g2+im`29,*qp2`bm; m/}#` ibqmaa · b: ⊂ s#2 bm#bt +2x +qmibmmqmbk t`: s→ bb+ hh2/ `2i` +ibqmb7`...

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�H;2#`�B+ hQTQHQ;v kyky aT`BM;! aG

G2+im`2 9, *Qp2`BM; �M/ }#`�iBQM AA

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:@T`BM+BT�H +Qp2`BM;

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.2}MBiBQMG2i : #2 � /Bb+`2i2 ;`QmTX � +QMiBMmQmb �+iBQM : × s → s Bb+�HH2/ T`QT2`Hv /Bb+QMiBMmQmb B7 7Q` �Mv t ∈ s- i?2`2 2tBbib �M QT2MM2B;?#Q`?QQ/ l Q7 t bm+? i?�i

;(l) ∩ l = ∅, ∀; ̸= 1 ∈ :.

q2 /2}M2 i?2 Q`#Bi bT�+2

s/: = s/ ∼

r?2`2 t ∼ ;(t) 7Q` �Mv t ∈ s, ; ∈ :X

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S`QTQbBiBQM�bbmK2 : �+ib T`QT2`Hv /Bb+QMiBMmQmbHv QM s- i?2M i?2 [mQiB2MiK�T s → s/: Bb � +Qp2`BM; rBi? }#2` :X

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.2}MBiBQM� H27i U`B;?iV :@T`BM+BT�H +Qp2`BM; Bb � +Qp2`BM; T : 1 → " rBi? �H27i U`B;?iV T`QT2`Hv /Bb+QMiBMmQmb :@�+iBQM QM 1 Qp2` "

1 ; !!

T""❃

❃❃❃❃

❃❃1

T##"""""""

"

, ∀; ∈ :

bm+? i?�i i?2 BM/m+2/ K�T 1/: → " Bb � ?QK2QKQ`T?BbKX

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1t�KTH22tT : R1 → a1 Bb � Z@T`BM+BT�H +Qp2`BM; 7Q` i?2 �+iBQMM : i → i + M, ∀M ∈ ZX

1t�KTH2aM → RPM ≃ aM/Z2 Bb � Z2@T`BM+BT�H +Qp2`BM;X

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S`QTQbBiBQMG2i T : 1 → " #2 � :@T`BM+BT�H +Qp2`BM;X h?2M i`�MbTQ`i +QKKmi2brBi? :@�+iBQM- BX2X-

h[γ] ◦ ; = ; ◦ h[γ], ∀; ∈ :, γ � T�i? BM ".

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h?2Q`2KG2i T : 1 → " #2 � :@T`BM+BT�H +Qp2`BM;- 1 T�i? +QMM2+i2/-2 ∈ 1, # = T(2)X h?2M r2 ?�p2 �M 2t�+i b2[m2M+2 Q7 ;`QmTb

1 → π1(1, 2) → π1(", #) → : → 1.

AM Qi?2` rQ`/b- π1(1, 2) Bb � MQ`K�H bm#;`QmT Q7 π1(", #) �M/

: = π1(", #)/π1(1, 2).

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h?Bb +�M #2 BHHmbi`�i2/ #v

ai�#2(π1(", #)× :)T`1

$$♠♠♠♠♠♠

♠♠♠♠♠♠

♠ T`2

% %❖❖❖❖❖

❖❖❖❖❖❖

❖❖

π1(", #) :

T`1 Bb �M BbQKQ`T?BbK �M/ T`2 Bb �M 2TBKQ`T?BbK rBi?

F2`(T`2) = ai�#2(π1(", #)) = π1(1, 2).

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1t�KTH2�TTHv i?Bb *Q`QHH�`v iQ i?2 +Qp2`BM; 2tT : R1 → a1- r2 }M/ �;`QmT BbQKQ`T?BbK U/2;`22 K�TV

/2; : π1(a1) → Z.

1t�KTH2�b r2 rBHH b22- aM Bb bBKTHv +QMM2+i2/ B7 M > 1X Ai 7QHHQrb i?�i

π1(RPM) = Z2, M > 1.

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�TTHB+�iBQMb

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.2}MBiBQMB : � ⊂ s #2 � bm#bT�+2X � +QMiBMmQmb K�T ` : s → � Bb +�HH2/ �`2i`�+iBQM B7 ` ◦ B = 1�X Ai Bb +�HH2/ � /27Q`K�iBQM `2i`�+iBQM B77m`i?2`KQ`2 r2 ?�p2 � ?QKQiQTv B ◦ ` ≃ 1sX q2 b�v � Bb �U/27Q`K�iBQMV `2i`�+i Q7 s B7 bm+? � U/27Q`K�iBQMV `2i`�+iBQM 2tBbibX

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S`QTQbBiBQMA7 B : � ⊂ s Bb � `2i`�+i- i?2M `∗ : π1(�) → π1(s) Bb BMD2+iBp2X

*Q`QHH�`vG2i .2 #2 i?2 mMBi /BbF BM R2X h?2M Bib #QmM/�`v a1 Bb MQi �`2i`�+i Q7 .2X

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h?2Q`2K U"`Qmr2` }t2/ TQBMi h?2Q`2KVG2i 7 : .2 → .2X h?2M i?2`2 2tBbib t ∈ .2 bm+? i?�i 7(t) = tX

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h?2Q`2K U6mM/�K2Mi�H h?2Q`2K Q7 �H;2#`�VG2i 7(t) = tM + +1tM−1 + · · ·+ +M #2 � TQHvMQKB�H rBi?+B ∈ C, M > 0X h?2M i?2`2 2tBbib � ∈ C bm+? i?�i 7(�) = 0X

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S`QTQbBiBQM U�MiBTQ/2VG2i 7 : a1 → a1 #2 �M �MiBTQ/2@T`2b2`pBM; K�T- BX2X 7(−t) = 7(−t)Xh?2M /2;(7) Bb Q//X AM T�`iB+mH�`- 7 Bb LPh MmHH ?QKQiQTB+X

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h?2Q`2K U"Q`bmF@lH�KVG2i 7 : a2 → R2X h?2M i?2`2 2tBbib t ∈ a2 bm+? i?�i 7(t) = 7(−t)X

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*Q`QHH�`v U>�K a�M/rB+? h?2Q`2KVG2i �1,�2 #2 irQ #QmM/2/ `2;BQMb Q7 TQbBiBp2 �`2�b BM R2X h?2Mi?2`2 2tBbib � HBM2 r?B+? +mib 2�+? �B BMiQ ?�H7 Q7 2[m�H �`2�bX

x = 1

m

Sm

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*H�bbB}+�iBQM Q7 +Qp2`BM;b

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.2}MBiBQMh?2 mMBp2`b�H +Qp2` Q7 " Bb � +Qp2`BM; K�T T : 1 → " rBi? 1bBKTHv +QMM2+i2/X

h?2 mMBp2`b�H +Qp2` Bb mMB[m2 UB7 2tBbibV mT iQ ?QK2QKQ`T?BbKXh?Bb 7QHHQrb 7`QK i?2 HB7iBM; +`Bi2`BQM �M/ i?2 mMB[m2 HB7iBM;T`QT2`iv Q7 +Qp2`BM; K�TbX

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.2}MBiBQM� bT�+2 Bb b2KB@HQ+�HHv bBKTHv +QMM2+i2/ B7 7Q` �Mv t0 ∈ s- i?2`2 Bb� M2B;?#Qm`?QQ/ l0 bm+? i?�i i?2 BK�;2 Q7 i?2 K�TB∗ : π1(l0, t0) → π1(s, t0) Bb i`BpB�HX

q2 `2+�HH i?2 7QHHQrBM; i?2Q`2K 7`QK TQBMi@b2i iQTQHQ;vX

h?2Q`2K U1tBbi2M+2 Q7 i?2 mMBp2`b�H +Qp2`V�bbmK2 " Bb T�i? +QMM2+i2/ �M/ HQ+�HHv T�i? +QMM2+i2/X h?2MmMBp2`b�H +Qp2` Q7 " 2tBbib B7 �M/ QMHv B7 " Bb b2KB@HQ+�HHv bBKTHv+QMM2+i2/ bT�+2X

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.2}MBiBQMq2 /2}M2 i?2 +�i2;Q`v *Qp(") Q7 +Qp2`BM;b Q7 " r?2`2! �M Q#D2+i Bb � +Qp2`BM; K�T T : 1 → "! � KQ`T?BbK #2ir22M irQ +Qp2`BM;b T1 : 11 → " �M/

T2 : 12 → " Bb � K�T 7 : 11 → 12 bm+? i?�i i?2 7QHHQrBM;/B�;`�K Bb +QKKmi�iBp2

117 !!

T1""❅

❅❅❅❅

❅❅12

T2##⑦⑦⑦⑦⑦⑦⑦

"

.2}MBiBQMG2i " #2 +QMM2+i2/X q2 /2}M2 *Qp0(") ⊂ *Qp(") iQ #2 i?2bm#+�i2;Q`v r?Qb2 Q#D2+ib +QMbBbi Q7 +QMM2+i2/ +Qp2`BM;b Q7 "X

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S`QTQbBiBQMG2i " #2 +QMM2+i2/ �M/ HQ+�HHv T�i? +QMM2+i2/X h?2M �MvKQ`T?BbK BM *Qp0(") Bb � +Qp2`BM; K�TX

117 !!

T1""❅

❅❅❅❅

❅❅12

T2##⑦⑦⑦⑦⑦⑦⑦

"

AM Qi?2` rQ`/b- B7 " Bb +QMM2+i2/ �M/ T1, T2 �`2 +Qp2`BM;b- i?2M 7 Bb�HbQ � +Qp2`BM;X

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.2}MBiBQMq2 /2}M2 i?2 +�i2;Q`v : @a2i r?2`2! �M Q#D2+i Bb � b2i a rBi? :@�+iBQM! KQ`T?BbKb �`2 :@2[mBp�`B�Mi b2i K�Tb- BX2X 7 : a1 → a2 bm+?

i?�i 7 ◦ ; = ; ◦ 7- 7Q` �Mv ; ∈ :X

:Bp2M � +Qp2`BM; T : 1 → "- # ∈ "- i?2 i`�MbTQ`i 7mM+iQ` BKTHB2b

T−1(#) ∈ π1(", #) @a2i.

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G2KK�G2i " #2 T�i? +QMM2+i2/X h?2M π1(", #) �+ib i`�MbBiBp2Hv QMT−1(#) B7 �M/ QMHv B7 1 Bb T�i? +QMM2+i2/X

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S`QTQbBiBQM�bbmK2 " Bb T�i? +QMM2+i2/ �M/ HQ+�HHv T�i? +QMM2+i2/X G2iT1, T2 ∈ *Qp(")X h?2M i?2`2 Bb � b2i BbQKQ`T?BbK

>QK*Qp(")(T1, T2) ≃ >QKπ1(",#) @a2i(T−11 (#), T−1

2 (#))

7Q` �Mv # ∈ "X

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h?2Q`2K�bbmK2 " Bb T�i? +QMM2+i2/- HQ+�HHv T�i? +QMM2+i2/ �M/b2KB@HQ+�HHv bBKTHv +QMM2+i2/X # ∈ "X h?2M i?2`2 2tBbib �M2[mBp�H2M+2 Q7 +�i2;Q`B2b

*Qp(") ≃ π1(", #) @a2i.

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h?2 2[mBp�H2M+2 Bb `2�HBx2/ #v i?2 7QHHQrBM; 7mM+iQ`b

*Qp(")6 &&

π1 @a2i.:

''

! G2i T : 1 → " #2 � +Qp2`BM;- r2 /2}M2

6(T) := T−1(#).

! G2i a ∈ π1 @a2i- r2 /2}M2

:(a) := !" ×π1 a = !" × a/ ∼.

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.2}MBiBQMG2i " #2 T�i? +QMM2+i2/ �M/ T : 1 → " #2 � +QMM2+i2/ +Qp2`BM;X� /2+F i`�Mb7Q`K�iBQM UQ` +Qp2`BM; i`�Mb7Q`K�iBQMV Q7 T Bb �?QK2QKQ`T?BbK 7 : 1 → 1 bm+? i?�i T ◦ 7 = TX

1 7 !!

T""❃

❃❃❃❃

❃❃1

T##"""""""

"

G2i �mi(T) /2MQi2 i?2 ;`QmT Q7 /2+F i`�Mb7Q`K�iBQMX

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LQi2 i?�i �mi(T) �+ib 7`22Hv QM 1 #v i?2 lMB[m2M2bb Q7 GB7iBM;X

S`QTQbBiBQMG2i " #2 T�i? +QMM2+i2/ �M/ T : 1 → " #2 � +QMM2+i2/ +Qp2`BM;Xh?2M �mi(T) �+ib T`QT2`Hv /Bb+QMiBMmQmb QM 1X

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h?2Q`2K�bbmK2 " Bb T�i? +QMM2+i2/- HQ+�HHv T�i? +QMM2+i2/X G2iT : 1 → " #2 � +QMM2+i2/ +Qp2`BM;- 2 ∈ 1, # = T(2) ∈ "

: = π1(", #),> = π1(1, 2).

h?2M�mi(T) ≃ L:(>)/>

r?2`2L:(>) : = {` ∈ : | `>`−1 = >}

Bb i?2 MQ`K�HBx2` Q7 > BM :Xh?Bb i?2Q`2K Bb � /B`2+i +QMb2[m2M+2 Q7 i?2 7QHHQrBM; +QKTmi�iBQM

�mi(T) ≃ >QK: @a2i(:/>,:/>) = L:(>)/>.

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1t�KTH26Q` i?2 mMBp2`b�H +Qp2` T : "̃ → "- i?Bb BKTHB2b i?�i

�mi(T) = π1(", #).

h?2`27Q`2 T Bb � π1(", #)@T`BM+BT�H +Qp2`BM;X

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.2}MBiBQMq2 /2}M2 i?2 Q`#Bi +�i2;Q`v P`#(:)! Q#D2+ib +QMbBbi Q7 UH27iV +Qb2i :/>- r?2`2 > Bb � bm#;`QmT Q7 :! KQ`T?BbKb �`2 :@2[mBp�`B�Mi K�Tb, :/>1 → :/>2X

P`#(:) Bb � 7mHH bm#+�i2;Q`v Q7 : @a2i +QMbBbiBM; Q7 bBM;H2 Q`#BibX

_2K�`F:/>1 �M/ :/>2 �`2 BbQKQ`T?B+ BM P`#(:) B7 �M/ QMHv B7 >1 �M/ >2

�`2 +QMDm;�i2 bm#;`QmTb Q7 :X

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A7 r2 `2bi`B+i iQ +QMM2+i2/ +Qp2`BM;b- r2 }M/ �M 2[mBp�H2M+2

*Qp0(") ≃ P`#(π1(", #)).

π1(", #)

((●●●

●●●●

●●!! π̃1(", #)/>

))✉✉✉✉✉✉✉✉✉✉

1

⇐⇒ !" 7 !!

**❁❁❁

❁❁❁❁

❁!"/>

++⑥⑥⑥⑥⑥⑥⑥⑥

"

h?2 mMBp2`b�H +Qp2` !" → " +Q``2bTQM/b iQ i?2 Q`#Bi π1(", #)X 6Q`i?2 Q`#Bi π1(", #)/>- Bi +Q``2bTQM/b iQ

1 = !"/> → ".

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� KQ`2 BMi`BMbB+ 7Q`KmH�iBQM Bb �b 7QHHQrbX :Bp2M � +Qp2`BM;T : 1 → "- r2 Q#i�BM � i`�MbTQ`i 7mM+iQ`

hT : Π1(") → a2i .

:Bp2M � +QKKmi�iBp2 /B�;`�K

117 !!

T1""❅

❅❅❅❅

❅❅12

T2##⑦⑦⑦⑦⑦⑦⑦

"

r2 }M/ � M�im`�H i`�Mb7Q`K�iBQM

τ : hT1 4⇒ hT2 , τ = {7 : T−11 (#) → T−1

2 (#)|# ∈ "}.

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h?2 �#Qp2 bi`m+im`2 +�M #2 bmKK�`Bx2/ #v � 7mM+iQ`

h : *Qp(") → 6mM(Π1("),a2i) .

h?2Q`2K�bbmK2 " Bb T�i? +QMM2+i2/- HQ+�HHv T�i? +QMM2+i2/ �M/b2KB@HQ+�HHv bBKTHv +QMM2+i2/X h?2M

h : *Qp(") → 6mM(Π1("),a2i)

Bb �M 2[mBp�H2M+2 Q7 +�i2;Q`B2bX

g2+im`29,*qp2`bm; m/}#` ibqmaa · b: ⊂ s#2 bm#bt +2x +qmibmmqmbk t`: s→ bb+ hh2/ `2i` +ibqmb7`...

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