EMBEDDINGS OF REAL PROJECTIVE SPACES
DONALD M. DAVIS
1. Statement of results
The question of finding the smallest Euclidean space in which real projective*
* space
P ncan be (differentiably) embedded was the subject of intense investigation du*
*ring
the 1960s and 1970s. The purpose of this paper is to survey the status of the q*
*uestion,
and add a little bit to our knowledge by proving one new family of embeddings, *
*using
old methods of obstruction theory. Our new result is given in the following the*
*orem.
Theorem 1.1. If n = 2i+ 3 11, then P ncan be embedded in R2n4.
As far as I can tell, this improves on previous embeddings by 1 dimension. In*
*deed,
Berrick's 1979 table ([4]) lists the best embedding for P nto be in R2n3 when *
*n =
2i+ 3, from [19], and I know of no embedding results for P nproved subsequent to
Berrick's table. (There are, however, subsequent nonembedding results, notably *
*those
of [3].)
The following table lists the best nonembedding and embedding results for P n*
*of
which I am aware, for n 63. Of course, most of these results fit into infinite*
* families.
Here we use the symbol to refer to differentiable embeddings.
___________
1991 Mathematics Subject Classification. 57R40.
Key words and phrases. embeddings, real projective spaces, obstruction theory.
1
2 D. DAVIS
_n_6__ref_____ref_ 
 n 6 ref  ref   32 63 [20] 64 [11] 
___________________    
 2 3[20] 4[11]  33 64[14] 65[11] 
 3 4[14] 5[11]  34 64[14] 65[18] 
 4 7[20] 8[11]  35 64[14] 66 new 
 5 8[14] 9[11]  36 65[3] 69[15] 
 6 8[14] 11[8]  37 70[2] 71[15] 
 7 8[14] 12[17]  38 70[2] 71[19] 
 8 15[20]16[11]  39 70[2] 71[19] 
 9 16[14]17[11]  40 70[2] 74[21] 
10 16[14]17[18] 41 77[1] 79[15] 
11 16[14]18new  42 77[1] 80[21] 
12 17[3] 21[15]  43 78[2] 80[21] 
13 22[2] 23[15]  44 78[2] 82[21] 
14 22[2] 23[19]  45 78[2] 84[21] 
15 22[2] 24[19]  46 79[3] 85[6] 
16 31[20]32[11]  47 79[3] 86[5] 
17 32[14]33[11]  48 83[3] 90[21] 
18 32[14]33[18] 49 93[1] 95[15] 
19 32[14]34new  50 93[1] 96[21] 
20 33[3] 37[15]  51 94[2] 96[21] 
21 38[2] 39[15]  52 94[2] 98[21] 
22 38[2] 39[19]  53 94[2] 100[21] 
23 38[2] 39[19]  54 95[3] 101[6] 
24 38[2] 42[21]  55 95[3] 102[5] 
25 45[1] 47[15]  56 99[3] 106[21] 
26 45[1] 48[21]  57 99[3] 108[21] 
27 46[2] 48[21]  58 103[7] 110[21] 
28 46[2] 50[21]  59 103[7] 111[6] 
29 46[2] 52[21]  60 107[3] 114[21] 
30 47[3] 53[6]  61 107[3] 116[21] 
_31_52[12]_54[5]___ 62 107[3] 117[5] 
_63_114[12]_117[5]___
Note that some of the nonembedding results (those of [1], [12], and [7]) are ac*
*tually
obtained from nonimmersion results.
2.Proof of theorem
In this section, we prove Theorem 1.1. Our method is that used by Mahowald in
[15]. A main tool is the following result, which was proved in [15], following *
*[8]. In
EMBEDDINGS OF REAL PROJECTIVE SPACES 3
Section 3, we will add a few details to the argument given in [15]. Let q denot*
*e the
Hopf bundle over P q, and let " denote the trivial bundle.
Theorem 2.1. Assume that P qembeds in Rp with normal bundle .
o If q has n linearly independent sections and P n1embeds in Sm1 , then
P q+nembeds in Rp+m .
o ( q) (q + 1)" (p + 1)q.
Here the hypothesized embeddings need only be topological, and the embedding *
*in
the conclusion is only topological. We then use the result of Haefliger ([10]) *
*that a
topological embedding of an nmanifold in Rd can be approximated by a different*
*iable
embedding provided 2d 3(n + 1), in our application of this theorem.
To prove Theorem 1.1, we apply Theorem 2.1 with q = 2i+ 2, p = 2i+1+ 1, n = 1,
i+2
and m = 1. Throughout this section, we assume i 3. The embedding of P 2
i+1+1
in R2 was proved by Nussbaum in [18]. Theorem 1.1 will then follow from the
following result (with = ) together with the embedding of P 0in S0.
i+2
Proposition 2.2. If is an orientable (2i 1)plane bundle over P 2 such that
(2i+ 3)" (2i+1+ 2);
then has a nonzero section.
Proof.We apply obstruction theory to the following diagram.
i2
S2
@R
BSO(2i 2)


?
i+2 ___________ i
P 2 BSO(2  1)
The desired lifting of to BSO(2i 2) is proved using modified Postnikov towe*
*rs
(MPTs), as introduced in [15] and refined in [9]. In the diagram below, Hn = K(*
*Z; n)
4 D. DAVIS
and Kn = K(Z2; n). We write Z=2 and Z2 interchangeably. Each vertical map is pa*
*rt
of a fiber sequence with the diagonal maps on either side of it in the diagram;*
* e.g.,
K2ix K2i+1! E3 ! E2 ! K2i+1x K2i+2
is a fiber sequence.
BSO(2i 2)


K2ix K2i+1 H Hj ?
E3 HHj
 K2i+2
K2i1x K2i+1 H Hj ?
E2 HHj
 K2i+1x K2i+2
H2i2H Hj ?
E1 Hk1
 Hj K2ix K2i+2

?
i+2________ i
P 2 BSO(2  1)
H Hj
H2i1
The kinvariants correspond to elements in an Adams resolution of the stable
sphere, where we kill the initial Z all at once. We need the relations which g*
*ive
rise to the kinvariants in the MPT. These are computed by the method initiated
in [9] and used in many subsequent papers such as [7] and [13]. It is a matter*
* of
building a minimal resolution using MasseyPeterson algebras. The relations are*
* as
in the table below, with fi denoting the Bockstein.
__________________________________
 fiw2i2 
_________________________________ 
 k1i: (Sq 2+w2)(fiw2i2) 
 2 
 k1i : (Sq 4+w4)(fiw2i2) 
__2_+2___________________________ 
 k2i : (Sq 2+w2)k1i 
 2 +1 2 
 k2i : Sq1k1i + (Sq 2Sq1+w3)k1i
__2_+2_______2_+2_______________2_
 k3i : Sq1k2i + (Sq 2+w2)k2i 
__2_+2_______2_+2____________2_+1_ 
EMBEDDINGS OF REAL PROJECTIVE SPACES 5
We illustrate how these relations are used, using k22i+1. Its relation means*
* that,
with the action map,
k22i+1
(K2i1x K2i+1) x E2 ! E2 ! K2i+1
corresponds to the class 1 k22i+1+ Sq22i1 1 + 2i1 w2. It also means that in
H*(E1),
(2.3) Sq2 k12i+ w2k12i= 0:
i1 2i+2 2i+2 * 1 *
* 2i
Since H2 (P ; Z) = 0, lifts to a map `1 : P ! E1. Suppose `1(k2i) = *
*ffl0x
i+2
and `*1(K12i+2) = ffl2x2 . By (2.3),
i 2i 2i+2
0 = ffl0Sq 2x2 + w2()ffl0x = 0 + ffl0x ;
and so ffl0 = 0. If ffl2 = 1, then let `01denote the composite
i+2x2i2x`1
P 2  ! H2i2x E1 ! E1:
This will satisfy
i2 * 1 4
`01*(k12i+2)= (x2 x `1) (1 k2i+2+ Sq 2i2 1 + 2i2 w4)
i+2 4 2i2 2i2
= x2 + Sq x + x . w4()
i+2 2i+2
= x2 + x + 0 = 0:
We don't have to worry about whether varying through H2i2changes `*(k12i) beca*
*use
we have already shown that this is 0 for any lifting. Thus a lifting to E1 can *
*be chosen
i+2
which sends both level1 kinvariants to 0, and hence lifts to a map `2 : P 2 *
*! E2.
To show that `*2(k22i+1) = 0, we need the following result, whose proof will *
*appear
at the end of this section.
i+30 i
Lemma 2.4. The map extends to a map P 2 ! BSO(2  1).
*
* i+3
Since our tower has no kinvariants in dimension 2i+3, 0lifts to a map `02: P*
* 2 !
E2 by the same analysis used to lift . Note that the MPT was constructed through
dimension 2i+2, i.e. the space at the top agrees with BSO(2i2) through dimensi*
*on
2i+ 2. Usually we would say that even though the space at the top may not agree
with BSO(2i 2) through dimension 2i+ 3, that is not of concern because in the *
*end
i+2 i
we are just using the lifting of P 2 ; the (2 + 3)cell is just used to detect*
* relations
6 D. DAVIS
i2
at early stages of the lifting. However, in the case at hand, since ss2i+2(S2 *
* ) = 0,
the space at the top of the tower agrees with BSO(2i 2) through dimension 2i+ *
*3.
i+1 0 * 2 2i+2 *
* 3
If (`02)*(k22i+1) = ffl1x2 and (`2) (k2i+2) = ffl2x , then by the relati*
*on for k2i+2,
we have
i+2 2 2i+1 2 2i+1 2i+3
0 = ffl2Sq 1x2 + ffl1Sq x + x . ffl1x = 0 + 0 + ffl1x ;
and so ffl1 = 0. This is where we need Lemma 2.4. If ffl2 6= 0, then varying `2*
* through
K2i+1will yield a new lifting which sends k22i+2to 0, thanks to the term Sq1k12*
*i+2in
the relation k22i+2.
i+2
Thus there is a lifting `002: P 2 ! E2 sending both kinvariants trivially, *
*and hence
i+2
a lifting `3 : P 2 ! E3. If `3 sends the lone kinvariant at height 3 nontri*
*vially,
then varying `3 through K2i+1gives a new lifting which sends the kinvariant to*
* 0,
and hence lifts to BSO(2i 2), as desired. 
Proof of Lemma 2.4.By [13, 3.1], (2i+1+ 2)2i+3 has 2i + 4 linearly independent
i+3 i i+1
sections, and hence there is a map f0 : P 2 ! BSO(2 2) which classifies (2 *
*+2).
Let f = iOf0, where i : BSO(2i2) !fBSO(2i1)iis the inclusion. By the hypothes*
*is
i+2
of Proposition 2.1, our map and ffifiP 2 differ by a map ffi which can be fac*
*tored as
i+2 i
P 2 ! V2i1! BSO(2  1);
since V2i1is the fiber of BSO(2i 1) ! BSO. By Lemma 2.5, any such ffi extends
i+3 2i+3
over P 2 , and hence = f + ffi is the sum of two maps which extend over P *
*. 
Lemma 2.5. The restriction
i+3 2i+2
[P 2 ; V2i1] ! [P ; V2i1]
of sets of homotopy classes of maps is surjective.
Proof.We will show that the target group is Z=8, generated by the class of the
composite
i+2c 2i+2
P 2 ! P2i1,! V2i1;
with c the collapse map. This composite clearly extends to
i+3 2i+3
P 2 ! P2i1,! V2i1;
EMBEDDINGS OF REAL PROJECTIVE SPACES 7
which will establish the lemma.
We begin by observing that ss*(V2i1) sss*(P2i1), the stable homotopy group*
*s of
a stunted projective space, through dimension at least 2i+ 3. These groups can *
*be
computed by the Adams spectral sequence, as done for example in [16, Table 8.16*
*],
to begin ssj(V2i1) Z2 if j = 2i 1, 2i+ 1, or 2i+ 2, with Adams filtrations 0,
1, and 2, respectively, and ss2i(V2i1) = 0. By obstruction theory, this impli*
*es that
i+2
the order of [P 2 ; V2i1] is no greater than 8. The claim is that these three*
* classes
extend cyclically.
i+2 *
*2i+3
One way to see this begins by noting that the desired group equals [P 2 ; P2*
*i1]
i+3 2i
and is in the stable range. There is a degree1 map P22i1! S . Let F be its f*
*iber.
We will show that in the exact sequence
i+2 2i+2 2i+3 2i+2 2i
(2.6) [P 2 ; F ] ! [P ; P2i1] ! [P ; S ]
the first group is 0, and the third group is Z=8. Since we have already observe*
*d that
the middle group has order no greater than 8, and has an element of Adams filtr*
*ation
0, the claim made in the first line of this proof will be established.
i
The third group in (2.6) is isomorphic to ss1(2 P2i3), using Sduality. T*
*his
is Z=8 by an elementary Adams spectral sequence calculation; it is the group in*
* the
column labeled 2 in [16, Table 8.14]. It is not difficult to check that H*(P2i*
*3^F ) is,
through dimension 0, a free module over the Steenrod algebra on classes of dime*
*nsion
4, 2, and 0. Thus the Adams spectral sequence or elementary obstruction theory
implies that ss1(P2i3^ F ), which equals the first group in (2.6), is 0. 
3. Review of Mahowald proof
In this section, we prove Theorem 2.1, merely fleshing out a few details in t*
*he proof
given in [15]. The second part of 2.1 is elementary:
( ) (q + 1)" ( (q + 1))
( o ")
(p + 1)"
(p + 1)
8 D. DAVIS
The first part utilizes the following result. Throughout this section, embed*
*dings
() are not necessarily differentiable.
Proposition 3.1. If nq Rp and P n1 Sm1 , then P q+n Rp+m .
Proof of first part of Theorem 2.1.By hypothesis, n" . Tensoring with yields
the first part of
n Rp;
with the second part clear since can be taken to be a tubular neighborhood of *
*P q
in Rp. The theorem now follows immediately from Proposition 3.1. 
To prove Proposition 3.1, we will need the description of P n+qgiven in the f*
*ollowing
lemma.
Lemma 3.2. There is a homeomorphism
(3.3) P n1[g D(nq) P n+q;
where g : S(nq) ! P n1 is defined by g([x; y]) = [x]. Here we use the model
S(nq) = (Sn1 x Sq)=(x; y) ~ (x; y).
Proof.The space on the LHS of (3.3) is
(3.4) Dn x Sq=((x; y) ~ (x; y); (x; y) ~ (x; y0) ifx = 1):
q_______
The desired homeomorphism sends [(x; y)] to [(x; 1  x2y)] in Sn+q=z ~ z. *
*
Proof of Proposition 3.1.Let f denote the embedding of D(nq) in Rp, and h the
embedding of P n1in Sm1 . Using the descriptions of P n+qgiven by the LHS of
(3.3) and (3.4), we define our embedding by
OE([(x; y)])= (OE1(x; y); OE2(x; y))
OE1(x; y)= (1  x)f(x; y)
OE2(x; y)= xh([x=x]):
Here x 2 Dn and y 2 Sq. We first show OE is welldefined. Because of the (1  *
*x)
factor, the second relation in (3.4) is not a problem, and because of the xf*
*actor, we
have continuity at x = 0, even though x=x is not defined. Injectivity is prov*
*ed as
EMBEDDINGS OF REAL PROJECTIVE SPACES 9
follows. If OE2(x; y) = OE2(x0; y0), then x0= x. If also OE1(x; y) = OE1(x0; y0*
*), then either
x = 1 and hence [(x; y)] = [(x; y0)], or (x0; y0) = (x; y) so that [(x; y)] =*
* [(x0; y0)].

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Lehigh University, Bethlehem, PA 18015
Email address: dmd1@lehigh.edu