Note on ^

R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

A. J

(Warszawa)

Note on ^

-equivariant immersions

1. Preliminaries. Let Rn,k be the Z2-space (Rn+k, T ) , where Rn+k is Euclidean (тг + fc)-space and

T : Rn+k -> Rn-i-k is an involution defined by

T ( t ! , .. •} t n , t n ^_i j ... ; itn + k ) ~ (^l ) • • * f t n > ^ n + 1 } • ‘ ) ^ n + k ) •

Let A be the antipodal involution on the standard sphere 8 m. In this note we will deal with Z2-equivariant immersions of { 8 m, A) into Rn,k.

Let us first consider some examples.

E

1. There is standard immersion of (Sm, A) into R 0,m+1.

E

. There is an immersion of (S1, A) into J21*1. This is shown in Fig.

.

E

3. If there is an immersion of RP{m) into Rk, then there exists an Z2-equivariant immersion of ( 8 ™, A) into Rk,°.

Let Tc be the fixed integer. We will give a lower bound for the set of integers n such that there exists an Z2-equivariant immersion of ($m, A) into Rn'k.

6 — R o c z n i k i P T M — P r a c e M a t e m a t y c z n e X V I I

We will approach the problem following Atiyah [2].

2. Reviev of KO theory. Let Vect(X) be the semigroup of equivalence classes of orthogonal vector bundles over compact space X. There is a na­

tural homomorphism

rj: Vect(A) -> KO( X) .

An element x e K O ( X ) is called positive if and only if it belongs to Im^. The geometric dimension of an element y of KO(X) is the least integer n such that y + (n) is positive, where (n) is the class of n-dimensional trivial bundle.

There are non-additive operations

f : K O ( X ) - > K O { X ) and Atiyah [2] had proved that

угх = 0 for i > g .àxmx.

The structure of KO[ RP( m)) has been described by Adams [1].

Additively

KO[RP(m)) = Z ® Z 2<p{m), where

(i) Ф(т) is the number of integers s such that 0 < s < m and s = 0, 1, 2, 4 mod8.

(ii) Z is generated by (1), the class of trivial line bundle.

(iii) ^2ф(т) is generated by x = £ — (1), where £ is the class of the Hopf bundle over BP(m).

The multiplicative structure is given by x 2 = — 2 x.

The / structure has been computed by Atiyah [2]. We have уг( — sx) =

We will recall that on the category of compact Z2-spaces there is defined a functor

KOz2(-)

by means of Z2-YecioT bundles. If X is free Z2-space there is an isomorphism (?: KOZ 2 ( X ) - * K O ( X ! Z 2)

induced by the isomorphism

q : VectZ2 (X) -* Yect ( X/ Z2).

The later is defined as follows : for E X a Z 2-vector bundle we define a vector bundle over X fZ 2 by E jZ2 -> X jZ2. For the details we référé to [3].

3. Results. Let (X , T) be any Z

-space. We define a Z2-line bundle 6 X>T over X as follows:

®( x , t ) — {E X), where

(i) E = X x M,

(ii) p is the projection onto first factor, (iii) T : E E is given by

T(x, t) — (Tx, —t).

L emma 3.1. In KO(RP(m))

where F is the Hopf bundle over BP(m).

Proof. Let E = в(^т>А). We will show that E/ Z2 -> RP(m) is iso­

morphic to F. Eecall that the Hopf bundle has the trivializing covering Ui — {[ж0: ... : Жщ]) Œi Ф 0}

and the transition functions

given by

(Pij•' Ui n Uj-> GL( 1) (Pij([x0: ... : жш]) - [xjxj].

Consider the local trivialization of E / Z 2 defined by means of the same covering as above and the functions

P

u, Ui

where щ{[х, /]) = ([a?], x^t). One can easily check that it is well defined and has the same transition functions as F. Thus E/ Z 2 and F are isomorphic.

L emma 3.2. Let f: ( Y , S ) - ^ { X , T ) be an equivariant map. Then

/ •

—

•

P roof. This is obvious.

L emma 3.3. The tangent Z 2-bundle to Bn,k is equal to (n) + M^n,^

Proof. We have t Rn+k = (Bn+k x Bn+k Bn+k). The action of T on Тдп+fc is defined by

T(u, v) = (Tu , dTu v) and '

The action of Z % on (n) + M (Rn,k) is defined in the same way.

Let us define the number x(s, m) to be the largest integer j such that

2 ^ - 1

Ф 0 mod2<P(m),

Now we can state our theorem.

T heorem 3.4. Let m , k be non-negative integers and s — m + 1 — ft.

There does not exist a Z^-immersion of ( 8 m, A) into j^H(s>m) - 2+8>km P roof. Let us suppose that there is a Z2-immersion

/ : ( / 8 й 1 ,

A) - + B n’k and let vf be the normal Z2-bundle of / . We have

r (Sm,A ) ® vf — Л - г ц п *

and it results from Lemmas 3.1-3.3 that in KO{BP(m)) we have xRP(m)^>Q{Vf) = (») + &£.

Since тДР(т) = ( m — l and x — f — 1 we get

Hence

(ft — m —1)®+ (k — m — l) + (n + l) = g(vf) f

—«С + (Л + 1 — 8 ) = Q(vf).

g . dim ( —sx) < — s + n + 1 . On the other hand from definition of x(s, m) we have

yx{s'm){ - s x ) Ф 0.

Therefore it results from Theorem 1.1 that

</.dim(— sx) > x(s, m) and we get

Thus

—s + n + 1 ^ x( 8 , m).

n Ф x(Sj m) + s — 1

and the theorem follows.

For some values of s the numbers x(s, m) can be computed. We have

»(1, m) = Ф(т), «(m + 1, m) = o'(m),

where cr(m) has been defined by Atiyah [2]. Therefore we get a corollary.

COROLLARY 3 . 5 .

(i) ( 8 m, A) cannot be Z 2-immersed in

(ii) A) cannot be Z 2-immersed in Jg°(jw)+λ -1»0.

Е е mark. The existence of Z2-immersion of (Sm, A) in B n,° is equi­

valent to the existence of immersion of BP(m) in Bn.

Thus (ii) of the above corollary is equivalent to (i) of Theorem 5.1 of [

].

References

[1] J. F. A d a m s , Vector fields on spheres, Bull. Amer. Math. Soc. 68 (1962), p. 39 -4 1 . [2] M. F. A t i y a h , Immersions and embeddinds of manifolds, Topology, Vol 1 (1961),

p. 125-132.

[3] M. F. A t i y a h and G. S e g a l , Equivariant К -theory, lecture notes, Oxford 1965.