Lecture 13 ; Tuesday tooth Farvardin 1399 .

#

P,so

t' n Ror- Simplex = ?

.

Citi - l Zeit Pier"

a

Ri=o

.

Pro

P.

- R. linearly independent .' P.

→oh

Pa

2- Simplex Las. Cop. -1919+919 I Cotcitczel , Ci ER-1 } .

- -

Barycentric coordinatesfoe .

%

• Pa→face .

Complex K { di } is if dick , Ojloi → g. EK-

2) if Oi , Oj Ek → bing.ie/OEQ.noje6i,6j

Def : face: if 8. ( P. ,% , R Pr ) → 8. - LP, R R) is a face f 8. e--

I 81=4919 R) is'

ansthufaufoacp.p.PK) Ps .

"simple et here, simplex .

€ "

④ GPD , 4919193, ( BBB) . (BBB) 2-faces

P. Pik

t - faces3-Simplex 4919 ) 4919) ( Pok) - - - .

(B) 49 ) CPD CB)O -face

:::i:÷:* .

K =L CP. 1919149191119191,49191, CRR) . ( 197,497,497,497,493}a- ma

@ BPD n E) = 49 ) ¥ Bk)

o , noo- een hot . oh

#-

yIT = { Oi , Or , 6, . - J . f. (K) = Jyj

Finitely generated abelian Gop .

Cr E GCK ) a. ⇐ a. ¢"

sie k

\r- chain

1%14914%1 and its faces} .

C. (K ) = { c. P. + after I a- EZ} .

Chain Groups.← / C

'( K) =L 90491%+949173+041919) I a. ez}

Crack) = I b. CREB) I b. EZ} .

C. ( K ) EZ-02*7 GCR) = ZAZIE acts = Z

⑧

Orientation : CBP, e) ELK; Ra Po.) if Lilia is as anevenpamttaf

- -

LD l r- i )

P ,°

/P.

. ok e- CB - - B) Jim E.&, I

• P,e.Is c'%

pr . l " " "

Here . Nj#-

GCK) : LRP.> ⇐ - EP. EBBE 19¥ - l Riki K:S

if Cci in ) is an odd

#penttf @ in D .

D : Cr CK )- Chick)-

r

dmdr= e- ein a complex f dimm n :

Crick) Crick) Crack)€s . - - C.We.cm

His -- Ij

" 963-08%1919> = eke> -CBR>+49197

2.2lb = epis-497 - ( < Rs-Cps) + (497-497)=0

thin : Omar - o proof .

. Or RP. - RATE.

⇐I' else E. Pm - e>

drier 40.19 Pr> = ¥.

( Tpi. . Hi ee -- E . En - - E. Rei - e>

+ ¥⇐,

l Pok Pi Piti B '

' Pac - - Pr>f .

- - -- - = .fi#ee.? ,

Ip,

Definition : c. = IT)÷,

c- GCK) K={ 1¥,

i -- J .

④ B) +49N

a'. go.TYg.pe, each

Dc,= d (CP. 197+49197) = Pop - Hot E- P, = R- B

DC,'=P, - P.

Zelda Kera e Crete ) FCK# { Cre Eck) I or -_ of .

↳ E Fck) 7 of Zick)

FCK) w a group . Prof : if 3 , ,2EZ . →

92=0 932=0 → dL3ie3u) -o → 3 , -132 C- Z (K)

Br =L be !!! !. .!:{"oh

p.

""

then ec.cn .

'

b'E- CRB>+ CBR>KRP,> E C, CK ) .

b =D BR) deep. Papi) ⇐ Cars - CP. KALBA>=

b't to ? a 49197+49197+47197

Question : es'

n an equivalence relation ?1) crncr - G- Cr-0=20 .

2) Cr- Cr'

- Cr- Cr' = @ Cre, → Cr'- Cr -0C - Cre .) - Cr

'- Cr

3) Crr Cri Cri - Cr "

ter- a. =3Crete. '- a'Ida;

Cr- Cr" =D ( Cratchit ) - Cr - C.

"

turn. -2k£.

=L "" ""'s

Ie.IIE

[ Cid + Ici ] I Crtci] .

{ - Ici se f - ah .

thick) ⇐ r-th homology Gopfk .

]

wig mix

if."

oa=o,a--a"

↳,

Definition : if f. K- X n'ahomeomojsafhxdgo.g.eg.ae.

'th Ks.

calkdatriangdihfx .

€?÷÷¥Q

Is