Steenrod coalgebras

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by  Justin Smith

Steenrod Coalgebras
Justin R. Smith
5 Department ofMathematics
1 DrexelUniversity
0
Philadelphia, PA19104
2

r
a
M

6
Abstract

]
T This paper shows that a functorial version of the “higher diago-
A
nal” of a space used to compute Steenrod squares actually con-
.
h tains far more topological information — including (in some cases)
t
a the space’s integral homotopy type.
m
[

1. Introduction
6
1
v It is well-known that the Alexander-Whitney coproduct is func-
8
torial with respect to simplicial maps. If X is a simplicial set, C(X)
1
6 is the unnormalized chain-complex and RS is the bar-resolution of
2
3 Z (see [12]), it is also well-known that there is a unique homotopy
. 2
1 class of Z -equivariant maps (where Z transposes the factors of
0 2 2
4 the target)
1 ξ :RS ⊗C(X) → C(X)⊗C(X)
X 2
:
v
and that this extends the Alexander-Whitney diagonal. We will call
i
X
such structures, Steenrod coalgebras and the map ξ the Steenrod
X
r
a diagonal. In his construction of cup-i products, Steenrod defined a
kind of dual of this map in [24].
With some care (see appendix B), one can construct ξ in a
X
manner that makes it functorial with respect to simplicial maps al-
though this is seldom done since the homotopy class of this map is
what is generally studied. Essentially, [18, 20, 21] show that C(X)
Email address: [email protected](Justin R. Smith)
URL: http://vorpal.math.drexel.edu(Justin R. Smith)
Preprintsubmitted to Elsevier March 9, 2015
1 INTRODUCTION
possesses the structure of a functorial coalgebra over an operad S
(see example 2.9) and that the arity-2 portion of this operad-action
is a functorial version of ξ . Throughout this paper, we will assume
X
this functorial version of ξ .
X
It is natural to ask whether ξ encapsulates more information
X
about a topological space than its cup-product and Steenrod
squares. The present paper answers this question affirmatively
for degeneracy-free simplicial sets. Roughly speaking, these
are simplicial sets whose degeneracies do not satisfy any
relations other than the minimal set of identities all face- and
degeneracy-operators must satisfy — see definition A.4 and
proposition A.5. Every simplicial set is canonically homotopy
equivalent to a degeneracy-free one (see proposition A.6). The
only place degeneracy-freeness is used in this paper is lemma 5.3.
Theorem. 5.9LetX andY bepointed, reduceddegeneracy-free(see
definition A.4 in appendix A) simplicial sets with normalized chain-
complexes N(X) and N(Y), let R = Z for some prime p or a subring
p
of Q, and let
f:N(X)⊗R → N(Y)⊗R
be a (purely algebraic) chain map that makes the diagram
RS ⊗N(X)⊗R 1⊗f //RS ⊗N(Y)⊗R (1.1)
2 2
ξX⊗1 ξY⊗1
(cid:15)(cid:15) (cid:15)(cid:15)
N(X)⊗R ⊗N(X)⊗R //N(Y)⊗R ⊗N(Y)⊗R
f⊗f
commute exactly (i.e., not merely up to a chain-homotopy). Then f
induces a simplicial map
f :R X → R Y
∞ ∞ ∞
where R denotes the R-completion, (see [1] or [6] for this concept)
∞
2
1 INTRODUCTION
that makes the diagram
X Y
φX φY
(cid:15)(cid:15) (cid:15)(cid:15)
R X f∞ //R Y
∞ ∞
qX qY
(cid:15)(cid:15) (cid:15)(cid:15)
R˜X //R˜Y
Γf
commute. If f is surjective, then f is a fibration, and if f is also a
∞
homology equivalence, then f is a trivial fibration.
∞
Corollary. If f is a surjective homology equivalence, R = Z, and X
and Y are nilpotent then there exists a homotopy equivalence
¯
f:|X| → |Y|
of topological realizations. Corollary 5.12 states that nilpotent,
degeneracy-free spaces are homotopy equivalent if and only if there
exists a homology equivalence of their chain-complexes that make
diagram 1.1 commute.
Here, R˜∗ is a pointed version of the R-free simplicial abelian
group functor — see definitions 4.2 and 4.5.
Becauseofthecanonicalhomotopyequivalencebetweenallsim-
plicial sets and degeneracy-free ones, the result above implies:
Corollary. 5.13 If X and Y are pointed reduced simplicial sets and
f:C(X) → C(Y)
is a morphism of Steenrod coalgebras — over unnormalized chain-
3
1 INTRODUCTION
complexes — then f induces a commutative diagram
X Y
OO OO
gX gY
d◦f(X) d◦f(Y)
φ(d◦f(X)) φ(d◦f(Y))
(cid:15)(cid:15) (cid:15)(cid:15)
R (d◦f(X)) f∞ //R (d◦f(Y))
∞ ∞
q(d◦f(X)) q(d◦f(Y))
(cid:15)(cid:15) (cid:15)(cid:15)
R˜(d◦f(X)) //R˜(d◦f(Y))
Γ˜f
whereg andg arehomotopyequivalencesifX andY areKancom-
X Y
plexes— and homotopyequivalencesof their topologicalrealizations
otherwise. In particular, if X and Y are nilpotent, R = Z, and f is an
integral homology equivalence, then the topological realizations |X|
and |Y| are homotopy equivalent.
Here, f and d are functors defined in definition A.2 in ap-
pendix A. Singular simplicial sets are always Kan complexes.
The reader might wonder how the Steenrod diagonal can con-
tain any information beyond the structure of a space at the prime
2. The answer is that it forms part of an operad structure that
contains information about all primes — and the only part of this
complex operad structure needed to compute, for instance, Steen-
rod pth powers is the Steenrod diagonal.
For example, let X be a simplicial set with functorial higher
diagonal
h:RS ⊗C(X) → C(X)⊗C(X)
2
let ∆ = h([]⊗∗):C(X) → C(X)⊗C(X) — the Alexander-Whitney di-
agonal — and let ∆ = h([(1,2]⊗∗):C(X) → C(X)⊗C(X). A straight-
2
forward calculation shows that
(1⊗∆)◦∆ :C(X) → C(X)⊗3
2
4
1 INTRODUCTION
has the property that
∂{(1⊗∆)◦∆ } = (1⊗∆)◦∂∆
2 2
= (1⊗∆)◦{(1,2)−1}∆
= (1,2,3)(∆⊗1)◦∆−(1⊗∆)◦∆
= {(1,2,3)−1}(1⊗∆)◦∆ (1.2)
where (1,2,3) is a cyclic permutationof the factors. It follows that ∆
and ∆ incorporate information about X at the prime 3. Although
2
the argument in equation 1.2 is elementary, the author is unaware
of any prior instance of it.
This paper’s general approach to homotopy theory is the end re-
sult of a lengthy research program involving some of the 20th cen-
tury’s leading mathematicians. In [15], Daniel Quillen proved that
the category of simply-connected rational simplicial sets is equiv-
alent to that of commutative coalgebras over Q. In [25], Sullivan
analyzed the algebraic and analytic properties of these coalgebras,
developing the concept of minimal models and relating them to
de Rham cohomology. That work was dual to Quillen’s and had
the advantage of being far more direct.
Since then, a major goal has been to develop a similar theory for
integral homotopy types.
In [17], Smirnov asserted that the integral homotopy type of
a space is determined by a coalgebra-structure on its singular
chain-complex over an E -operad. Smirnov’s proof was somewhat
∞
opaque and several people known to the author even doubted the
result’s validity. In any case, the E -operad involved was complex,
∞
being uncountably generated in all dimensions.
In [21], the author showed that the chain-complex of a space
was naturally a coalgebra over an E -operad S and that this could
∞
be used to iterate the cobar construction. The paper [19] applied
those results to show that this S-coalgebra determined the integral
homotopy type of a simply-connected space.
In [13]1, Mandell showed that the mod-p cochain complex of
a p-nilpotent space had a algebra structure over an operad that
1Based on Mandell’s 1997 thesis.
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2 DEFINITIONS
determined the space’s p-type. In [14], Mandell showed that the
cochains of a nilpotent space whose homotopy groups are all fi-
nite have an algebra structure over an operad that determined its
integral homotopy type.
The paper [18] showed that the S-coalgebra structure of a
chain-complex had a “transcendental” structure that determines a
nilpotent space’s homotopy type (without the finiteness conditions
of [14]). It essentially reprised the main result of [19], using a
very different proof-method. The present paper shows that this
transcendental structure even manifests in the sub-operad of S
generated by its arity-2 component, RS .
2
I am indebted to Dennis Sullivan for several interesting discus-
sions.
2. Definitions
Given a simplicial set, X, C(X) will always denote its unnormal-
ized chain-complex and N(X) its normalized one (with degenera-
cies divided out).
Remark 2.1. Throughout this paper R denotes a fixed ring satisfy-
ing
Z for some prime p or
R = p
(R ⊂ Q
Definition 2.2. We will denote the category of R-free chain chain-
complexes by Ch and ones that are bounded from below in dimen-
sion 0 by Ch .
0
We make extensive use of the Koszul Convention (see [8]) re-
garding signs in homological calculations:
Definition 2.3. If f:C → D , g:C → D aremaps, anda⊗b ∈ C ⊗C
1 1 2 2 1 2
(where a is a homogeneous element), then (f ⊗ g)(a⊗ b) is defined
to be (−1)deg(g)·deg(a)f(a)⊗g(b).
Remark 2.4. If f , g are maps, it isn’t hard to verify that the Koszul
i i
convention implies that (f ⊗g )◦(f ⊗g ) = (−1)deg(f2)·deg(g1)(f ◦f ⊗
1 1 2 2 1 2
g ◦g ).
1 2
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2 DEFINITIONS
The set of morphisms of chain-complexes is itself a chain com-
plex:
Definition 2.5. Given chain-complexes A,B ∈ Ch define
Hom (A,B)
Z
to be the chain-complex of graded R-morphisms where the degree
of an element x ∈ Hom (A,B) is its degree as a map and with differ-
Z
ential
∂f = f ◦∂ −(−1)degf∂ ◦f
A B
As a R-module Hom (A,B) = Hom (A ,B ).
Z k j Z j j+k
Remark. Given A,B ∈ ChSn, weQcan define Hom (A,B) in a corre-
ZSn
sponding way.
Recall the concept of algebraicoperad in [11] or [10]: a sequence
of ZS chain-complexes {V(n)} for n ≥ 0 with structure maps
n
γ :V(n)⊗V(i )⊗···⊗V(i ) → V(i +···i )
i1,...,in 1 n 1 n
for n,i ,...,i ≥ 0.
1 n
Definition 2.6. We will call the operad V = {V(n)} Σ-cofibrant if
V(n) is ZS -projective for all n ≥ 0.
n
Remark. The operads we consider here correspond to symmetric
operads in [22].
The term “unital operad” is used in different ways by different
authors. We use it in the sense of Kriz and May in [10], mean-
ing the operad has a 0-component that acts like an arity-lowering
augmentation under compositions. Here V(0) = R.
The term Σ-cofibrant first appeared in [3].
We also need to recall compositions in operads:
Definition 2.7. If V is an operad with components V(n) and V(m),
define the ith composition, with 1 ≤ i ≤ n
◦ :V(n)⊗V(m) → V(n+m−1)
i
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2 DEFINITIONS
by
V(n)⊗V(m)
V(n)⊗Zi−1 ⊗V(m)⊗Zn−i
1⊗ηi−1⊗1⊗ηn−i
(cid:15)(cid:15)
V(n)⊗V(1)i−1 ⊗V(m)⊗V(1)n−i
γ
(cid:15)(cid:15)
V(n+m−1)
Here η:Z → V(1) is the unit.
Remark. Operads were originally called composition algebras and
defined in terms of these operations — see [5].
It is well-known that the compositions and the operad
structure-maps determine each other — see definition 2.12 and
proposition 2.13 of [22].
A simple example of an operad is:
Example 2.8. For each n ≥ 0, S (n) = ZS , with structure-map a
0 n
Z-linear extension of
γ :S ×S ×···×S → S
α1,...,αn n α1 αn α1+···+αn
defined by
γ (σ ×θ ×···×θ ) = T (σ)◦(θ ⊕···⊕θ )
α1,...,αn 1 n α1,...,αn 1 n
with σ ∈ S and θ ∈ S where T (σ) ∈ S is a permutation
n i αi α1,...,αn Pαi
that permutes the n blocks
{1,...,α },{α +1,α +α },...,
1 1 1 2
{α +···+α +1,α +···+α }
1 n−1 1 n
via σ. See [21] for explicit formulas and computations.
Another important operad is:
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2 DEFINITIONS
Example 2.9. The operad, S, defined in [21] is given by S(n) = RS
n
— the normalized bar-resolution of Z over ZS . This is well-known
n
(like the closely-related Barrett-Eccles operad defined in [2]) to be
a Hopf-operad, i.e. equipped with an operad morphism
δ:S → S⊗S
and is important in topological applications. See [21] for formulas
for the structure maps.
For the purposes of this paper, the main example of an operad
is
Definition 2.10. Given any C ∈ Ch, the associated coendomor-
phism operad, CoEnd(C) is defined by
CoEnd(C)(n) = Hom (C,C⊗n)
Z
Its structure map
γ :Hom (C,C⊗n)⊗Hom (C,C⊗α1)⊗···⊗Hom (C,C⊗αn) →
α1,...,αn Z Z Z
Hom (C,C⊗α1+···+αn)
Z
simply composes a map in Hom (C,C⊗n) with maps of each of the n
Z
factors of C.
This is a non-unital operad, but if C ∈ Ch has an augmen-
tation map ε:C → R then we can regard ε as the generator of
CoEnd(C)(0) = R ·ε ⊂ Hom (C,C⊗0) = Hom (C,R).
Z Z
We use the coendomorphism operad to define the main object of
this paper:
Definition 2.11. A coalgebra over an operad V is a chain-complex
C ∈ Ch with an operad morphism α:V → CoEnd(C), called its struc-
ture map. We will sometimes want to define coalgebras using the
adjoint structure map,
α:C → Hom (V(n),C⊗n) (2.1)
ZSn
n≥0
Y
9
2.1 Coalgebrasoveroperads 2 DEFINITIONS
where S acts on C⊗n by permuting factors or the set of chain-maps
n
α :C → Hom (V(n),C⊗n)
n ZSn
for all n ≥ 0 or even
β :V(n)⊗C → C⊗n
n
It is not hard to see how compositions (in definition 2.7) relate to
coalgebras
Proposition 2.12. If the maps β :V(n)⊗C → C⊗n for all n ≥ 0 define
n
a coalgebra over an operad V, for any x ∈ V(n) and any n ≥ 0 define
∆ = β (x⊗∗):C → C⊗n
x n
If x ∈ V(n) and y ∈ V(m), then
∆ = 1⊗···⊗1⊗∆ ⊗1⊗···⊗◦∆
y◦ix y x
ithposition
| {z }
Proof. Immediate, from definitions 2.7 and 2.10.
2.1. Coalgebras over operads
Example 2.13. Coassociative coalgebras are precisely the coalge-
bras over S (see 2.8).
0
Definition 2.14. Comm is an operad defined to have one basis el-
ement {b } for each integer i ≥ 0. Here the arity of b is i and the
i i
degree is 0 and the these elements satisfy the composition-law:
γ(b ⊗b ⊗···⊗b ) = b , where K = n k . The differential of this
n k1 kn K i=1 i
operad is identically zero. The symmetric-group actions are trivial.
P
Example 2.15. Coassociative, commutative coalgebras are the
coalgebras over Comm.
We can define a concept dual to that of a free algebra generated
by a set:
10