aW\

D0bWG2H

A7B

, N × [0, 1]

9

, (x, 0) ∼ (f(x), 1)

?0c+d.eQf

3

D)ghFi

kj

f

, M = N × [0, 1]/ ∼

^Aml%nIJ#3

. M

:

C ^∞

^T7U7V7X

?I@po

, mapping torus

^A!q2r

f 3

. (1)

^L<SAN

0 → H ^{k− 1} (N, R )

Image(id −f ^∗ ) → H ^k (M, R ) → H ^k (N, R ) ^{f ∗} → 0

A!sut

;%=Wv+w

kxSy

.

^zW{B

f ^∗

:

, f

^)|7x4JR3~}

D)E

f ^∗ : H ^k (N, R ) → H ^k (N, R )

?

@po

, H ^{k− 1} (N, R ) ^{f ∗}

:

, f ^∗

?2

l

ef

3~€W\7‚AKJ3

. (2) N = S ¹ , f (e ^iθ ) = e ^−iθ

^A7Bkƒ

,

^„)

d9

o

, M

^~l+n4J3

. M

:

Klein

^F†

?I@

34LIA0!‡ˆBkƒ

,

‰7IŠ0‹+ŒP7Ž! !R

.

‘%’

.

2

?I:

,

“2”•BFz'–0—

?™˜

–

ef

ƒ2s2z€2\QšpB#›œRJ34LIA

.

^‰+€W\

9

BkY>ž

:FŸ

1 Z f4¡

s

.

Š<‹WŒP%Ž!

‘, R ’

^{–%J34L4A J#3}

.

1.

•B

9

™JR3

. (1) α

=

, β

^A

δ

P?I@

3 ¡

ZQr

, γ

:

P?4@

3

. c ∈ Ker γ

^{A J3}

.

1. δf 3 (c) = f 3 ⁰ γ (c) = 0

?4@o

, δ

:

{>Y•Z

, f 3 (c) = 0

?4@

3

. C

9

yI3

;%=

Y

Z

c = f 2 (b)

^A

¡

3I#t

¡

b ∈ B

^4J3

. 2. f 2 ⁰ β(b) = γf 2 (b) = γ (c) = 0

?4@

3

. B ⁰

9

y23

;7=

YpZ

, β(b) = f 1 ⁰ (a ⁰ )

^A

¡

34

t ¡

a ⁰ ∈ A ⁰

^IJ3

. α

=

{>YpZ

, α(a) = a ⁰

^A

¡

3#t

¡

a ∈ A

^IJ3

. 3. βf 1 (a) = f 1 ⁰ α(a) = f 1 ⁰ (a ⁰ ) = β(b)

?I@

3

. β

S9

o

, f 1 (a) = b

?4@

3

. 4. c = f 2 (b) = f 2 f 1 (a)

?P@

3

,

;7=

(

9+:Wd

0

9%¡

3PLPA

?

\

)

9 o

, c = 0

?I@

3

.

)ƒ

, γ

P?I@

3ILIA‡

eFf

z

. (2) ε

, β

^A

δ

=P?4@

3 ¡

Z~r

, γ

:F=P?I@

3

. c ⁰ ∈ C ⁰

^{A J3}

.

1. f 3 ⁰ (c ⁰ )

</W143RA

, δ

=P?I@

34LA YpZ

, δ(d) = f 3 ⁰ (c ⁰ )

^A

¡ 3

d ∈ D

^IJ3

. 2. εf 4 (d) = f 4 ⁰ δ(d) = f 4 ⁰ f 3 ⁰ (c ⁰ )

?

,

^L

f#:

0

?P@

3

. ε

:

#?4@

3YZ

, f 4 (d) = 0

?I@

3

. D

9

yW3

;%=

YpZ

f 3 (c) = d

^A

¡ 3

c

^IJ3

.

3. f 3 ⁰ γ(c) = δf 3 (c) = δ(d) = f 3 ⁰ (c ⁰ )

?I@

30YpZ

, f 3 ⁰ (c ⁰ − γ(c)) = 0

?I@

3

. C ⁰

9

y3

;

=M9

o

, c ⁰ − γ(c) = f 2 ⁰ (b ⁰ )

^A

¡ 3

b ⁰ ∈ B ⁰

^IJR3

. 4. β

=?I@

30YZ

, b ⁰ = β(b)

^A

¡ 3

b ∈ B

^PJ3

. γf 2 (b) = f ⁰ 2 β(b) = f ⁰ 2 (b ⁰ ) = c ⁰ − γ(c)

?I@

3

.

^{BFz <ƒ}

, c ⁰ = γ(f 2 (b) + c)

^A

¡o

,

^)ƒ

γ

:F=4?I@

3

.

!

6"

2ƒ

,

^#%$™

ef

z

.

2. (1)

&('

“2”pBFz'–—

:)*

,+

o

:

- ‚

[0, 1]

I + = (0, 1), I − = [0, 1] \ { ¹ ₂ }

^A]\My

,

^.0/ˆBkƒ

M = M + ∪ M −

^A]\My3

. M + 3 [(x, t)] 7→ (x, t) ∈ N × I +

9 o

, M +

:

N × I +

^{A a+\}

D<b4?@po

, M −

:

M − 3 [(x, t)] 7→

( (x, t) 0 ≤ t < ¹ ₂

^SAN

(f ^{− 1} (x), t − 1) ¹ ₂ < t ≤ 1

^SAN

∈ N × (− 1 2 , 1

2 )

9 o

, N × (− ¹ ₂ , ¹ ₂ )

^{A a%\}

DFb?2@

3

.

^z'{B

[(x, 0)] = [(f (x), 1)]

^21N43(

,

^„+25

I?

O * ,5

?

O

(x, 0)

^A

¡

~ƒ

well-defined

?W@

3IL2A 9,67

B>4t

. N × I + , N × (− ¹ ₂ , ¹ ₂ )

:

,

⁸

9

N

9

‹+Œ:9<;(=>

?I@

3

.

^B!Y.B4?@.

-,A

JR3'z7

9 &B

N + ,

^C

B

N −

?

D("

J.LA

9

J3

.

, I + ∩ I − = (0, ¹ ₂ ) t ( ¹ ₂ , 1)

^{A 3}

,

^{R \}

I 0 ,

^C

I 1

^J

.

^L0

A™N

M + ∩ M −

:

,

^aW\

Db

M + ∼ = N × I +

^+~ƒ

, N × I 0 t N × I 1

^A]aW\

D0bP?I@

3

. N × I 0 , N × I 1

:

,

⁸

9

N

^{A ‹7Œ 9<;0= >}

?@

3

.

^B!YB4?@

-2A

J3>z%

9 &B

N 0 ,

C B

N 1

?D("

JLA

9

J3

. Mayer-Vietoris

;'=Ww

o

H ^k (M ) ^- H ^k (N + ) ⊕ H ^k (N − ) ϕ

- H ^k (N 0 ) ⊕ H ^k (N 1 ) H ^{k− 1} (M ) ^- H ^{k− 1} (N + ) ⊕ H ^{k− 1} (N − )

ϕ

- H ^{k− 1} (N 0 ) ⊕ H ^{k− 1} (N 1 )

d ^∗

S3

.

^L<SAN

ϕ

¹

wD

‡4J3A

ϕ =

id id id f ^∗

A ¡ 3

. (

^š#–

)

^)ƒ

Ker ϕ = {u ⊕ v ∈ H ^k (N + ) ⊕ H ^k (N − ) | u + v = 0, u + f ^∗ (v ) = 0} ∼ = H ^k (N ) ^{f ∗} , Coker ϕ = H ^k (N 0 ) ⊕ H ^k (N 1 )

{x ⊕ y ∈ H ^k (N 0 ) ⊕ H ^k (N 1 ) | x = u + v, y = u + f ^∗ (v)} ∼ = H ^k (N ) Image(id −f ^∗ )

S3

.

^{0ƒ S3}

.

——————————

^–—

" o

——————————

ϕ

^!œ4FJ43

.

^–)—I

?'?

ƒ#NFzQa'\

Db

M + → N + ×I + , M − → N − ×(− ¹ ₂ , ¹ ₂ )

,

^‰

f™f

F + , F −

?D("

J

.

ϕ

M ±

)ƒ N>JpA

,

H ^k (M + ) ⊕ H ^k (M − ) −−−→ ^Φ H ^k (M + ∩ M − )

F ₊ ^∗ ⊕F − ^∗

x





∼ = ∼ =

x



 ( ^{F + | M+ ∩ M−} ) ^∗ H ^k (N + × I + ) ⊕ H ^k (N − × (− ¹ ₂ , ¹ ₂ )) H ^k (N 0 × (I 0 t I 1 ))

π ^∗ ₊ ⊕π ^∗ −

x





∼ = ∼ =



 y ^s

∗ 0 ⊕s ^∗ ₁

H ^k (N + ) ⊕ H ^{k− 1} (N − ) −−−→

ϕ H ^k (N 0 ) ⊕ H ^k (N 1 )

A ¡ 3

.

^„2

5

Φ

:

[α + ] ⊕ [α − ] 7→ h

α − | _{M + ∩M −} − α + | _{M + ∩M −} i

?4@•o

, π + : N + × I + → N + , π ₋ : N ₋ × (− ¹ ₂ , ¹ ₂ ) → N ₋

:

d

\..3

G2H#?I@o

, s 0 , s 1

:

, N 0 → N 0 × (I 0 t I 1 )

?

,

^‰

f ™f

x 7→ (x, ¹ ₄ ), x 7→ (x, ³ ₄ )

?>Ÿ

1uZ

f

3PO<

?4@

3

. (Poincar´e

9™?

ƒNz

D<E+GWH

)l7n #s2{+‰t

.)

9 ˆBkƒ2sˆ›™A

ϕ([β + ] ⊕ [β − ])

= (s ^∗ 0 ⊕ s ^∗ 1 ) ◦

F + | M + ∩M −

− 1 ∗

◦ Φ ◦ (F + ^∗ ⊕ F ₋ ^∗ ) ◦ (π ^∗ + ⊕ π ₋ ^∗ )([β + ] ⊕ [β − ])

= (s ^∗ 0 ⊕ s ^∗ 1 ) ◦

F + | M + ∩M −

− 1 ∗

◦ Φ[(π + ◦ F + ) ^∗ β + ] ⊕ [(π − ◦ F − ) ^∗ β − ]

= (s ^∗ 0 ⊕ s ^∗ 1 ) ◦

F + | M + ∩M −

− 1 ∗ h

(π − ◦ F − ) ^∗ β − | _{M + ∩M −} − (π + ◦ F + ) ^∗ β + | _{M + ∩M −} i

= (s ^∗ 0 ⊕ s ^∗ 1 ) ◦

F + | M + ∩M −

− 1 ∗ h

( π − ◦ F − | _{M + ∩M −} ) ^∗ β − − ( π + ◦ F + | _{M + ∩M −} ) ^∗ β +

i

= h

( π − ◦ F − | _{M + ∩M −} ◦ F + | M + ∩M −

− 1

◦ s 0 ) ^∗ β − − (π + ◦ F + | _{M + ∩M −} ◦ F + | M + ∩M −

− 1

◦ s 0 ) ^∗ β +

i

⊕ h

( π − ◦ F − | _{M + ∩M −} ◦ F + | M + ∩M −

− 1

◦ s 1 ) ^∗ β − − ( π + ◦ F + | _{M + ∩M −} ◦ F + | M + ∩M −

− 1

◦ s 1 ) ^∗ β +

i

A ¡ 3

.

+d7GWH

~l+n

9

BFz )ƒ ˆBkƒ43A

, π − ◦ F − | _{M + ∩M −} ◦ F + | M + ∩M −

− 1

◦ s 1 = f ^{− 1}

?

,

^‰7O<

:

id

?I@

3

.

^{BFz <ƒ}

ϕ([β + ] ⊕ [β − ]) = ([β − ] − [β + ]) ⊕ (f ^{− 1} ) ^∗ [β − ] − [β + ]

A ¡ 3

.

^{L< 1}

wD

‡4J

f r

,

− id id

− id (f ^{− 1} ) ^∗

?I@

3

,

9

YZ 1 w

"!Sy

f r

,

^„2#O<

9%¡

3

.

(2)

^–