Remark concerning -theory oî classifying spaces

A N N A L E S SO CTETA TIS M A TH EM A TIC A E PO LO N A E Series I : CO M M EN TA TIO N ES M A TH EM A TIC A E X I X (1977) R O C Z N IK I P O L SK IE G O TO W A RZYSTW A M A TEM A TYCZNEGO

Séria I : P R A C E M A TEM A TY CZN E X I X (1977)

Andrzej Jankowski| (Sopot)

Remark concerning K -theory oî classifying spaces

In [6] J. W. Vick proved the following theorem:

Theorem

(V). For any finite group G the compact group K°(BG) is isomorphic to the character group of the discrete group K X{BG).

He asked if this duality results from some natural pairing of the theories into the rational numbers modulo the integers.

Let ( , ) be the pairing defined in [6] and Pc' U *(')-> K *(’ )

the Eiemann-Eoch transformation. The transformation pc was originally defined in [2]. For the definition and properties we référé to [3]. We will define a pairing

{ > ^odd (■B Z J® K °(B Z m)->QIZ, such that the diagram

Uoid(BZm)^K °(BZ m) \ <t>

m \

Qiz

is commutative.

The pairing < , ) has quite clear geometrical meaning and the state­

ment above gives the geometrical interpretation of the pairing ( , ) which was defined in purely algebraic way. Our approach works only for cyclic groups. I have no idea how to extend it to general case.

1. The definition of the pairing < , ). Let F be a canonical line bundle over GP(oo). It is known, see for instance [5], that the space BZm can be identified with the sphere bundle 8 F m, where F m is m-th tensor power of the bundle F. The corresponding disc bundle is de­

noted by D F m.

We will work with Z2-graded üZ-theory and Z7-theory. We have the exact triangle

U*{DFm, S F m)

X /

\ Ü *{8F m) ^

The map i* is zero since U *{BZm) is a torsion group and V *(D F m) is torsion-free. Therefore the triangle splits into

0 -> Ü* (.D Fm) U* (D F m, 8 F m) -±-> Ü* (BZm) -> 0.

Let

/ : {M, d M )-> {D F m, 8 F m) represents the element

fi€V *{D F m, 8 F m) such that

d(3 = a

and let

r)eK°(BZm).

Consider the diagram:

K °(D F m, 8 F m)~

ch

Hev(DFm, 8 F m)

->K°(DFm) ~

ch

-> K ev{DFm)

->K°(BZm)

The map i l is an epimorphism and j * is an isomorphism. Let rjeK°{D Fm)

he such that and let be such that

i'r] — r}.

x eE ev(DFm, 8 F m)

j* x = chrj.

Let T(M ) stand for the Todd class of the manifold M, it is a poly­

nomial with rational coefficients in Chern classes of the tangent bundle

of M. For the details see [4].

К -theory o f classifying spaces 231

We define pairing < , > by the formula:

<a, г]} = ( f x T(M ), [M]>modZ.

Th e o r e m 1.

The map

( 1 У ' ^odd (BZm) <s>K° (BZm) ->Q[Z is well defined.

P roof. We have to show that < , ) does not depend on the choices made and that it is bilinear.

(i) Does not depend on the choice of rj: if rrj — i lfj, then

Ц = r j+ j'p , PeK°(D Fm, S F m).

Therefore

x — x' + chf}

and we get

< fx T ( M ), [M J> -< f*x 'T (M ), [Jf]>

= <fch fi T(M ), [Jf]> = (chfp T(M ), [Jfj]>.

The last term is an integer from the integrality theorem.

(ii) Does not depend on the choice of (Ж ,/) in the class /3:

if 0, then there exists ( W ,F ) such that h: M ^ W , f = F \ M, W closed and

F {W \M ) c S F m.

Therefore

(f*x T {M ), [Ж]> = <Jfc*(P*a>T(W)), [Ж]> = <.F *x T {W ), й*[Ж]> = 0.

(iii) Does not depend on the choice of the element f3 such that d/5 = a:

indeed, let d(3' — a; then

P = P + j * y , yeÜ *(D Fm)' Let y — (N, g), we have

< f*x T (M '), [Ж ']> - < /*« Т (Ж ), [Ж]>

= (g* x T (S ), [# ]> = (chg'îjT(N )t [ДГ]>

is an integer from the integrality theorem.

(iv) The map a, rjь-><а, yi} is clearly bilinear.

Recall, that

* i ( - ) ^ V

{ - ) *

Z,

where Z is given the U-module structure via the homomorphism T d: U*->Z,

where

Td{M) = <T(M), [Ж]>

is called a genus of M, see [4].

Under the identification above the map f*C: ^odd( ' ’ ) corresponds to

X - + X ® ^{1 .}

Pr o p o sit io n 1.

The pairing

< , >: Uodd(BZm)®K°(BZm)->QIZ factors to a pairing

< , >0: K-l (BZm)®K°(BZm)->QIZ.

P roof. It is sufficient to show that, for XeU,

<Aot, if) = Td(X)(a, ф .

Let a be represented by (Ж ,/) and X by N. Then Xa is represented by (M x N , f p ), where p : Ж x N~>M is a projection. Therefore

<Aa, ф = ((fp )*x T {M x N ), [M x N ]>

= < fx T (M )® T d (N ), [Ж]®[Ж]>

= [M-]>(T(N), [JV]> = <a,

2. Some lemmas. Let < , U*(X) ® K *(X )-+Z be the pairing defined by

<a, ф в = (chfr]T(M ), [Ж]>

for

a = (Ж ,/)<=и*(Х ) and r)tK *(X ).

Let ? Ук: K *(X )® K *(X )-+Z be a Kronecker pairing in IT-theory.

Then the following lemma holds.

Lem m a 1.

The diagram

I \

HQ ® id Z

K .( X ) e K * ( X ) / < ’>x

is commutative.

К -theory o f classifying spaces 233

P roof. Let MTJ be the Thom spectrum which represents the theory

£7*( ) and let UeK*(MTJ) be the Thom class. The map /ла : U *(')-> K *(‘ )

is defined as follows. Given the element oce U*(X) we take a map Z -^ > M U л X

representing x and we define yc (œ) as an element represented by the composition

MU л X л X ,

where the map of spectra

q

: M U -+BU

represents the element U. Therefore, if yeK *(X ) is represented by the map X — >B U

the element (/лсх,у')к in n*(BU) is represented by

Z - b + M U a X ^ ^ B U ^a B U ^ B U ,

where m is a multiplication in the spectrum BU. Let cJieE**(BU ,Q )

be the universal Chern character class. It is known that the map

given by

{S , B U f k->Z S }

is an isomorphism and that the pairing < , yK is the pairing to n*{BU) followed by the above map. See [1].

Therefore

<РсЯуУ>к = <(m Q A fy f x)* ch, [ £ ] >

= < 0 е л / , ) У л , и >

= <£(<? л / у)*<л®<л,[Г]>

= f*(<pT®chy),

because

Q*ch — cpT, where T is a universal Todd class

T eH **{B U ,Q )

nd cp is a Thom isomorphism (here B U is the space, not the spectrum).

Now if the element a?e U*(X) is represented by the singular manifold / : M -^X

the map f x is the composition

£ = MU

X .

Now j* [ $ ] = <p([M])f where cp is a homology Thom isomorphism, and we get:

</*(

T®

% ), [27]>

= <T(fvxfp)*(<pT®chy),(p[M]>

= < {f,x fp )m(Techy), [Jf]> = (cJifyT (M ), [Jf]> . Let A be an abelian torsion group and let

0 * — — 0

be a free resolution of A. Let

F t = Н о т (W2, Z), F 2 = Hom(Wi, Z) and let

< , >: F ^ W ^ Q be a pairing such that

(a), dw) = æ(w).

We have a commutative diagram

0

0

0 0

Л ^

Ext {A, Z )--- '—>0

-f ' f

> F

* d*

Horn {W2,Q)

ïïom(Wr1,Ç)

* 0

> Horn(W2, Q/Z)

■ ^HomCWi, QjZ) I

i

> H o m {A, Q/Z)

•>0

-*0

0 0

К -theory o f classifying spaces 235

and the ker-coker homomorphism

<3: Hom(A, Q[Z)->~Ext(A, Z) is an isomorphism.

Lem m a 2.

I f

(ô~1æ)(a) — ( x 7 w}.

P roof. It results from the definition of ô that for sreExt(A ,Z ) we have

ô~lx = <p where ye Horn (A, Q/Z) is suoh that

4<P) =Р(У ) and

y € Horn ( W!, Q) satisfies

d%{y) =

x,

x.

Therefore

ô~1x(a) — (p(a) = p(y){w) —y(w)modZ, where n(w) = a. There exists an integer N such that

Nw = d(w) for some we W2 and we have

y(w) — N ~ly{dw) = N ~1dQ(y)(w) — N ~lx{w) = N ~l (x, dwy

=

3. The main result. In Section 1 we defined a pairing

< , >0: K 1(BZm)9K?(BZm)-+QIZ.

Now we are in position to show the following theorem:

Th e o r e m 2.

The pairing

,

is non-singular and coincides with the pairing defined in [6].

P roof. Consider the sequences

0^& *(C P{<x>))^K t (CP(oo))-*Km{BZm)-+Q, 0 + -K *(B Z J< -K *(C P (°o ))^ -K * (GP( оо))ч-0.

Both are exact because the Gysin exact sequences in K * and K * for the bundle

BZm = S F m->CP(

)

5 Prace Matematyczne 19 z. 2

split by the argument similar to that used in [5] in the case of ZJ*( ) theory. Let

Wt = K„{CP( oo)), Wz = Ê 0(CP(oо)).

Therefore

F x = Horn (W2tZ) = K °(C P{°o)), F 2 = Horn(Wl t Z) = K°(CP( oo))

and we get a diagram of the type considered in Lemma 2 with А {B Z J , Ш Ц А , г ) = К ° (BZm)

and the pairings

F x® Wz->Z, F 2® Wx-+Z

corresponding to the Kronecker pairing in ^-theory. We identify the group Wx with

K 0{D Fm, S F n) via the Thom isomorphism. Define a pairing

< , >: U *(P F m,jSFm)® K *(D Fm)->Q by the formula

for w = { M J ) € U A P Fm, 8 F m) and x e H *{B F m, 8 F m) such that j * X = chrj.

This clearly factors to give a pairing

< >: W1®F1~^Q.

We have to show that, for

w'e TJb{DFm) we have

<rj, dw'y =

But it is just Lemma 1. Therefore Lemma 2 applies and we get a commutative diagram:

E x (BZm) ® K ° (BZr

K x (BZm) ® Ext

id ® a

% ( B Z J ,Z )

id ® <5—1

к , ( B Z J ® Horn (.К, (B Z J, Q/zy

>0

\ Qiz

К -theory o f classifying spaces 237

The inspection of paper [6] shows that the non-singular pairing ( , ) is just the composition

( , ) = et? (id® <5-1)(id®

Therefore the proof of the theorem is completed.

References

[1] M. F. A tiy a h and F. H irzebru ch , Vector bundles and homogeneous spaces, Diff. Geometry, Proe. Symp. Pure Math. Amer. Math. Soc. 3 (1961), p. 7-38.

[2] P. E. Conner and E. E. F lo y d , Relations of eobordism to К -theory, Lecture Notes in Math. 28, Springer-Yerlag, 1966.

[3] P. E. Conner and L. Sm ith , On the complex bordism of finite complex, Publ.

Math. IH ES 37 (1969), p. 117-221.

[4] F. H irzeb ru ch , Topological methods in algebraic geometry, Springer-Verlag, 1966.

[5] D. Q uillen, Elementary proof of some results of eobordism theory using Steenrod operations, Advances in Math. 7 (1971), p. 29-56.

[6] J . W. Y ick, Pontryagin duality in К -theory, Proc. Amer. Math. Soc. 24 (1970), p. 611-616.