Homotopy groups of certain highly connected manifolds via loop space homology

0.17 MB·

by  Samik Basu

HOMOTOPY GROUPS OF CERTAIN HIGHLY CONNECTED
MANIFOLDS VIA LOOP SPACE HOMOLOGY
SAMIK BASU ANDSOMNATHBASU
Abstract. For n ≥ 2 we consider (n−1)-connected closed manifolds of dimension at
6 most (3n−2). We prove that away from a finite set of primes, the p-local homotopy
1 groupsofM aredeterminedbythedimension ofthespaceofindecomposable elementsin
0 thecohomology ringH∗(M). Moreover, we show that thesep-local homotopy groups can
2
be expressed as direct sum of p-local homotopy groups of spheres. This generalizes some

n of the results of our earlier work [1].
a
J

8 1. Introduction
1
In this document we consider (n−1)-connected closed manifolds of dimension at most

] (3n−2) and prove analogous results to those for (n−1)-connected 2n-manifolds in [1]. We
T
shall prove the following result (cf. Theorem 3.6, Theorem 3.7).
A
. Theorem 1.1. Let M be a closed (n−1)-connected d-manifold with n≥ 2, d ≤ 3n−2 and
h
t dimH∗(M) > 4. Let r denote the dimension of the space of indecomposables in H∗(M;Q).
a
m Then there is a finite set of primes Π such that for p ∈/ Π,
(a) the p-local homotopy groups of M are determined by r;
[
(b) the p-local homotopy groups of M can be expressed as a direct sum of p-local homotopy

1
groups of spheres.
v
3
If a generator of Hd(M;Q) is indecomposable then it follows from Poincar´e duality that
1
4 the rational cohomology of M is that of Sn. In this case dimH∗(M;Q) = 2 and r = 1.
4 Conversely, supposethat r = 1 and M does not have the cohomology of a sphere. It follows
0 that M is a manifold of dimension 2n with H∗(M;Q) = Q[x]/(x3) and n is even.
.
1 By the assumptions on M, the cup product of any three cohomology classes (of positive
0 degree) is zero. Now we assume that dimH∗(M;Q) > 4. If any class α= a∪b ∈ Hi(M;Q)
6
1 (with i < d) is reducible then by Poincar´e duality there exists β ∈ Hd−i(M;Q) such that
: α ∪ β is a generator of Hd(M;Q). Thus, a ∪ b ∪ β 6= 0 and this violates our previous
v
i observation. Therefore, if r > 1 we deduce that
X
r r = dim(⊕ Hi(M;Q)) = dimHi(M;Q) = dimH∗(M;Q)−2.
0<i<d
a
0<Xi<d
We note that the condition dimH∗(M) > 4 is equivalent to r ≥ 3. In terms of rational
homotopy groups, M is rationally hyperbolic if and only if r > 2. In this case we have the
following result (cf. Theorem 3.8).
Theorem 1.2. Let M be a closed (n−1)-connected d-manifold with n≥ 2, d ≤ 3n−2 and
dimH∗(M;Q) > 4. Then the homotopy groups of M has unbounded p-exponents for all but
finitely many primes.
2010 Mathematics Subject Classification. Primary:55P35,55Q52; secondary:16S37, 57N15.
Key words and phrases. Homotopy groups, Koszul duality, Loop space, Moore conjecture, Quadratic
algebra.
1
2 SAMIKBASUANDSOMNATHBASU
The above result verifies the Moore conjecture (see the discussion before Theorem 3.8 as
well as [5] pp. 518) for such spaces in the rationally hyperbolic case. The low rank cases,
i.e., when r = 1,2 are discussed in §3.3 (see Theorem 3.9).
The authors would like to thank Alexander Berglund for certain helpful discussions.
2. Homology of the loop space
In [2], the homology of ΩM is computed for M a (n−1)-connected (3n−2) manifold.
Let us recall it. Let n ≥ 2 and suppose that M is an (n−1)-connected closed manifold
of dimension d ≤ 3n − 2 such that dimH∗(M) > 4. Choose a basis x ,··· ,x for the
1 r
indecomposables of H∗(M). If we choose an orientation class [M] for M then let c =
ij
hx x ,[M]i. Consider the homology ring H (ΩM) of the based loop space, equipped with
i j ∗
the Pontrjagin product. This ring is freely generated as an associative algebra by classes
u ,··· ,u whose homology suspensions are dual to the classes x ,··· ,x (in particular
1 r 1 r
|u | = |x |−1), modulo the single quadratic relation
i i
(−1)|ui|c u u = 0.
ji i j
Xi,j
Let M be a closed (n−1)-connected d-manifold with d≤ 3n−2. The cohomology of M
is finitely generated and has p-torsion only for a finite set of primes p. Let Σ be the set of
primes such that the cohomology of M has p-torsion. Define
R = Z[1|p ∈Σ].
Σ p
Then we may deduce the following facts.
(a) H∗(M;R ) is a free R -module.
Σ Σ
(b) The natural map H∗(M;R )⊗ Q → H∗(M;Q) is an isomorphism.
Σ RΣ
The first fact follows from Universal Coefficient Theorem for cohomology and the defining
property of R . The second fact is clear.
Σ
Asnotedearlier, theonlynon-trivialproductsofpositivedimensionalclasses aregiven by
theintersectionform. ThereforethemoduleofindecomposablesA(M) = ⊕ Hi(M;R ).
0<i<d Σ
Let x ,··· ,x be a basis of A(M). Fix a choice of an orientation class [M] ∈ Hd(M;R )
1 r Σ
of M. Let c = hx x ,[M]i. Let u ,··· ,u denote classes whose homology suspensions
ij i j 1 r
are dual to the classes x ,··· ,x (in particular |u | = |x |−1). With coefficients in a field
1 r i i
k = Q or a quotient field of R we have the following result for the homology of the loop
Σ
space with respect to the Pontrjagin product (cf. [2], Theorem 1.1).
Proposition 2.1. As associative rings,
H (ΩM;k) ∼= T (u ,··· ,u )/ (−1)|ui|+1c u u
∗ k 1 r ji i j
(cid:0)P (cid:1)
This directly leads us to the following integral version.
Proposition 2.2. As associative rings,
H (ΩM;R ) ∼=T (u ,··· ,u )/ (−1)|ui|+1c u u
∗ Σ RΣ 1 r ji i j
(cid:0)P (cid:1)
Proof. Since M is an orientable manifold the homology H (M −pt) matches H M in all
∗ ∗
degrees upto (d−1). From the conditions on M we deduce that H∗(M −pt) is free on the
classes x∗ which are dual to the classes x and the products are all zero. It follows that
i i
H (Ω(M −pt)) is a tensor algebra on the classes u . Therefore we have a map
∗ i
∼=
φ :T (u ,··· ,u ) −→ H (Ω(M −pt))−→ H (ΩM).
RΣ 1 r ∗ ∗
3
In the dimension range 0≤ ∗ ≤ d−1, we may compute H (ΩM) using the Serre spectral
∗
sequence associated to the path-space fibration ΩM → PM → M. This has the form
E2 = H (M)⊗H (ΩM) =⇒ H (pt)
p,q p q ∗
with coefficients in R . From the multiplicative structure on the dual cohomology spectral
Σ
sequence it follows that the indecomposable elements (with basis x∗) lie in the image of the
j
transgression. Therefore the classes x are transgressive and the transgress onto the classes
j
u . Thus in the spectral sequence we have d(x ) = u .
j j j
The homology of M being torsion-free implies that the cohomology of M is just the
dual. From the dual spectral sequence, we deduce that the classes x ⊗u are mapped by
k j
differentials onto the classes u ⊗u on the vertical 0-line. It follows that in degrees ≤ d−1,
k j
H (ΩM) are generated by the classes u ,u u . The differential on the class [M] hits a linear
∗ i i j
combination of x ⊗u . Hence, in this range of degrees H (ΩM)∼= T (u ,··· ,u )/(l) for
k j ∗ RΣ 1 r
some element l of homogeneous degree 2 in u .
i
Let H (ΩM)(2) denote the free R -submodule generated by the homogeneous degree 2
∗ Σ
elements which is isomorphic to R {u ⊗u }/(l). The computations of [2] as quoted above
Σ i j
imply that
R {u ⊗u }/(l)⊗ k ∼= R {u ⊗u }/ (−1)|ui|+1c u u ⊗ k
Σ i j RΣ Σ i j ji i j RΣ
(cid:0)P (cid:1)
for k being either the fraction field of R , or R/(π) for primes π in R . The first case
Σ Σ
implies that there are a,b ∈R such that
Σ
al = b (−1)|ui|+1c u u
ji i j
(cid:0)P (cid:1)
and the second cases imply that a and b are non-zero and differ by a unit modulo π for
every prime π. Thus a and b are forced to be units after possible cancellations, and we may
take l = (−1)|ui|+1c u u . Thus
ji i j
P
H (ΩM)(2) ∼= R {u ⊗u }/ (−1)|ui|+1c u u
∗ Σ i j ji i j
(cid:0)P (cid:1)
so that the element (−1)|ui|+1c u u ∈ T (u ,··· ,u ) goes to 0 under φ above. Thus
ji i j RΣ 1 r
we obtain a ring mapP
T (u ,··· ,u )/ (−1)|ui|+1c u u )→ H (ΩM;R
RΣ 1 r ji i j ∗ Σ
(cid:0)P (cid:1)
which is an isomorphism after tensoring with the fraction field of R or going modulo a
Σ
prime from Proposition 2.1. The result now follows. (cid:3)
3. Homotopy groups of certain (n−1)-connected manifolds
In this section we deduce results about the homotopy groups of (n−1)-connected mani-
folds of dimension d ≤ 3n−2 after inverting finitely many primes. We use the computation
of thehomology oftheloop spaceinSection 2. Note fromProposition 2.2 thatH (ΩM)is a
∗
quadratic algebra. We prove that this possesses a nice basis and so does the corresponding
quadratic Lie algebra. The basis of the Lie algebra is used to express π (M) as a direct
∗
sum of homotopy groups of spheres after inverting finitely many primes.
3.1. Algebraic preliminaries. We start by recalling some algebraic preliminaries on qua-
dratic algebras and quadratic Lie algebras. For further details we refer to [1, 11, 10]. Let
A be a commutative ring (usually a principal ideal domain (PID)). If V is a free A-module
then we shalldenote by T (V) (often abbreviated as T(V)) the tensor algebra generated by
A
V. The notation Lie(V) (respectively Liegr(V)) denotes the free Lie algebra (respectively
graded Lie algebra) on the A-module V.
4 SAMIKBASUANDSOMNATHBASU
Definition 3.1. For R ⊂ V ⊗ V, the associative algebra A(V,R) = T(V)/(R) is called a
A
quadratic A-algebra.
If R ⊂ V ⊗ V lies in Lie(V), the Lie algebra L(V,R) = Lie(V)/((R)) is called a
A
quadratic Lie algebra over A. In the graded case this is denoted Lgr(V,R).
Itmay beobserved that the universal enveloping algebra of L(V,R) is A(V,R) and in the
graded case the universal enveloping algebra of Lgr(V,R) is A(V,R) as graded modules. If
inaddition themodulesA(V,R) andL(V,R) arefree, thereis aPoincar´e-Birkhoff-Witt the-
orem which may be stated as E (A(V,R)) ∼= A[L(V,R)]. The notation A[L(V,R)] denotes
0
the polynomial A-algebra on the module L(V,R) and E (A(V,R)) denotes the associated
0
graded for the filtration of A(V,R) induced by the weight filtration on the tensor algebra.
A similar statement holds for the graded case where one interprets the polynomial algebra
as the polynomial algebra on even degree classes tensored with the exterior algebra on the
odd degree classes. Finally from [4], [8] one may deduce that for a PID A, L(V,R) is a free
module if A(V,R) is free and V has finite rank.
Next we recall the Diamond lemma from [3]. Suppose that the free module R can be
given a basis where each element is of the form W −f where W is a monomial. Call a
i i i
monomial R-indecomposable if it does not possess any submonomial which occurs as W
i
in the above chosen basis. The Diamond lemma states certain sufficient conditions under
which the R-indecomposable monomials form a basis of A(V,R). The following implication
of the Diamond lemma suffices for this paper.
Proposition 3.2. Suppose that R is generated by a single element of the form
x ⊗x = a x ⊗x
a β i,j i j
X
(i,j)6=(α,β)
with α 6= β. Then the R-indecomposable elements form a basis for A(V,R).
There is an analogous construction for Lie algebras L(V,R) defined by generators and
relations (see [7]). This is called a Lyndon basis. Start with a basis of V and an order
on the basis set. We call a word in elements of V a Lyndon word if it is lexicographically
smaller than its cyclic rearrangements. For a Lyndon word l there are unique Lyndon
words l and l so that l = l l and l is the largest possible Lyndon word occuring in the
1 2 1 2 2
right in l. Inductively we may associate to the Lyndon word l the free Lie algebra element
b(l) = [b(l ),b(l )]. One verifies that this is a basis of the free Lie algebra on V. In the case
1 2
R 6= 0, we say a Lyndon word is R-standard if it cannot be further reduced using relations
in R (with respect to the chosen order on V). From [7] we recall the following result (also
see [1], Theorem 2.17).
Proposition 3.3. Supposethat A is a localization of Z and R as in Proposition 3.2. Then,
the R-standard Lyndon words give a basis of L(V,R).
3.2. Homotopy groups using loop space homology. Let us denote by l(M) the sum
(−1)|ui|+1c u u . It is clear that l(M) lies in the free graded Lie algebra on the classes
ji i j
Pu1,··· ,ur which we denote by Liegr(u1,··· ,ur). Consider the graded Lie Algebra Lgkr(M)
(over Z) given by
Liegr(u ,··· ,u )
1 k
(l(M))
where (l(M)) denotes the graded Lie algebra ideal generated by l(M). This is a quadratic
graded Lie algebra. In this respect we denote
l(M) = l [u ,u ]gr = l (u ⊗u −(−1)|ui||uj|u ⊗u ).
i,j i j i,j i j j i
Xi<j Xi<j
5
We make an analogous ungraded construction. Consider the element
lu(M) = l [u ,u ]= l (u ⊗u −u ⊗u ).
i,j i j i,j i j j i
Xi<j Xi<j
This element lies in Lie(u ,...,u ). We shall make use of the following notation:
1 r
T (u ,...,u ) Lie(u ,...,u )
Au(M) = RΣ 1 r , Lu(M) = 1 r .
r (lu(M)) r (lu(M))
The Lie algebra Lu(M) and the associative algebra Au(M) possess an induced grading.
r r
Proposition 3.4. Lu(M) and Lgr(M) are free over R . The Lyndon basis gives a basis of
r r Σ
Lu(M).
r
Proof. From the formulas in Proposition 2.2, it is clear that H (ΩM) is the universal en-
∗
veloping algebra of the Lie algebra Lgr(M) in the graded sense. Analogously, Au(M) is the
r
universal enveloping algebra of Lu(M). As R is a PID, for the first statement it suffices
Σ
to show that H (ΩM) and Au(M) are free R -module. We verify this last fact by proving
∗ r Σ
l(M) and lu(M) satisfies the hypothesis of Proposition 3.2. Now Proposition 3.3 implies
the second statement as well.
Since the coefficients of l(M) and lu(M) differ only by a sign, it suffices to write the
element l(M) as
u u = terms not containing u u .
i j i j
Thisis equivalent to a change of basis of the x so thatsome c equals 1. We have seen that
i ji
this is possible for d = 2n in [1]. For d > 2n note that n and d−n are not the same. Now
pick the basis x so that the dual classes in Hn(M) are Poincar´e dual to those in Hd−n(M)
i
(this is possible as n 6= d). For example we may start with a basis of Hn(M) and then
2
the dual basis of Hd−n(M) and extend to a basis of H0<∗<d(M). Order the basis so that
x∗ ∈ Hn(M) and x∗ ∈ Hd−n(M) is the dual element. Then c = 1 and thus
1 r 1,r
l(M) = (−1)|ui|+1c u u = u u +combination of other terms
ji i j 1 r
X
This completes the proof. (cid:3)
Next we enlarge the set of primes so that the classes u are in the image of the Hurewicz
i
homomorphism.
Proposition 3.5. There exists a finite set of primes Γ containing Σ such that the classes
u lie in the image of the Hurewiciz homomorphism π (ΩM)⊗R → H (ΩM;R ).
i ∗ Γ ∗ Γ
Proof. Consider the commutative diagram
π (ΩM)⊗R Hur // H (ΩM;R ) .
∗ Σ ∗ Σ
(cid:15)(cid:15) (cid:15)(cid:15)
π (ΩM)⊗Q Hur // H (ΩM;Q)
∗ ∗
Since H (ΩM;R ) is a free R -module, the right vertical arrow is injective; it takes u to
∗ Σ Σ i
the corresponding element u . We know from the Milnor-Moore theorem that H (ΩM;Q)
i ∗
is the universal enveloping algebra on the rational homotopy Lie algebra π (M)⊗ Q. It
∗
follows by standard methods (for e.g., from formality of M and minimal models) that x ’s
i
are the generators of homotopy Lie algebra in the appropriate degrees. Thus, the element
u lie in the Hurewicz homomorphism for Q coefficients. It follows that for every i there is
i
an integer d so that d u lies in the image of the Hurewicz homorphism. Define Γ as Σ plus
i i i
all the prime factors of d for 1 ≤ i≤ r. Consequently, over R , all the u are in the image
i Γ i
of the Hurewicz homomorphism. (cid:3)
6 SAMIKBASUANDSOMNATHBASU
Our goal is to compute the homotopy groups π (M) ⊗ R . We work in the R -local
∗ Γ Γ
category: thatis,thecategoryobtainedfromspacesbylocalizingwithrespecttoH (−;R )-
∗ Γ
equivalences. Let Sn denote the R -local sphere. We know that if a map between simply
Γ Γ
connected spaces (or, more generally, simple spaces) is a H (−;R )-equivalence then it
∗ Γ
induces an isomorphism on π (−)⊗R .
∗ Γ
FromProposition3.5, thereareelements inπ (M)⊗R whichloopsdowntou underthe
∗ Γ i
Hurewiczmap. By iterated Whiteheadproductswemay mapspheresintoM corresponding
to chosen elements of the Lie algebra Lu(M). We describe this in a precise fashion below.
Let d(u ) denote the degree of u . We fix a map Sd(ui) → ΩM which maps to u under
i i i
the Hurewicz homomorphism. By adjunction we have a map α : Sd(ui)+1 → M with the
i
property that after looping α the generator of the Pontrjagin ring maps to u .
i i
There exists a Lyndon basis for Lu(M) by Proposition 3.4. Enlist these elements in
order as l < l < ... and define the height of a basis element by h = h(l ) = k + 1 if
1 2 i i
b(l ) ∈ (Lu(M)) the kth-graded piece. Then h(l ) ≤ h(l ). Note that b(l ) represents an
i k i i+1 i
element of Lie(u ,...,u ) and is thus represented by an iterated Lie bracket of u . Define
1 r i
λ : Shi → M as the Whitehead product replacing each u in the bracket by α .
i Γ i i
Theorem 3.6. There is an isomorphism
π (M)⊗R ∼= π Shi ⊗R
∗ Γ ∗ Γ
Xi≥1
and the inclusion of each summand is given by λ .
i
Observe that the right hand side is a finite sum in each degree.
Proof. The maps Ωλ : ΩShi → ΩM for i = 1,...,n can be multiplied using the H-space
i Γ
structure on ΩM to obtain a map from S(n) = n ΩShi → ΩT. Letting n vary S(n)
i=1 Γ
gives a directed system arising from the inclusionQof subfactors using the basepoint. Fix
an associative model for ΩT (for example using Moore loops) and observe that the various
maps from S(n) induces a map on the homotopy colimit
Λ :S := hocolim S(n) −→ ΩM.
n
Note that homotopy groups of S is the right hand side of the expression in the Theorem
shifted in degree by 1. Hence it suffices to prove that Λ is aweak equivalence after inverting
the primes in Γ. As both the domain and codomain are simple spaces, it suffices to show
that this is a R -homology isomorphism.
Γ
The homology of S is a polynomial algebra with a generator for each copy of ΩShi
Γ
H (S) ∼= T (c )⊗T (c )... ∼= R [c ,c ,...]
∗ RΓ h1−1 RΓ h2−1 Γ h1−1 h2−1
and Λ c is the Hurewicz image of λ ∈H (ΩM). Denote ρ as
∗ hi−1 i hi−1
ρ:π (X) ∼= π (ΩX) −H−u→r H (ΩX)
n n−1 n−1
We know from [6] that
(1) ρ([a,b]) = ρ(a)ρ(b)−(−1)|a||b|ρ(b)ρ(a).
Now from Theorem 2.2 that H ΩM ∼= T (u ,··· ,u )/(l(M)) the universal enveloping
∗ RΓ 1 r
gr
algebra of L (M) (in the graded sense). From the Poincar´e-Birkhoff-Witt theorem for
r
graded Lie algebras we have
E T (u ,··· ,u )/(l(M)) ∼= E(Lgr(M)odd)⊗P(Lgr(M)even)
0 RΓ 1 r r r
7
The map ρ carries each α to u . The element b(l ) is mapped inside H (ΩM) to the
i i i ∗
element corresponding to the graded Lie algebra element by equation (1). We prove that
T(a ,...,a )/(l(M)) has a basis given by monomials on ρ(b(l )),ρ(b(l )),....
1 r 1 2
gr
Observe inductively that all the elements in L (M) can be expressed as linear combi-
r
nations of monomials in ρ(b(l )). It is clear for elements of weight 1. For the weight 2
i
elements note that they are generated by [u ,u ]gr for i < j, (i,j) 6= (1,2) and u2 if d(u(i))
i j i
is odd. The former are the Lyndon words and the latter is the square of a monomial. In
the general case, a graded Lie algebra element is either a monomial or the square of a lower
odd degree class; from one of the conditions in the definition of a graded Lie algebra the
bracket with a square can be expressed as a bracket. Such a monomial may be obtained by
applying ρ on the corresponding ungraded element. This is a linear combination of certain
b(l ) and something in the ideal generated by lu(M). Applying ρ we obtain a combination
i
of ρ(b(l )) and something in the ideal generated by l(M) as ρ(lu(M)) = l(M) which verifies
i
theinductionstep. Asanapplication ofthePoincar´e-Birkhoff-Witt Theorem,weknowthat
Λ is surjective.
∗
Hence we have that the graded map
R [ρ(b(l )),ρ(b(l )),...] → T (u ,...,u )/(l(M))
Γ 1 2 RΓ 1 r
is surjective. We also know
R [b(l ),b(l ),...] → T (u ,...,u )/(lu(M))
Γ 1 2 RΓ 1 r
is an isomorphism. Now both T (u ,...,u )/(l(M)) and T (u ,...,u )/(lu(M)) have
RΓ 1 r RΓ 1 r
bases given by the Diamond lemma and thus are of the same graded dimension. It follows
thatthegradedpiecesofR [ρ(b(l )),ρ(b(l )),...]andT (u ,...,u )/(l(M))havethesame
Γ 1 2 RΓ 1 r
rank which is finite. Thus on graded pieces one has a surjective map between free R -
Γ
modules of the same rank which must be an isomorphism. (cid:3)
We may now compute the number of copies on of Sk in the expression of Theorem 3.6
from the rational cohomology groups of M. Let
q (t) = 1− b (M)ti +td
M i
n−X1<i<d
Then 1 is the generating series for ΩM (see [9], Theorem 3.5.1) and the fact that
qM(t)
H (ΩM) is Koszul as an associative algebra (cf. [2]). Let
∗
η := coefficient of tm in log(q (t)).
m M
We may repeat the proof of Theorem 5.7 of [1] to deduce the following result.
Theorem 3.7. The number of groups π Sj ⊗R in π (M)⊗R is
s Γ s Γ
η
(j−1)/d
l = − µ(d)
j−1
d
X
d|j−1
where µ is the M¨obius function.
Recall that simply connected, finite cell complexes either have finite dimensional rational
homotopy groups or exponential growth of ranks of rational homotopy groups (cf. [5], §33).
The former are called rationally elliptic while the latter are called rationally hyperbolic.
From [2] we note that the (n−1)-connected manifolds of dimension at most (3n−2) with
H∗(M)havingrankatleast4areallrationallyhyperbolic. Onemayalsoverifythisdirectly.
Since the rank of H∗(M) is at least 4 the number of generating u is at least 3. Then one
i
observes that after switching the ordering appropriately the word
u u u u u u
1 2 1 2 1 3
8 SAMIKBASUANDSOMNATHBASU
is a Lyndon word in degree > 2d as each u has degree > d. So these manifolds cannot be
i 3
rationally elliptic. This forces by the Milnor-Moore Theorem, that Lgr(M)⊗Q has infinite
rank. Hence, Lu(M) also has infinite rank.
There are many conjectures that lie in the dichotomy between rationally elliptic and
hyperbolic spaces. We verify such a conjecture by Moore ([5], pp. 518) below. For a
rationally hyperbolicspace X the Moore conjecture states that there are primes p for which
the homotopy groups do not have any exponent at p, that is, for any power pr there is an
element α ∈π (X) of order pr. We verify the following version.
∗
Theorem 3.8. If p ∈/ Γ, the homotopy groups of M do not have any exponent at p.
Proof. We have noted above that Lu(M) has infinite rank. Thus there are elements of the
Lyndon basis of arbitrarily large degree. Hence for arbitrarily large l, π Sl occurs as a
∗
summand of π M. The proof is complete by observing that any ps may occur as the order
∗
of an element in π Sl for arbitrarily large l. This follows from the fact that the same is true
∗
for the stable homotopy groups and these can be realized as πs ∼= π Sl for l > k+1. Now
k k+l
torsion of order ps for any s occurs in the image of the J-homomorphism (cf. [12], Theorem
1.1.13). (cid:3)
3.3. The low rank cases. We end by demonstrating the above computations when the
rank of H (M;Q) is at most 4. Since H0 and Hd are always Q, we have to consider three
∗
possibilities: rank 2,3,4. Our main techniques involve determining the rational homotopy
type of M, of dimension d, and using it to compute the homotopy type at all but finitely
many primes.
In the rank 2 case we know that rationally M is a sphere so that M ≃ Sd. Let Σ
Q Q
denote the finite set of primes which occur as torsion in the homology of M. With R -
Σ
coefficients, the R localization M of M is a homology d-sphere. Thus H∗(M;R) ∼=
Σ Σ
H∗(Sd;R) for any ring R lying between R and Q. Let α stand for the common notation
Σ
foragenerator of H (M;R)foranysuchR. AsM ≃ Sd,αlies intheimage oftherational
∗ Q Q
Hurewicz homomorphism. It follows that with R -coefficients, there is an integer k so that
Σ
kα lies in the image of the Hurewicz homomorphism. Let Γ denote the union of Σ and
the prime divisors of k. Then Γ is a finite set and α lies in the image of the R Hurewicz
Γ
homomorphism. It is now clear that there is a map Sd → M which is an isomorphism
Γ
with R -coefficients. Therefore, we have a homotopy equivalence M ≃ Sd. Note that the
Γ Γ Γ
torsion in the homology can be quite varied. So this is precisely the sort of result we are
looking for.
For the next case, let J Sn be the second stage of the James construction which is
2
obtained as the mapping cone of the Whitehead product [id,id]. If H∗(M;Q) has rank
3, by Poincar´e duality the cohomology ring is forced to be Q[x ]/(x3) where d = 2s. By
s s
gradedcommutativity sis forcedtobeeven. Therational homotopy of suchaspacemay be
computed directly from thecohomology ringstructureas theringstructureforces thespace
tobeformal. Theminimalmodelis given by Λ(x ,y )withd(x )= 0andd(y )= x3.
s 3s−1 s 3s−1 s
Thus the rational homotopy groups of M are given by
Q if k = s,3s−1
Q
π (M) =
k (cid:26) 0 otherwise.
It follows that on rationalizations we have map Ss → M which is an isomorphism on
Q Q
πQ for ∗ ≤ 2s−2. In degree 2s−1 the homotopy groups are πQ Ss ∼= Q{[id,id]} and
∗ 2s−1
Q
π (M) = 0. Therefore the composite
2s−1
S2s−1 −[i−d−,i→d] Ss −→ M
Q Q Q
9
is null-homotopic and thus factors through the cofibre (J Ss) . Therefore we obtain a map
2 Q
(J Ss) → M which is an isomorphism on Hs and by cup products also on Hd. As a
2 Q Q
result, one obtains that M ≃ J Ss.
Q 2 Q
Nextweupgradetherationalhomotopyresulttoonewhichisvalidafterinvertingfinitely
many primes, that is, over a set Γ of finitely many primes that the Γ-localizations of the
abovetwospacesareweakly equivalent. LetΣdenotealltheprimeswhichappearastorsion
in the homology of M. As M is a compact CW-complex, the homotopy groups of M are
finitely generated. LetΓ denotetheprimes inΣ together with thefinitelist of primeswhich
appear as torsion in π (M) and those which need to be inverted so that x lies in the
2s−1 s
image of the Hurewicz homomorphism. We consider the following commutative diagram:
S2s−1 [id,id]// Ss // M
Γ Γ Γ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
S2s−1 [id,id]// Ss // M
Q Q Q
The composite in the bottom row is 0. The composite in the top row gives an element in
π (M)⊗R which injects into π (M)⊗Q by our choice of Γ. The latter group is 0
2s−1 Γ 2s−1
from our choices. It follows that the composite of the top row is 0 and thus we obtain a
map from the mapping cone J Ss → M which is an isomorphism in cohomology with R
2 Γ Γ Γ
coefficients by the same argument as that for Q. Thus we deduce that
M ≃ (J Ss) .
Γ 2 Γ
Itremainstoconsiderthelastcasewhentotalrankis4. Let#2J (n)denotethemapping
2
cone of
[id1,id1]+[id2,id2]: S2n−1 → Sn ∨Sn
Then,
H∗(#2J (n);Q) ∼= Q[x ,y ]/(x3,y3,x y ,x2 = y2).
2 n n n n n n n n
If H∗(M;Q) has rank 4, then by Poincar´e duality the rational cohomology ring is forced to
be one of the following:
(a) {1,x ,y ,x2 = y2} (where d= 2s with s even),
s s s s
(b) {1,x ,y ,x ·y }.
k d−k k d−k
Notice that (a) is the rational cohomology ring of #2J (s) while (b) is the rational coho-
2
mology ring of Sk ×Sd−k. We now deduce that the rational homotopy type of M must
indeed be one of these.
The rational homotopy groups may be computed directly as the ring structure forces the
space to be formal. The minimal model for type (a) is given by
Λ(x ,y ,u ,v ),d(x ) = d(y )= 0,d(u ) = x2−y2,d(v )= x y .
s s 2s−1 2s−1 s s 2s−1 s s 2s−1 s s
Thus the rational homotopy groups of M are given by
Q if k = 0
πQ(M) =  Q2 if k = s,2s−1
k
 0 otherwise.
Therefore, on rationalizations we havea map Ss ∨Ss → M representing x and y which
Q Q Q s s
Q
is an isomorphism on π for ∗ ≤ 2s−2. In degree 2s−1 the homotopy group
∗
πQ (Ss ∨Ss) ∼= Q{[id1,id1],[id1,id2],[id2,id2]}.
2s−1
10 SAMIKBASUANDSOMNATHBASU
Inthecomputation ofhomotopygroupsusingminimalmodelsoneknowsthatthequadratic
part of the differential represents the Whitehead product, and so it follows that the element
[id1,id1]+[id2,id2] goes to 0 in M. Therefore the composite
S2s−1 −[−id−1−,id−1−]+−[−id−2−,id−→2] Ss −→ M
Q Q Q
is null-homotopic and thus factors through the cofibre (#2J (s)) . Therefore we obtain a
2 Q
map (#2J (s)) → M which is an isomorphism on Hs and by cup products also on Hd.
2 Q Q
It follows that M ≃ #2J (s) .
Q 2 Q
If the cohomology algebra is of type (b), the minimal model matches that for the prod-
uct Sk × Sd−k. Thus, on rationalizations we have a map Sk ∨ Sd−k → M which is an
Q Q Q
Q
isomorphism on π for ∗ ≤ 2s−2. In degree 2s−1 the class [id ,id ] generates a copy
∗ k d−k
of Q in πQ (Sk ∨Sd−k). As in the argument above, one shows that [id ,id ] goes to 0
2s−1 k d−k
in M. Therefore we obtain a map (Sk×Sd−k) → M which is an isomorphism on H≤d−1
Q Q
and by cup products also on Hd. It follows that M ≃ (Sk ×Sd−k) .
Q Q
As in the r = 3 case we upgrade the rational homotopy result to one which is valid after
inverting finitely many primes. Let Σ denote all the primes which appear as torsion in
the homology of M and Γ denote the primes in Σ together with the finite primes which
appear as torsion in π (M), and so that the generators of H lie in the image of the
2s−1 ≤d−1
R -Hurewicz homomorphism. Let φ denote [id1,id1]+[id2,id2] or [id1,id2] accordingly as
Γ
H∗(M) is of type (a) or (b). We consider the following commutative diagram
S2s−1 φ // Ss ∨Ss // M
Γ Γ Γ Γ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
S2s−1 φ // Ss ∨Ss // M
Q Q Q Q
The composite in the bottom row is 0. The composite in the top row gives an element in
π (M)⊗R which injects into π (M)⊗Q by our choice of Γ. The class φ maps to 0
2s−1 Γ 2s−1
in the latter group as proved above. It follows that the composite of the top row is 0 and
thus we obtain a map from the mapping cone
Cone(φ) → M
Γ Γ
which is an isomorphism in cohomology with R coefficients by the same argument as that
Γ
for Q. Thus we deduce that
M ≃ Cone(φ) .
Γ Γ
We summarize all the above computations and observations in the result below.
Theorem 3.9. Let M be a (n−1)-connected d-manifold with d ≤ 3n−2. Suppose that
the total rank of H∗(M;Q) is at most 4.
(i) If the rank is 2, then there is a finite set of primes Γ such that M ≃ Sd.
Γ Γ
d/2
(ii) If the rank is 3, then there is a finite set of primes Γ such that M ≃ J S .
Γ 2 Γ
(iii) If the rank is 4, then there is a finite set of primes Γ such that M ≃ (#2J (d)) or
Γ 2 2 Γ
M ≃ (Sk ×Sd−k) .
Γ Γ
References
[1] S. Basu and S. Basu, Homotopy groups of highly connected manifolds. available at
http://arxiv.org/abs/1510.05195.
[2] A. Berglund and K. Bo¨rjeson,Free loop space homology of highly connected manifolds.available at
http://arxiv.org/abs/1502.03356.