Symmetric topological complexity
of projective and lens spaces
Jes'us Gonz'alez and Peter Landweber
Dedicated to the memory of Bob Stong
Abstract
For real projective spaces, (a) the Euclidean immersion dimension, (b)
the existence of axial maps, and (c) the topological complexity are known *
*to
be three facets of the same problem. This paper describes the corresponding
relationship between the symmetrized versions of (b) and (c) to the Euclid*
*ean
embedding dimension of projective spaces. Extensions to the case of 2e tor*
*sion
lens spaces and complex projective spaces are discussed.
2000 MSC: 55M30, 57R40.
Keywords and phrases: Euclidean embeddings, symmetric axial maps, symmetric
topological complexity.
1 Introduction and main result: real projective
spaces
The Euclidean immersion and embedding questions for projective spaces were topi*
*cs
of intense research during the beginning of the second half of the last century*
*. In
the case of real projective spaces, the immersion problem has recently received*
* a
fresh push, partly in view of a surprising reformulation in terms of a basic co*
*ncept
arising in robotics, namely, the motion planning problem of mechanical systems.*
* In
this first section we start with a brief review of such an interpretation for t*
*he immer
sion problem of real projective spaces, and then we continue to describe our ma*
*in
result (Theorem 1.7 below) on the corresponding interpretation for the embedding
problem.
1.1 Axial maps. Activities were launched with Hopf's early work [17] constructi*
*ng,
for n > r, a Euclidean ndimensional embedding for the rdimensional real proje*
*ctive
1
space Pr out of a given symmetric nonsingular bilinear map ff :Rr+1xRr+1 ! Rn+1.
By restricting to unit vectors (and normalizing), this yields a (symmetric) Z=2
biequivariant map eff:SrxSr ! Sn covering a (symmetric) axial map bff:PrxPr !
Pn, that is, a map which is homotopically nontrivial over each axis (and satis*
*fies
the extra condition bff(x, y) = bff(y, x) for x, y 2 Pr).
Using Hirsch's characterization of (Euclidean) immersions in terms of the no*
*rmal
bundle's geometric dimension, the relevance of (not necessarily symmetric) axial
maps was settled in [1] by showing:
Theorem 1.1. For n > r, the existence of an axial map Prx Pr ! Pn is equivalent
to the existence of a smooth immersion Pr # Rn. 
The hypothesis n > r is needed only for r = 1, 3, 7. In those cases Pr is p*
*ar
allelizable and has an optimal Euclidean immersion in codimension 1; however the
complex, quaternion, and octonion multiplications yield axial maps with n = r.
The connection with robotics was established almost 30 years latter with M. *
*Far
ber's work (initiated in [7, 8]) on the motion planning problem.
1.2 Topological complexity. The Schwarz genus ([19]) of a fibration p: E ! B,
denoted by genus(p), is the smallest number of open sets U covering B in such a
way that p admits a (continuous) section over each U. The topological complexity
of a space X, TC(X), is defined as the genus of the endpoints evaluation map
ev: P (X) ! X x X, where P (X) is the free path space X[0,1]with the compact
open topology. TC(X) is a homotopy invariant of X. Thinking of X as the space
of configurations of a given mechanical system, TC(X) gives a measure of the to*
*po
logical instabilities in a motion planning algorithm for X _a perhaps discontin*
*uous
section of the map ev. We refer the reader to [9] for a very useful survey of r*
*esults
in this area.
Remark 1.2. As shown in [12], for r 6= 1, 3, 7, TC(Pr) can be characterized as *
*the
smallest integer n such that there is an axial map Pr x Pr ! Pn1. Consequently,
TC(Pr)1 is the smallest dimension of Euclidean spaces where Pr can be immersed.
This assertion holds for the three exceptional values of r provided TC(Pr)  1 *
*is
replaced by TC(Pr).
1.3 Symmetric axial maps. As shown in [4], the embedding problem for projective
spaces can be closely modeled by keeping Hopf's original symmetry condition for
axial maps. We give below a quick review of some of the main ideas in [4], but *
*for the
sake of fluidity in reading, we first state the following basic result of Haefl*
*iger's ([15,
Th'eor`eme 10]).
2
Theorem 1.3. Let 2m 3(n + 1). For a smooth compact ndimensional manifold
M, there is a surjective map from the set of isotopy classes of smooth embeddin*
*gs
M Rm onto the set of Z=2 equivariant homotopy classes of maps M* ! Sm1 .
Here Z=2 acts antipodally on Sm1 , and by interchanging coordinates on M* =
M x M  M , where M stands for the diagonal in M x M. 
All we need from Theorem 1.3 is the fact that, under the stated hypothesis, *
*the
existence of a smooth embedding M Rm is equivalent to the existence of a Z=2
equivariant map M* ! Sm1 . Although it is not relevant for our purposes, it is
worth remarking that the surjective map in Theorem 1.3 is explicit, and that it*
* is
in fact bijective when 2m > 3(n + 1).
As for the application in [4], we start by observing that a symmetric axial *
*map
bff:Pr x Pr ! Ps is covered by a map eff:Sr x Sr ! Ss satisfying
eff(x, y) = eff(x, y) = eff(x, y) and eff(x, y) = eff(y, x)(1)
for x, y 2 Sr _what we call a symmetric Z=2 biequivariant map. Under these
conditions it is elementary to check that the composite
` '
x + y x  y
Vr+1,2! Sr x Sr eff!Ss, (x, y) = _____p_, _____p_ (2)
2 2
is a D4 equivariant map. Here D4 is the dihedral group written as the wreath
product (Z=2 x Z=2) o Z=2 where Z=2 acts on Z=2 x Z=2 by interchanging factors.
This group acts on Ss via the canonical projection (Z=2 x Z=2) o Z=2 ! Z=2, and
on Vr+1,2(the Stiefel manifold of orthonormal 2frames in Rr+1) via the restric*
*ted
D4 action in Sr x Sr, where Z=2 x Z=2 and Z=2 act on Sr x Sr by the product
antipodal action and by switching coordinates, respectively.
On the other hand, setting e = {(x, y) 2 Sr x Sr  x 6= y}, the map H :(Sr x
Sr  e ) x [0, 1] ! Sr x Sr  e defined by H(u1, u2, t) = (eu1, eu2) where
u1 + t(v1  u1) u2 + t(v2  u2)
eu1= _________________ eu2= _________________
 u1 + t(v1  u1)   u2 + t(v2  u2) 
v1 = w1 + w2 v2 = w1  w2
u1 + u2 u1  u2
w1 = _____________p_ w2 = _____________p_
1 + 1 
gives a D4 equivariant deformation retraction of Sr x Sr  e onto Vr+1,2. (Figu*
*re 1
depicts the case in which the angle between u1 and u2 is less than 90 degrees; *
*the
situation for an angle between 90 and 180 degrees is similar, but lowering the *
*angle
3
v2___ w1 __v1_
@@I____________  ___`_________
_________ 6 _________
@ _________ _________
@ ___u2__u1___________________________
@ K__________________________~
_____CO__A~K_______
@ __eAuC2 euR__1
@ AC 
@ AC

@AC
@AC______________w2
Figure 1: The D4 equivariant deformation retraction H
to be 90 degrees.) Then, composing the retraction H(, 1) with effO and passi*
*ng
to Z=2 x Z=2 orbit spaces, we get a Z=2 equivariant map (Pr)* ! Ss.
In view of Theorem 1.3, the above construction settles the first part in The*
*o
rem 1.4 below. The bulk of the work in [4] uses Haefliger and Hirsch's fundamen*
*tal
work [15, 16] on embeddings and immersions in the stable range to establish the
(unrestricted) converse.
Theorem 1.4. For 2s > 3r, the existence of a symmetric axial map Pr x Pr ! Ps
implies the existence of a smooth embedding Pr Rs+1. Conversely, the existence
of a smooth embedding Pr Rs implies the existence of a symmetric axial map
Pr x Pr ! Ps. 
The arguments in [4] go a bit further; using the full power of Theorem 1.3, *
*they
explicitly relate, for instance, isotopy classes of embeddings to symmetric hom*
*otopy
classes of symmetric axial maps. We will not make use of these more complete
results, though.
From [1, Lemma 2.1], an axial map PrxPr ! Ps can only hold within Haefliger's
stable range 2s > 3r, when r > 15. In fact, we will need to consider the follow*
*ing
slight improvement (to be proved at the end of Section 2).
Proposition 1.5. For r 2 {8, 9, 13} or r > 15, an axial map Pr x Pr ! Ps can
hold only when 2s > 3r + 2.
Our main interest in Theorem 1.4 arises as follows. Let E(r) denote the Eu
clidean embedding dimension of Pr, and let aS(r) denote the smallest integer k *
*for
which there exists a symmetric axial map Pr x Pr ! Pk. It is immediate from
Theorem 1.4 that, at least in the range of Proposition 1.5,
E(r) = aS(r) + ffi with ffi = ffi(r) 2 {0, 1}. (3)
To the best of our knowledge, the explicit value of ffi (as a function of r) re*
*mains
to be an open question. As an alternative, our main result, Theorem 1.7 below,
4
will avoid this ffi indeterminacy by replacing aS(r) with the symmetric topolog*
*ical
complexity of Pr.
1.4 Symmetric topological complexity. A slightly weaker form (Theorem 1.6 below*
*) of
our main result extends the E vs. aS relation in (3) within the topological com*
*plexity
viewpoint, giving our symmetric interpretation of the first statement in Remark*
* 1.2.
We start by recalling the basic definitions in [10].
For a topological space X, let ev1 :P1(X) ! X x X  X be the restriction of
the fibration ev in Subsection 1.2 where, as in Theorem 1.3, X is the diagonal*
* in
X x X, and P1(X) is the subspace of P (X) consisting of paths fl : [0, 1] ! X w*
*ith
fl(0) 6= fl(1). Note that ev1is a Z=2 equivariant map, where Z=2 acts freely on*
* both
P1(X) and XxX X , by running a path backwards in the former, and by switching
coordinates in the latter. Let P2(X) and B(X, 2) denote the corresponding orbit
spaces, and let ev2:P2(X) ! B(X, 2) denote the resulting fibration. The symmetr*
*ic
topological complexity of X is defined by TC S(X) = genus(ev2) + 1.
Corollary 9 in [10] establishes the inequality
TC TC S. (4)
The proof of this result suggests that the "+1" part in the definition of TCS is
intended to cover (a neighborhood of) the diagonal X when describing symmetric
local sections s for ev with the property that, for any x 2 X, s(x, x) is the c*
*onstant
path at x 2 X.
Theorem 1.6. E(r)+1 TC S(Pr) aS(r)+2, the first inequality holding provided
2 TC S(Pr) > 3r + 4.
Remark 1.2, Proposition 1.5, and (4) assure that the condition 2 TC S(Pr) > *
*3r+4
holds for all r > 15 as well as for r 2 {8, 9, 13}. In those cases, and in view*
* of (3),
the end terms in the inequalities of Theorem 1.6 are off at most by 1.
We can actually do a bit better, but we have chosen to give Theorem 1.6 first
in order to compare with the work in [12] for the nonsymmetric case. Let , be *
*the
Hopf line bundle over Pr and consider the exterior tensor product , , over Prx*
*Pr.
Let I(r) denote the smallest integer k such that the iterated (k + 1)fold Whit*
*ney
multiple of , , admits a nowhere vanishing section. Finally, let a(r) denote *
*the
smallest integer k for which there is an axial map Pr x Pr ! Pk. The main resul*
*ts
in [12], Corollary 5 and Proposition 17, give
I(r) + 1 T C(Pr) a(r) + 1, (5)
an assertion a bit sharper than its symmetric analogue in Theorem 1.6. The punch
line then comes from the classical fact that, for n 6= 1, 3, 7, both I(r) and a*
*(r) agree
5
with the dimension of the smallest Euclidean space where Pr admits an immersion.
However, there is no sectioningWhitneymultiples interpretation available for *
*the
symmetric version of (5). Instead, an adaptation of the ideas in [4] allows us*
* to
refine Theorem 1.6.
Theorem 1.7. E(r) + 1 = TC S(Pr) when 2 TC S(Pr) > 3r + 4.
This is our symmetric interpretation of the second statement in Remark 1.2. *
*As
discussed after Theorem 1.6, the hypothesis in Theorem 1.7 holds for all r > 15*
* as
well as for r 2 {8, 9, 13}. Of course, Theorem 1.6 is a direct consequence of (*
*3) and
Theorem 1.7.
The paper is organized as follows. Section 2 is devoted to the proof of The*
*o
rem 1.7. The situation for 2e torsion lens spaces is discussed in Section 3. He*
*re our
results are weaker than the case e = 1, due in part to the fact that, as the to*
*rsion
increases, the end terms in (5) start measuring different phenomena; this preve*
*nts
us from closing the cycle of inequalities. The numerical value of the symmetric
topological complexity of complex projective spaces is computed in Section 4.
2 Proof of Theorem 1.7
As a warm up, we start by showing that the second inequality in Theorem 1.6 can
be settled with a straightforward adaptation of the idea in the first part of t*
*he proof
of [12, Proposition 17].
Let eff:Sr x Sr ! Ss satisfy (1), with s = aS(r). For 1 i s + 1, let ffi*
*:Sr x
Sr ! R be the ith real component of eff, and set Ui to be the open subset of Pr*
* x
Pr  Pr consisting of pairs (L1, L2) of lines with ffi(`1, `2) 6= 0, for repre*
*sentatives
`j 2 Lj \ Sr, j = 1, 2. Consider the function si:Ui ! P1(Pr) defined as follow*
*s:
given (L1, L2) 2 Ui, choose elements `j as above with ffi(`1, `2) > 0. The two
such possibilities (`1, `2) and (`1, `2) give the same orientation for the 2*
*plane
P (L1, L2) generated by L1 and L2. Under these conditions si(L1, L2) is the pa*
*th
rotating L1 to L2 in the oriented plane P (L1, L2).
Evidently si is a continuous section of the fibration ev1 over Ui. It is als*
*o Z=2
equivariant, in view of the last condition in (1). Therefore, it induces a corr*
*esponding
(continuous) section ~siof the fibration ev2 over the image of Ui under the can*
*onical
(open) projection Pr x Pr  Pr ! B(Pr, 2). But Pr x Pr  Pr is covered by the
Ui's, so we deduce genus(ev2) s + 1. Adding 1, we get the second inequality in
Theorem 1.6. 
In proving Theorem 1.7, it will be convenient to recall a few of the constru*
*ctions
and methods used in [12] for handling the nonsymmetric topological complexity
(see Remark 1.2).
6
Let Sr xZ=2Sr denote the quotient space of Sr x Sr by the diagonal action of
Z=2. The 2fold Cartesian product of the canonical projection Sr ! Pr factors
through SrxZ=2Sr yielding a Z=2 covering space ss :SrxZ=2Sr ! Prx Pr. It is well
known that ss is the sphere bundle associated to , , ! Prx Pr, the line bundl*
*e in
Subsection 1.4 (see for instance [21, Lemma 3.1]). Now, using Theorem 3 together
with the final remarks in Chapter II of [19], we know that the Whitney multiple
k(, ,) admits a global nowhere trivial section for k = genus(ss). The relevan*
*ce of
these observations comes from the standard fact that the smallest such k agrees*
* with
the smallest n for which there is an axial map Prx Pr ! Pn1 (see for instance *
*the
proof of [14, Proposition 2.7]). In these terms, half of the work in [12] comes*
* from
observing that the topological complexity of Pr is bounded from below by genus(*
*ss)
_the other half being a sharpening of the argument at the beginning of this sec*
*tion.
This lower bound is easily settled in [12, Theorem 3] from a fiberpreserving m*
*ap of
the form
f r r
P (Pr)_________________//_S xZ=2S
KK pp
KKK pppp
evKKK%%KK wwpppsspp (6)
Pr x Pr
In fact, as a byproduct of the methods in [12], we know
TC (Pr) = genus(ss). (7)
But for the the symmetric situation it will be necessary to settle the analogous
equality in a more direct way. To this end, we start by recalling the definiti*
*on
of f. For a path fl 2 P (Pr), let bfl:[0, 1] ! Sr be any lifting of fl through*
* the
canonical projection Sr ! Pr, and then set f(fl) to be the class of (bfl(0), bf*
*l(1)).
For our purposes, all we need to know is that (6) is a commutative Z=2 equivari*
*ant
diagram, where the Z=2 action on Sr xZ=2Sr switches coordinates (and the Z=2
actions on P (Pr) and Pr x Pr are the obvious extensions of the respective Z=2
actions on P1(Pr) and Pr x Pr  Pr described in Subsection 1.4). In particula*
*r,
by restricting to Pr x Pr  Pr and then passing to Z=2 orbit spaces, (6) yields
corresponding triangles
f1 r f2
P1(Pr) ________________________//E1 P2(P ) _________________//_E2
OO qqqq KKK vv
OOOO qqq KKK vvv
ev1OOOO''OO xxqqss1qqq ev2KKK%%K vss2vvv (8)
Pr x Pr  Pr B(Pr, 2)
The next result is the symmetric analogue of (7).
Proposition 2.1. For i 2 {1, 2}, genus(evi) = genus(ssi).
7
Proof. It suffices to construct a fiberpreserving Z=2 equivariant map g1: E1 !
P1(Pr) running backwards in the left triangle of (8). To this end, we use the i*
*dea at
the beginning of this section. An explicit model for E1 is the quotient of SrxS*
*r e
by the relation (x, y) ~ (x, y), where e SrxSr is defined in Subsection 1.3*
*. In
these terms, the Z=2 action on E1 interchanges coordinates. Then, the required *
*map
g1 takes the class of a pair (x1, x2) into the curve [0, 1] ! Sr ! Pr, where th*
*e second
map is the canonical projection, and the first map is given by t 7! (tx2+ (1 *
* t)x1).
Here :Rr+1  {0} ! Sr is the normalization map. 
Remark 2.2. The continuity problems avoided in the proof of [12, Proposition 17]
with their Lemma 11 are simply not an issue here, as the diagonal Pr has been
removed.
The next ingredient replaces the axial map characterization for the section
ing problem of multiples of , , (discussed in the paragraph previous to (6)) *
*by
Schwarz's identification of the space classifying Z=2 covers of a given genus.
Lemma 2.3 (Corollary 1 pg. 97 in [19]). The canonical Z=2 cover Sn1 ! Pn1
classifies Z=2 covers of genus at most n. *
* 
The word classifies in Lemma 2.3 is used in the sense that the Z=2 principal
actions on a space X which admit a Z=2 equivariant map X ! Sn1 are precisely
those for which the canonical projection X ! X=(Z=2) has genus n. A similar
(ab)use of this word is made in Lemma 3.7.
Proof of Theorem 1.7. Think of the space PrxPr Pr as the quotient of SrxSr e
by the relations
(x, y) ~ (x, y) ~ (x, y). (9)
Likewise, and just as in the proof of Proposition 2.1, E2 is the quotient of Sr*
*xSr e
by the relations
(x, y) ~ (x, y) ~ (y, x). (10)
In these terms, BerrickFederGitler's trick described in Subsection 1.3 transl*
*ates
into the observation that the extension of (2)
Sr x Sr  e ! Sr x Sr  e , (x, y) = (x + y) , (x  y), (11)
where :Rr+1  {0} ! Sr is the normalization map at the end of the proof of
Proposition 2.1, sends relations (9) into relations (10) and vice versa. Moreov*
*er, the
resulting maps 0:E2 ! Pr x Pr  Pr and 00:Pr x Pr  Pr ! E2 are easily
seen to be equivariant with respect to the Z=2 action on E2 coming from ss2 in *
*the
right triangle of (8), and on Pr x Pr  Pr coming from interchanging coordinat*
*es.
8
The result is now a direct consequence of Proposition 2.1 and Lemma 2.3, whi*
*ch
characterize TCS(Pr)1 as the smallest integer t for which there is a Z=2 equiv*
*ariant
map E2 ! St1, and from Theorem 1.3, which (under the current hypothesis)
characterizes E(r) as the smallest integer e for which there is a Z=2 equivaria*
*nt
map Pr x Pr  Pr ! Se1. 
Proof of Proposition 1.5. In view of Theorem 1.1, the axial map hypothesis can*
* be
replaced by an immersion Pr # Rs, and we need to prove that, for r as stated, t*
*he
smallest such s satisfies 2s > 3r +2. Cases with r 2 {8, 9, 13} follow from ins*
*pection
of [6]. For r > 15 we revisit the argument in the proof of [1, Lemma 2.1]. Pick*
* ae 4
with 2ae r < 2ae+1. Each of the cases
o r 2ae+ 3
o r = 2ae+1 1
o 2ae+ 2ae1+ 2 r 2ae+1 3
can be dealt with by the corresponding nonimmersion result stated in [1].
Assume 2ae+ 4 r 2ae+ 2ae1+ 1 and choose oe 2 {1, 2, . .,.ae  2} with
ae+2oe+2
2ae+ 2oe+ 2 r 2ae+ 2oe+1+ 1. From [5], P2 does not immerse in Euclidean
space of dimension 2ae+1+ 2oe+1 4. Therefore, in the optimal immersion Pr # Rs,
we must have s 2ae+1+ 2oe+1 3, and this easily yields the required inequality
2s > 3r + 2 when oe 3 or ae 5. For the smaller cases with ae = 4 and 1 oe*
* 2,
the required 2s > 3r + 2 follows, as above, from direct inspection of [6].
It remains to consider the case r = 2ae+12. As the case ae = 4 follows agai*
*n from
inspection of [6], we assume further ae 5. Let m = 2ae1+ 2ae2+ 2ae3. From *
*[5] we
know that P 2(m+ff(m)1)does not immerse R4m2ff(m), where ff(m) is the number *
*of
ones appearing in the dyadic expansion of m. Therefore, in the optimal immersion
Pr # Rs, we must have s 4m  2ff(m) + 1 = 2ae+1+ 2ae+ 2ae1 5, from which one
easily deduces the required inequality 2s > 3r + 2. *
* 
3 Lens spaces
Unlike the case of real projective spaces, the symmetric topological complexity*
* of a
lens space does not seem to be related to its embedding dimension. In retrospec*
*t,
the problem arises from the fact that the (nonsymmetric) topological complexity
of L2n+1(2e) actually differs from the immersion dimension of this manifold, and
the difference gets larger as e increases (until it attains a certain stable va*
*lue, see
Remark 3.9). Following the nonsymmetric lead, in this section we (a) indicate *
*how
9
one can characterize TCS(L2n+1(2e)), and (b) point out concrete differences with
respect to a similar characterization for the embedding dimension of L2n+1(2e).*
* To
better appreciate the picture, it will be convenient to make a summary of, and
compare to, the known situation in the nonsymmetric case.
3.1 eaxial maps, their symmetric analogues, and embeddings of lens spaces. The
well known relationship (in Theorem 1.1) between Euclidean immersions of real
projective spaces and (not necessarily symmetric) axial maps has been generaliz*
*ed
in [2] for 2e torsion lens spaces to prove that, with the possible exceptions o*
*f n =
2, 3, 5, the existence of an immersion L2n+1(2e) # Rm is equivalent to the exis*
*tence
of an eaxial map P2n+1 xZ=2e1P2n+1 ! Pm , that is, a map yielding a standard
axial map when precomposed with the canonical projection PnxPn ! PnxZ=2e1Pn
_the notation Pn xZ=2e1Pn refers to the usual Borel construction with respect *
*to
the standard free Z=2e1 action on P2n+1 with orbit space L2n+1(2e).
At the level of covering spaces, the eaxial map condition translates into h*
*aving
a map eff:S2n+1 x S2n+1 ! Sm satisfying the relations
eff(!x, y) = eff(x, !y) and eff(x, y) = eff(x, y) (12)
for x, y 2 S2n+1 and ! 2 Z=2e S1 _these correspond to the first group of
conditions in (1). Our first objective is to indicate how the slight variation *
*in (13)
below of the obvious symmetrization of these conditions describes the Euclidean
embedding dimension for 2e torsion lens spaces. To this end, start by observing
that the product action of Z=2e x Z=2e on the Cartesian product S2n+1 x S2n+1
extends to an action of the wreath product Ge = (Z=2e x Z=2e) o Z=2, where Z=2
acts on S2n+1 x S2n+1 by interchanging axes. This action is stable on the orbit
configuration space FZ=2e(S2n+1, 2) consisting of pairs in S2n+1 x S2n+1 genera*
*ting
different Z=2e orbits (this is the obvious generalization of the space Sr x Sr 
e found in Subsection 1.3 and in the proof of Proposition 2.1). The quotient
Fn,e= FZ=2e(S2n+1, 2)=(Z=2e x Z=2e) has an involution induced by the action of
Ge on the orbit configuration space, and this gives a Z=2 equivariant model for
L2n+1(2e) x L2n+1(2e)  L2n+1(2e), where Z=2 acts by switching coordinates. In
these terms, Theorem 1.3 translates into:
Lemma 3.1. Assume m 3(n + 1). L2n+1(2e) can be smoothly embedded in Rm if
and only if there is a Z=2 equivariant map Fn,e! Sm1 . *
*
Of course, having a Z=2 equivariant map as above is equivalent to having a
Ge equivariant map eff:FZ=2e(S2n+1, 2) ! Sm1 , where Ge acts on Sm1 via the
canonical projection (Z=2e x Z=2e) o Z=2 ! Z=2. Explicitly, effmust satisfy
eff(!x, y) = eff(x, y) = eff(x, !y) and eff(x, y) = eff(y,(x)13)
10
for x, y 2 S2n+1 and ! 2 Z=2e S1. In the case e = 1, the key connection be
tween (13) and the symmetrized version of (12) is given by the ideas in [4], wh*
*ich
teach us how to take care of the deleted "equivariant diagonal" in FZ=2e(S2n+1,*
* 2).
Unfortunately, we have not succeeded in obtaining such a connection for larger *
*val
ues of e. The major problem seems to be given by the apparent lack of a suitable
equivariant deformation retraction of L2n+1(2e) x L2n+1(2e)  L2n+1(2e)that pl*
*ays
the role of V2n+2,2in the e = 1 arguments of [4] described in Subsection 1.3. T*
*his
problem will reappear, in a slightly different form, in regards to a potential *
*char
acterization for the symmetric topological complexity of lens spaces in terms o*
*f the
Z=2e biequivariant maps of the next subsection (Remark 3.5 below). It is worth *
*re
marking that, in the symmetric e = 1 situation of Section 2, we do make an indi*
*rect
use of this equivariant deformation retraction through the extended map in (1*
*1).
3.2 Z=2e biequivariant maps, their symmetric analogues, and symmetric topologi
cal complexity of lens spaces. As shown in [14], the (nonsymmetric) topologic*
*al
complexity of L2n+1(2e) turns out to be (perhaps one more than) the smallest odd
integer 2k1 for which there is a Z=2e biequivariant map eff:S2n+1xS2n+1 ! S2k*
*1,
that is, a map satisfying the (stronger than (12)) requirements
eff(!x, y) = eff(x, !y) = ! eff(x, y),
for x, y 2 S2n+1 and ! 2 Z=2e S1. Alternatively, if c: S2n+1 ! S2n+1 stands
for complex conjugation in every complex coordinate, then by precomposing with
1 x c, a Z=2e biequivariant map as above can equivalently be defined through the
requirements
eff(!x, y) = ! eff(x, y) = eff(x, !1y). (14)
In analogy to the aS notation introduced at the end of Subsection 1.3 to mea*
*sure
the existence of symmetric axial maps, the following definition (which, up to c*
*om
position with 1 x c, corresponds to the symmetrized version of the number s(n, *
*e)
defined in [14]) is intended to measure the existence of symmetric Z=2e biequiv*
*ariant
maps.
Definition 3.2. For integers n and e, let bS(n, e) denote the smallest integer *
*k such
that there is a map eff:S2n+1 x S2n+1 ! S2k1 satisfying (14) and
eff(x, y) = eff(y, x) (15)
for x, y 2 S2n+1 and ! 2 Z=2e S1.
The next result gives our characterization for the symmetric topological com
plexity of lens spaces. The proof will be postponed to the end of the subsectio*
*n.
11
Theorem 3.3. The integral part of 1_2TCS(L2n+1(2e)) agrees with the smallest in*
*teger
m such that there is a map eff:FZ=2e(S2n+1, 2) ! S2m1 satisfying (14) and (15)*
* for
x, y 2 S2n+1 and ! 2 Z=2e S1.
Most of the work in [10] goes into the direction of giving strong lower boun*
*ds
for TCS. However, there seems to be a relative lack of suitable upper bounds; t*
*he
only ones we are aware of are derived, some way or other, from Schwarz's general
estimate for the genus of a fibration F ! E ! B in terms of the dimension of B *
*and
the connectivity of F ([19, Theorems 5 and 50]). For instance, in [10, Proposit*
*ion 10]
the upper bound
TC S(M) 2m + 1 (16)
is derived for any mdimensional closed smooth manifold M. In the case M =
L2n+1(2e), Corollary 3.4 below (which is an immediate consequence of Theorem 3.*
*3)
offers an alternative to (16) that takes not only dimension into account, but a*
*lso
torsion. Theorem 3.10 below gives a typical example (in the nonsymmetric setti*
*ng,
though) of the potential use of this kind of result.
Corollary 3.4. The integral part of 1_2TCS(L2n+1(2e)) is no greater than bS(n, *
*e).
Remark 3.5. In the direction of exploring a possible symmetric analogue of the
main result in [14], it would be useful to make precise how much the above upper
bound differs from being an equality. The main obstruction for such a goal seems
to be the apparent lack of an eanalogue of the map in (2) and (11).
We close this subsection with the proof of Theorem 3.3. As will quickly beco*
*me
clear, the details are formally the same as in the e = 1 case. The e analogue o*
*f (6),
first considered in [12, Theorem 3], reads
f 2n+1 2n+1 . e
P (L2n+1(2e))______________________________//S x S Z=2
RRRR ii
RRRR iiiiii
evRRRRRR iiiiiss
RR)) ttiii
L2n+1(2e) x L2n+1(2e)
The orbit space in the upper right corner is taken with respect to the Z=2e dia*
*gonal
action. The map f, whose definition is the obvious generalization of that in t*
*he
case e = 1, is Z=2 equivariant. In these conditions, the analogue of (8) and t*
*he
proof of Proposition 2.1 generalize in a straightforward way to produce the fol*
*lowing
alternative definition of TCS(L2n+1(2e)).
Proposition 3.6. TC S (L2n+1(2e)) 1 = genus ss2,e:E2,e! B(L2n+1(2e), 2).
Here E2,eis the quotient of FZ=2e(S2n+1, 2) by the relations (x, y) ~ (!x, !y) *
*and
(x, y) ~ (y, x). Moreover, ss2,eis a Z=2e cover with Z=2e acting on E2,eas !.(x*
*, y) 7!
(!x, y), for x, y 2 S2n+1 and ! 2 Z=2e S1. *
*
12
Theorem 3.3 is now a direct consequence of Proposition 3.6 and the following
e analogue of Lemma 2.3 (proved in full generality in [19, Corollary 1, pg. 97]*
*).
Lemma 3.7. The canonical Z=2e cover S2n1 ! L2n1(2e) classifies Z=2e covers of
genus at most 2n. 
3.3 Topological complexity of high torsion lens spaces. We say that a lens spa*
*ce
L2n+1(2e) is of high torsion when e is larger than ff(n), the number of ones in*
* the
dyadic expansion of n. The (nonsymmetric) topological complexity of a high tor*
*sion
lens space has recently been settled in [11].
Theorem 3.8. For e > ff(n), TC (L2n+1(2e)) = 4n + 2. 
Remark 3.9. This result is the analogue of the following situation. For a fixed
n, the immersion dimension of L2n+1(2e) is a bounded nondecreasing function of*
* e
which, therefore, becomes stable for large e. As explained in [13] and [14, Sec*
*tion 6],
the stable value of the immersion dimension is expected to be attained roughly *
*for
e > ff(n) (with an expected value close to the immersion dimension of the compl*
*ex
projective ndimensional space). A very concrete situation, which compares TC to
the immersion dimension of lens spaces, is illustrated in Example 3.11 below.
We extend Theorem 3.8 to the first case outside the hightorsion range by co*
*m
bining the techniques in [11] with the Z=2e biequivariant map characterization *
*of
TC(L2n+1(2e)) discussed at the beginning of Subsection 3.2. The result arose fr*
*om
an email exchange between the first author and Professor Farber in regards to *
*the
results in [11].
Theorem 3.10. For e = ff(n), TC (L2n+1(2e)) = 4n.
Proof. Proposition 2.2 and Theorem 2.9 in [14] yield TC (L2n+1(2e)) 4n. The
opposite inequality follows by taking m = 2ff(n), k = n, and ` = n  1 in [11,
Theorem 11], and using the easily verified fact that ff(n)  1 is the largest p*
*ower of
2 dividing the binomial coefficient 2n1n. *
* 
Example 3.11. The table below summarizes the topological complexity and im
mersion dimension for L2n+1(2e) and CPn in the case n = 2r + 1 with r 1. The
information is taken from [6, 12] in the case of P2n+1, from [13, 20] in the ca*
*se
of the immersion dimension of L2n+1(2e) for e 2, from [12, Corollary 2] in the
case of TC(CPn), and from [3, 18] in the case of the immersion dimension of CPn.
Note that in the case under consideration TC(CPn) is just half the stable value*
* of
TC(L2n+1(2e)) (i.e., for e 3). Such a behavior comes from the fact that CPn *
*is
simply connected and from Schwarz's estimates [19, Theorem 5] for the genus of a
fibration.
13
___________________________________________________________
  P2n+1 L2n+1(4) L2n+1(2e) e 3  CPn 
______________________________________________________
 TC 4n  3 (r 2)  4n  4n + 2 2n + 1 
     
 4n  4 (r = 1)    
______________________________________________________
 Imm  4n  4  4n  3  4n  2 4n  3 
______________________________________________________
We close this section with what we believe should be an accessible challenge:
Determine the symmetric topological complexity of high 2 torsion lens spaces. We
remark that the inequalities
4n + 2 TC S(L2n+1(2e)) 4n + 3,
for e > ff(n), follow from (4), (16), and Theorem 3.8.
4 Complex projective spaces
The (nonsymmetric) topological complexity of the ndimensional complex projec
tive space was computed in [12, Section 3] to be TC(CPn) = 2n + 1. In this brief
final section we show that the same value holds in the symmetric case.
Theorem 4.1. TCS(CPn) = 2n + 1.
Proof. In view of (4), we only need to show that TCS(CPn) 2n + 1. The diagram
of pullback squares
P (CPn)oo__________P1(CPn) __________//_P2(CPn)
ev ev1 ev2
fflffl fflffl fflffl
CPn x CPn oo___CPn x CPn  CPn ____//_B(CPn, 2)
where horizontal maps on the left are inclusions, and horizontal maps on the ri*
*ght
are canonical projections onto Z=2 orbit spaces, shows that the common fiber fo*
*r the
three vertical maps is the path connected space CPn. In particular, Theorem 50
in [19] applied to ev2 gives
dim (Y )
TC S(CPn) = genus(ev2) + 1 ________+ 2
2
where Y is any CW complex having the homotopy type of B(CPn, 2). The required
inequality follows since, as indicated below, there is such a model Y having di*
*m(Y ) =
4n  2. 
In the proof of [10, Proposition 10] it is observed that, for a smooth close*
*d m
dimensional manifold M, B(M, 2) has the homotopy type of a (2m  1)dimensional
14
CW complex. Although this is certainly enough for completing the proof of The
orem 4.1, we point out that an explicit (and smaller) model for B(CPn, 2) was
described by Yasui in [22, Proposition 1.6]. We recall the details. The unita*
*ry
group U(2) has the two subgroups
T 2:diagonal matrices, and ` '
0 z1 1
G :matrices in T 2together with those of the form for z1, z2 2 S .
z2 0
Consider the standard action of U(2) on the complex Stiefel manifold Wn+1,2of
orthonormal 2frames in Cn+1 with quotient the Grassmann manifold of complex 2
planes in Cn+1. Yasui's model for B(CPn, 2) is the corresponding quotient Wn+1,*
*2=G.
Note that dim (G) = dim(T 2) = 2, so that the dimension of Yasui's model is
dim (Wn+1,2)  2 = 4n  2.
References
[1]J. Adem, S. Gitler, and I. M. James, "On axial maps of a certain type", Bol*
*. Soc. Mat.
Mex. 17 (1972) 5962.
[2]L. Astey, D. M. Davis, and J. Gonz'alez, "Generalized axial maps and Eucli*
*dean
immersions of lens spaces", Bol. Soc. Mat. Mex. 9 (2003) 151163.
[3]M. F. Atiyah and F. Hirzebruch, "Quelques th'eor`emes de nonplongement po*
*ur les
vari'et'es diff'erentiables", Bull. Soc. Math. France 87 (1959) 383396.
[4]A. J. Berrick, S. Feder, and S Gitler, "Symmetric axial maps and embedding*
*s of
projective spaces", Bol. Soc. Mat. Mex. 21 (1976) 3941.
[5]D. M. Davis, "A strong nonimmersion theorem for real projective spaces", An*
*n. of Math.
(2), 120 (1984) 517528.
[6]D. M. Davis, "Table of immersions and embeddings of real projective spaces*
*", available
from http://www.lehigh.edu/~ dmd1/immtable
[7]M. Farber, "Topological complexity of motion planning", Discrete Comput. *
*Geom. 29
(2003) 211221.
[8]M. Farber, "Instabilities of robot motion", Topology Appl. 140 (2004) 2452*
*66.
[9]M. Farber, "Topology of robot motion planning", in: Morse theoretic methods*
* in nonlinear
analysis and in symplectic topology, 185230, NATO Sci. Ser. II Math. Phys.*
* Chem., 217,
Springer, Dordrecht, 2006.
[10]M. Farber and M. Grant, "Symmetric motion planning", in Topology and Robo*
*tics,
Contemp. Math. 438 (2007) 85104.
[11]M. Farber and M. Grant, "Robot motion planning, weights of cohomology class*
*es, and
cohomology operations", Proc. Amer. Math. Soc. 136 (2008) 33393349.
15
[12]M. Farber, S. Tabachnikov, and S. Yuzvinsky, "Topological robotics: motion *
*planning
in projective spaces", Int. Math. Res. Not. 34 (2003) 18531870.
[13]J. Gonz'alez, "Connective Ktheoretic Euler classes and nonimmersions of 2*
*klens spaces",
J. London Math. Soc. (2) 63 (2001) 247256.
[14]J. Gonz'alez, "Topological robotics in lens spaces", Math. Proc. Cambridge *
*Phil. Soc. 139
(2005) 469485.
[15]A. Haefliger, "Plongements diff'erentiables dans le domaine stable", Commen*
*t. Math. Helv.
37 (1962) 155176.
[16]A. Haefliger and M. W. Hirsch, "Immersions in the stable range", Ann. of Ma*
*th. 75
(1962) 231241.
[17]H. Hopf, "Systeme symmetrischer Bilinearformen und euklidische Modelle der *
*projektiven
R"aume", Vierteljschr. Naturforsch. Gesellschaft Z"urich 85 (1940) 165177.
[18]A. Mukherjee, "Embedding complex projective spaces in Euclidean space", Bul*
*l. London
Math. Soc. 13 (1981) 323324.
[19]A. S. Schwarz, "The genus of a fiber space", Amer. Math. Soc. Transl., Ser.*
* 2, vol. 55
(1966) 49140.
[20]T. A. Shimkus, "Some new immersions and nonimmersions of 2rtorsion lens sp*
*aces", Bol.
Soc. Mat. Mex. 9 (2003) 339357.
[21]B. Steer, "On the embedding of projective spaces in Euclidean spaces", Pr*
*oc. London
Math. Soc. (3) 21 (1970) 489501.
[22]T. Yasui, "The reduced symmetric product of a complex projective space and *
*the embedding
problem", Hiroshima Math. J. 1 (1971) 2740.
Jes'us Gonz'alez jesus@math.cinvestav.mx
Departamento de Matem'aticas, CINVESTAVIPN
Apartado Postal 14740 M'exico City 07000
Peter Landweber landwebe@math.rutgers.edu
Department of Mathematics, Rutgers University
110 Frelinghuysen Rd, Piscataway, NJ 088548019
16