4- periode homotopy groupsECHT

,

hectare II

21-5-2020

k - periode homotopy groups-

for spectra ( eventing plaat)

ste : rabiaat htpygpsconsorten [ $

"

,E) t met artsen of

p: §"→ §

stapt : p.to.sn , takealter p

al consider

[ Hp , E) ÷ aka, LE; #p)consider the Adang la Barrett ) map

Ed # is # (%!? !}not : v Ndmes rsomorphrsn

in Etten

form is-ie modp htpy groups

vindt ÷ ⇐"Ja dat:#p)⇐is

5¥ : consider u - torsen :

[ aten ,E)

be alt - map Edith)→ ook)

!

FijtDet : Afula spectrum V is oftypen of

Ki),✓ 4=0for ien

µ to for En

µ Mama klotenKo) > HA , Ko) = Hp , Kk) :-.

A ins as a map v :Edu→V

such thatKE), ✓ = ( a

not innl potent t i # n .

ey : 80 rot type 0 , p:[→ 5 is a mij

$} as of type 1, Adersmap

es au,-

map

ook) not type? rz - map?

Tb Antw) Typen we spectra exist th .

II (Hopkins- Snk) Any fade speak at

type in adits a a settings.( in settings me

"asymptotenUnique

" )

Def : Fix V we of typen , v an settings.

vi'# (E ; V) :-. #HIJ aftapt#

#Ei)

is mdipet of done of u

Det. A map → E ' is a

4-Peirce ofAndresa so on un-periode htpy

groups .

is bakt at done at V

¥ : vo - periode que. = rakend eqnìv.

4- periode egun = (Kp), - no's

localray uit.in - perioden's

Tln) i ' V

= contre Edu [drijfte

the Basketd located with) is

praat de laatste of Spat the hopende qua 's

nota : ¢ : ' ↳ 0 ..-0%)

"

locatietron net ¥0 vind -equal→ gat tonen of locations

4¥t

heks the

¥ 4) doornat

µ , filtratie↳€

[ → IE → En

lesjes : a kniepees Vis of typen preonly of Eve

for paid SpaceX, ← adat dekenin -periode htpy groups by

✓ ' Ma Mapa ( V, X)beter : box V

,en selfnap ui EN →V

deken tellen Imo)§ :b. → Sp

by map. =: #Nk = Ad

structure maps'# Xtkd

d kso

(Exlid →REIN

ÜNAL MIJ.EDU, ×)

de : na # X = in Mappy

erin :the Bousfield ¥,④n

:b,→ 6PM

has the proper : a) Er = FCV, En)Ik

✓ nek2) # Ie ITA)

construct : there ends a system ( Kub)Hi) → Vk) →Vs) → .

.

of k. type in spectra with a map

hij # → §what a a

Tt) , -eqni.ba min

do E- hij #va)

es : 5 '

} → 5'tje → 5'

# → -

E)pa → 8eqnin alter

~" topcell

'p-u-pkba.vn-periode htpygps ← na #

rij , Ids of calculators MahawatThompsonDavisBousfield

treedt'

I

Pets%.ie→

se→ is - ad

only natural layas Dpksl indeed by

paus of p

Én Presents tbh sequens

⇒ gettouur 0%159 with" associated granted

"

# rond) ⇐ La, Dulst)ik (Aron - Makende)

1) Lan) Dj) = 0 of kan

2) Ik tovenares in a- periode htpyGrap:

¥44 -7¥,

# Ppr = #Ppd)update git kite resolution do IN) by

spectra of Reform ↳ Apk

maat : E- ataobgyot Desk) ske over subalgebra AND

redt n rationeel ltpy Henry , could edad

Ra (X; A)"kan dus veel ideeen possible

of dt × , a) la algo model of it )

autogmmpraktijk (Behrens -Red)(Algar : augmented commutatieve

mg gaatR → 8

Hen is a adjunct← deemed ndeeaposabks

cdgat_ Sp[⇒ ⇐M " bromt squadron

rukt the exhals Sp m station at(Alg 7 ; TAL is wreed

toen for augmented rugspectra

Betrek : Xe & late

ftp.s anatural cantor mag

# X → Last#%)"

what a a gunde de X as

a spleet .

npshet : try to calculated X bywhisky eet ofcantate

mg HI, :hard bindt kanfree cnn.mg spectra?

④iT4=1) Smith -op nduees an quaker

Ipa) ¢,Sdu = ↳ Ed ' ' Npd(Nahuatl)

budgetsthe spectra stelt ,> , form a opened (dig)

the spectra Loe ophad

Rosalie : TAART is a speldIn algebra

rekentfactoren 1 Ís

↳trekpad

so that her alg . structure reproducerthe vlottend bracht a kind

A Carpenter mapEx → ↳ TALMT

is a mapof spectra be algebra,