Higher homotopy associativity of power maps on *p*-regular *H*-spaces
Yusuke Kawamoto (National Defense Academy of Japan)

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Algebraic and Geometric Models for Spaces and Related Topics 2014 September 18, 2014, Shinshu University

All spaces are assumed to be pointed, arcwise connected and of the ho-
motopy type of *CW* -complexes.

Let (*X, µ*) be a homotopy associative *H*-space. From the above assump-
tion, (*X, µ*) is a group-like space. The power maps *{Φ*^{X}_{λ} : *X* *→* *X}*_{λ}_{∈Z}
are deﬁned as follows:

*•* *Φ*^{X}_{0} (*x*) = *x*_{0}

*•* *Φ*^{X}_{λ} (*x*) = *µ*(*Φ*^{X}_{λ}_{−}_{1}(*x*)*, x*) for *λ >* 0

*•* *Φ*^{X}_{λ} (*x*) = *ι*(*Φ*^{X}_{−}_{λ}(*x*)) for *λ <* 0,

where *x*_{0} *∈* *X* and *ι*: *X* *→* *X* denote the homotopy unit and the homotopy
inverse on (*X, µ*), respectively.

*•* (*X, µ*) is homotopy commutative *⇐⇒ {*^{iﬀ} *Φ*^{X}_{λ} *}*_{λ}_{∈Z} are *H*-maps

*•* If *X* is a double loop space, then *{Φ*^{X}_{λ} *}*_{λ}_{∈Z} are loop maps
Theorem. [Sullivan 1974]

Let *p* be an odd prime and *t* *≥* 1. Then *S*^{2t}^{−}^{1}

(*p*) is a loop space

*⇐⇒*iﬀ *t|*(*p* *−* 1).

We denote the loop space *S*^{2t}^{−}^{1}

(*p*) by *W*_{t}.

Theorem 1. [Arkowitz-Ewing-Schiﬀman 1975]

Let *p* be an odd prime. The power map *Φ*^{W}_{λ} ^{p}^{−}^{1} on *W*_{p}_{−}_{1} is an *H*-map

*⇐⇒*iﬀ *λ*(*λ* *−* 1) *≡* 0 mod *p*.

Remark.

*•* When *t* *̸*= *p* *−* 1, all the power maps *{Φ*^{W}_{λ} ^{t}*}*_{λ}_{∈Z} on *W*_{t} are *H*-maps
since the multiplication on *W*_{t} is homotopy commutative.

*•* Theorem 1 is generalized to the case of several *p*-localized ﬁnite loop
spaces by [McGibbon 1980] and [Theriault 2013].

Theorem 2. [Lin 2012]

Let *p* be an odd prime and *t* *≥* 1 with *t|*(*p* *−* 1). The power map *Φ*^{W}_{λ} ^{t} on
*W*_{t} is a loop map *⇐⇒*^{iﬀ} *λ* = *α*^{t} for some *p*-adic integer *α* *∈* Z^{∧}_{p} .

Remark 3.

*•* When *λ* *̸≡* 0 mod *p*, Theorem 2 is proved by [Rector 1971] and [Arkowitz-
Ewing-Schiﬀman 1975].

*•* Theorem 2 can also be derived from [Adams-Wojtkowiak 1989] and [Wo-
jtkowiak 1990].

Corollary 4.

Let *p* and *t* be as in Theorem 2. Put *m* = (*p* *−* 1)*/t*. Assume *λ* *̸*= 0 and
write *λ* = *p*^{a}*b* with *a* *≥* 0 and *b* *̸≡* 0 mod *p*. The power map *Φ*^{W}_{λ} ^{t} on *W*_{t}
is a loop map *⇐⇒*^{iﬀ} *t|a* and *b*^{m} *≡* 1 mod *p*.

Deﬁnition. [Sugawara 1957], [Stasheﬀ 1963]

A space *X* is an *A*_{n}-space *⇐⇒*^{def}

*∃**{µ*_{i} : *K*_{i} *×* *X*^{i} *→* *X}*1*≤**i**≤**n*

with some relations, where *{K*_{i}*}*_{i}_{≥}_{1} denote the associahedra constructed
by [Stasheﬀ 1963].

*K*_{3}

(*xy*)*z* *x*(*yz*)

*µ*_{3}(*t, x, y, z*)

**t** **t**

*K*_{4}

((*xy*)*z*)*w* (*x*(*yz*))*w*

(*xy*)(*zw*) *x*((*yz*)*w*)

*x*(*y*(*zw*))

*xµ*_{3}(*t, y, z, w*)

*µ*_{3}(*t, x,* (*yz*)*, w*)
*µ*_{3}(*t, x, y, z*)*w*

*µ*_{3}(*t,* (*xy*)*, z, w*)
*µ*_{3}(*t, x, y,* (*zw*))

*µ*_{4}(*a, x, y, z*)

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*•* *X* is an *A*_{2}-space *⇐⇒*^{iﬀ} *X* is an *H*-space

*•* *X* is an *A*_{3}-space *⇐⇒*^{iﬀ} *X* is a homotopy associative *H*-space

*•* *X* is an *A*_{∞}-space *⇐⇒*^{iﬀ} *X* *≃* *Ω*(*BX*) for some space *BX* by [Sugawara
1957] and [Stasheﬀ 1963]

Deﬁnition. [Sugawara 1960], [Stasheﬀ 1970], [Iwase-Mimura 1989]

Let *X, Y* be *A*_{n}-spaces. A map *f* : *X* *→* *Y* is an *A*_{n}-map *⇐⇒*^{def}

*∃**{η*_{i}: *J*_{i} *×* *X*^{i} *→* *Y* *}*1*≤**i**≤**n*

with some relations, where *{J*_{i}*}*_{i}_{≥}_{1} denote the multiplihedra constructed
by [Iwase-Mimura 1989].

*J*_{2}

*f*(*xy*)
*f*(*x*)*f*(*y*)

*η*_{2}(*t, x, y*)

**t**
**t**

*J*_{3}

(*f*(*x*)*f*(*y*))*f*(*z*) *f*(*x*)(*f*(*y*)*f*(*z*))

*f*(*x*)*f*(*yz*)

*f*(*x*(*yz*))
*f*((*xy*)*z*)

*f*(*xy*)*f*(*z*)

*µ*^{Y}_{3} (*t, f*(*x*)*, f*(*y*)*, f*(*z*))

*f*(*x*)*η*_{2}(*t, y, z*)

*η*_{2}(*t, x,* (*yz*))
*f*(*µ*^{X}_{3} (*t, x, y, z*))

*η*_{2}(*t,* (*xy*)*, z*)
*η*_{2}(*t, x, y*)*f*(*z*)

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*•* *f* : *X* *→* *Y* is an *A*_{2}-map *⇐⇒*^{iﬀ} *f* is an *H*-map

*•* *f* : *X* *→* *Y* is an *A*_{3}-map *⇐⇒*^{iﬀ} *f* is an *H*-map preserving homotopy
associativity homotopically

*•* *f* : *X* *→* *Y* is an *A*_{∞}-map *⇐⇒*^{iﬀ} *f* *≃* *Ω*(*Bf*) for some map *Bf* : *BX* *→*
*BY* by [Sugawara 1960], [Stasheﬀ 1970] and [Iwase-Mimura 1989]

In this talk, we study the condition for the power map on an *A*_{n}-space
to be an *A*_{n}-map. The higher homotopy associativity of the power maps
*{Φ*^{X}_{λ} *}*_{λ}_{∈Z} measures a lack of higher homotopy commutativity of (*X, µ*).

Theorem A.

Let *p* be an odd prime and *t* *≥* 1 with *t|*(*p* *−* 1). Put *m* = (*p* *−* 1)*/t*.

The power maps *{Φ*^{W}_{λ} ^{t}*}*_{λ}_{∈Z} on *W*_{t} satisfy the following:

(1) *Φ*^{W}_{λ} ^{t} is an *A*_{m}-map for any *λ* *∈* Z.

(2) *Φ*^{W}_{λ} ^{t} is an *A*_{m+1}-map *⇐⇒*^{iﬀ} *λ*(*λ*^{m} *−* 1) *≡* 0 mod *p*.

Remark 5.

*•* If *t* = *p* *−* 1, then Theorem A (2) is the same as Theorem 1.

*•* When *t* = (*p* *−* 1)*/*2, Theorem A (2) is proved by [McGibbon 1982].

*•* When *λ* *̸≡* 0 mod *p*, *Φ*^{W}_{λ} ^{t} is an *A*_{m+1}-map *⇐⇒*^{iﬀ} *Φ*^{W}_{λ} ^{t} is a loop map by
Theorem A (2) and Corollary 4.

Theorem B.

Let *p, t* and *m* be as in Theorem A. Assume that *λ* *≡* 0 mod *p* and
2 *≤* *j* *≤* *t*. The power maps *{Φ*^{W}_{λ} ^{t}*}*_{λ}_{∈Z} on *W*_{t} satisfy the following:

(1) If *Φ*^{W}_{λ} ^{t} is an *A*_{(j}_{−}_{1)m+1}-map, then it is also an *A*_{jm}-map.

(2) *Φ*^{W}_{λ} ^{t} is an *A*_{jm+1}-map *⇐⇒*^{iﬀ} *λ* *≡* 0 mod *p*^{j}.

From Theorems A (2) and B (2) and Corollary 4, we have the following corollary:

Corollary 6.

Let *p, t* and *m* be as in Theorem A. The power map *Φ*^{W}_{λ} ^{t} on *W*_{t} is an
*A*_{p}-map *⇐⇒*^{iﬀ} *λ* *≡* 0 mod *p*^{t} or *λ*^{m} *≡* 1 mod *p*.

Deﬁnition.

*•* A space *X* is F*p*-ﬁnite *⇐⇒*^{def} *H*^{∗}(*X*; F*p*) is ﬁnite-dimensional as a vector
space over F*p*.

*•* A space *X* is F*p*-acyclic *⇐⇒*^{def} *H*e^{∗}(*X*; F*p*) = 0.

Theorem C.

Let *p* be an odd prime. Assume that *X* is a simply connected F*p*-ﬁnite
*A*_{p}-space and *λ* is a primitive (*p* *−* 1)-st root of unity mod *p*. If the
reduced power operations *{P*^{i}*}**i**≥*1 act trivially on the indecomposable
module *QH*^{∗}(*X*; F*p*) and the power map *Φ*^{X}_{λ} on *X* is an *A*_{n}-map with
*n >* (*p* *−* 1)*/*2, then *X* is F*p*-acyclic.

Remark 7.

*•* The condition for *λ* cannot be removed. In fact:

(1) If *λ* *≡* 0 mod *p*, then the power map *Φ*^{W}_{λ} ^{2} on *W*_{2} is an *A*_{(p+1)/2}-map
by Theorem A (2)

(2) Assume that *λ*^{k} *≡* 1 mod *p* for some *k* with 1 *≤* *k < p* *−* 1 and
*k|*(*p* *−* 1). Put *t* = (*p* *−* 1)*/k >* 1. Then the power map *Φ*^{W}_{λ} ^{t} on *W*_{t} is
a loop map by Corollary 4.

*•* Since the power maps *{Φ*^{W}_{λ} ^{2}*}*_{λ}_{∈Z} on *W*_{2} are *A*_{(p}_{−}_{1)/2}-maps by Theorem
A (1), the assumption “*n >* (*p* *−* 1)*/*2” cannot be relaxed in Theorem
C.

Deﬁnition.

An *H*-space is *p*-regular *⇐⇒*^{def}
*X*_{(p)} *≃* *S*^{2t}^{1}^{−}^{1}

(*p*) *× · · · ×* *S*^{2t}^{ℓ}^{−}^{1}

(*p*) (1 *≤* *t*_{1} *≤ · · · ≤* *t*_{ℓ}) *· · ·* (*∗*)
Theorem. [Hubbuck-Mimura 1987], [Iwase 1989]

Let *p* be an odd prime. If *X* is a connected *p*-regular *A*_{p}-space with (*∗*),
then *t*_{ℓ} *≤* *p*.

Theorem D.

Let *p* and *λ* be as in Theorem C. Assume that *X* is a simply connected
*p*-regular *A*_{p}-space with (*∗*). If the power map *Φ*^{X}_{λ} on *X* is an *A*_{n}-map
with *n >* [*p/t*_{ℓ}], then *X* is F*p*-acyclic.

Remark 8.

Since the power maps *{Φ*^{W}_{λ} ^{t}*}*_{λ}_{∈Z} on *W*_{t} are *A*_{m}-maps by Theorem A (1)
and [*p/t*] = *m*, the assumption “*n >* [*p/t*_{ℓ}]” cannot be relaxed in Theorem
D.

Proof of Theorem A (1).

By induction on *i*, we construct an *A*_{m}-form *{η*_{i}*}*_{1}_{≤}_{i}_{≤}_{m} on *Φ*^{W}_{λ} ^{t}. Put
*η*_{1} = *Φ*^{W}_{λ} ^{t}. Assume inductively that *{η*_{j}*}*1*≤**j<i* is constructed for some
*i* *≤* *m*. Let *Γ*_{i}(*W*_{t}) = *∂J*_{i} *×* (*W*_{t})^{i} *∪* *J*_{i} *×* (*W*_{t})^{[i]}, where *X*^{[i]} denotes the
*i*-fold fat wedge of a space *X* deﬁned as

*X*^{[i]} = *{*(*x*_{1}*, . . . , x*_{i}) *∈* *X*^{i} *|* *x*_{j} = *∗* for some *j* with 1 *≤* *j* *≤* *i}.*
Then (*J*_{i} *×* (*W*_{t})^{i})*/Γ*_{i}(*W*_{t}) *≃* *S*^{2ti}^{−}^{1}

(*p*) .

We deﬁne *η*e_{i}: *Γ*_{i}(*W*_{t}) *→* *W*_{t} using *{η*_{j}*}*1*≤**j<i*. The obstructions to
obtain *η*_{i} : *J*_{i} *×* (*W*_{t})^{i} *→* *W*_{t} with *η*_{i}*|*_{Γ}_{i}_{(W}_{t}_{)} = *η*e_{i} appear in the cohomology
groups

*H*^{k+1}(*J*_{i} *×* (*W*_{t})^{i}*, Γ*_{i}(*W*_{t}); *π*_{k}(*W*_{t})) *∼*= *H*e^{k}(*S*^{2ti}^{−}^{2}

(*p*) ; *π*_{k}(*W*_{t})) for *k* *≥* 1.

The above is non-trivial only if *k* is an even integer with *k <* 2*p* *−* 2 since
*ti* *≤* *tm* = *p* *−* 1. On the other hand, *π*_{k}(*W*_{t}) = 0 for any even integer *k*
with *k <* 2*p* *−* 2 by [Toda 1962]. Then we have a map *η*_{i}. This completes
the induction, and we have an *A*_{m}-form *{η*_{i}*}*1*≤**i**≤**m* on *Φ*^{W}_{λ} ^{t}.

Let *X* be an *A*_{n}-space. According to [Stasheﬀ 1963], we have the pro-
jective spaces *{P*_{i}(*X*)*}*_{0}_{≤}_{i}_{≤}_{n} with the following properties:

*•* There is a ﬁbration

*X* *−→* *Σ*^{i}^{−}^{1}*X*^{∧}^{i} *−−−→*^{γ}^{i}^{−}^{1} *P*_{i}_{−}_{1}(*X*) for 1 *≤* *i* *≤* *n*

*•* There is a long coﬁbration sequence:

*Σ*^{i}^{−}^{1}*X*^{∧}^{i} *−−−→*^{γ}^{i}^{−}^{1} *P*_{i}_{−}_{1}(*X*) *−−→*^{ι}^{i}^{−}^{1} *P*_{i}(*X*) *−→*^{ρ}^{i} *Σ*^{i}*X*^{∧}^{i} *−−−−→ · · ·*^{Σγ}^{i}^{−}^{1}
for 1 *≤* *i* *≤* *n*,
where *X*^{∧}^{i} denotes the *i*-fold smash product of *X*.

*•* *P*_{0}(*X*) = *{∗}* and *P*_{1}(*X*) = *ΣX*.

*•* When *X* is an *A*_{∞}-space, *P*_{∞}(*X*) = *BX*.

Theorem. [Stasheﬀ 1970], [Iwase-Mimura 1989], [Hemmi 2007]

Let *X, Y* be *A*_{n}-spaces.

(1) If *f* : *X* *→* *Y* is an *A*_{n}-map, then

*∃**{P*_{i}(*f*) : *P*_{i}(*X*) *→* *P*_{i}(*Y* )*}*_{1}_{≤}_{i}_{≤}_{n}

with *P*_{1}(*f*) = *Σf* and *P*_{i}(*f*)*ι*_{i}_{−}_{1} = *P*_{i}_{−}_{1}(*f*)*ι*_{i}_{−}_{1} for 2 *≤* *i* *≤* *n*.

(2) If *Y* is an *A*_{n+1}-space, then the converse of (1) also holds.

Put *ε*_{i}_{−}_{1} = *ι*_{i}_{−}_{1} *· · ·* *ι*_{1} : *ΣX* = *P*_{1}(*X*) *→* *P*_{i}(*X*) for *i* *≥* 2.

Proof of the “ only if ” part of Theorem A (2).

It is known that

*H*^{∗}(*P*_{m+1}(*W*_{t}); F*p*) *∼*= F*p*[* u*]

*/*(

**u**^{m+2}) with deg

*= 2*

**u***t*and

*P*^{1}(* u*) =

*ξ*

**u**^{m+1}with

*ξ*

*̸≡*0 mod

*p*.

If *Φ*^{W}_{λ} ^{t} is an *A*_{m+1}-map, then

*∃**P*_{m+1}(*Φ*^{W}_{λ} ^{t}) : *P*_{m+1}(*W*_{t}) *→* *P*_{m+1}(*W*_{t})
with *P*_{m+1}(*Φ*^{W}_{λ} ^{t})*ε*_{m} *≃* *ε*_{m}(*ΣΦ*^{W}_{λ} ^{t}). This implies

*P*_{m+1}(*Φ*^{W}_{λ} ^{t})^{∗}(* u*) =

*λ*

**u**.Since

*P*^{1}*P*_{m+1}(*Φ*^{W}_{λ} ^{t})^{∗}(* u*) =

*ξλ*

**u**^{m+1}and

*P*_{m+1}(*Φ*^{W}_{λ} ^{t})^{∗}*P*^{1}(* u*) =

*ξλ*

^{m+1}

**u**^{m+1}

*,*we have

*λ*(

*λ*

^{m}

*−*1)

*≡*0 mod

*p*.

Proof of the “ if ” part of Theorem A (2).

According to [Toda 1962], we have

*π*_{2t+2(p}_{−}_{1)}_{−}_{2}(*W*_{t}) *∼*= Z*/p{α}.*
Let *C*(*φ*) be the coﬁber of *φ* = *Σα*: *S*^{2t+2(p}^{−}^{1)}^{−}^{1}

(*p*) *→* *ΣW*_{t}. Then

*H*^{∗}(*C*(*φ*); F*p*) = F*p**{ z,*

*as an F*

**w**}*p*-algebra

with deg * z* = 2

*t*and deg

*= 2*

**w***t*+ 2(

*p*

*−*1) and

*P*^{1}(* z*) =

*ζ*with

**w***ζ*

*̸≡*0 mod

*p*.

Since *φ* = *Σα* is a suspension map, we have a map *Λ*: *C*(*φ*) *→* *C*(*φ*)
with the following commutative diagram:

*S*^{2t+2(p}^{−}^{1)}^{−}^{1}

(*p*)

*−→**φ* *S*^{2t}

(*p*) *−→* *C*(*φ*)

[*λ*]

y y^{[λ]} y^{Λ}
*S*^{2t+2(p}^{−}^{1)}^{−}^{1}

(*p*)

*−→**φ* *S*^{2t}

(*p*) *−→* *C*(*φ*)*,*
where [*λ*] denote the self-maps of degree *λ*.

Since *Φ*^{W}_{λ} ^{t} is an *A*_{m}-map,

*∃**P*_{m}(*Φ*^{W}_{λ} ^{t}) : *P*_{m}(*W*_{t}) *→* *P*_{m}(*W*_{t})
with *P*_{m}(*Φ*^{W}_{λ} ^{t})*ε*_{m}_{−}_{1} *≃* *ε*_{m}_{−}_{1}(*ΣΦ*^{W}_{λ} ^{t}).

Let *φ*e = *ε*_{m}_{−}_{1}*φ*: *S*^{2t+2(p}^{−}^{1)}^{−}^{1}

(*p*) *→* *P*_{m}(*W*_{t}). Since there is a ﬁbration
*W*_{t} *−→* *S*^{2t+2(p}^{−}^{1)}^{−}^{1}

(*p*)

*γ*_{m}

*−−→* *P*_{m}(*W*_{t})*,*
we have

*π*_{2t+2(p}_{−}_{1)}_{−}_{1}(*P*_{m}(*W*_{t})) *∼*= Z_{(p)}*{γ*_{m}*} ⊕* Z*/p{φ*e*}.*
Put *X* = *C*(*φ*), whereb *φ*b = *ι*_{m}*φ*e = *ε*_{m}*φ*: *S*^{2t+2(p}^{−}^{1)}^{−}^{1}

(*p*) *→* *P*_{m+1}(*W*_{t}).

Then *C*(*φ*) *⊂* *X* and *π*_{2t+2(p}_{−}_{1)}_{−}_{1}(*X*) = 0.

Since *P*_{m+1}(*W*_{t}) = *C*(*γ*_{m}), we have a map *Ψ*e: *P*_{m+1}(*W*_{t}) *→* *X* with
the following commutative diagram:

*S*^{2t}

(*p*) *ΣW*_{t} *−−−→*^{ε}^{m}^{−}^{1} *P*_{m}(*W*_{t}) *−−→*^{ι}^{m} *P*_{m+1}(*W*_{t})

[*λ*]

y ^{ΣΦ}^{Wt}_{λ} y y^{P}^{m}^{(Φ}^{Wt}_{λ} ^{)} y^{Ψ}^{e}
*S*^{2t}

(*p*) *ΣW*_{t} *−−−→*

*ε*_{m}_{−}_{1} *P*_{m}(*W*_{t}) *−−→*

e*ι*_{m} *X,*

where e*ι*_{m} denotes the composition of *ι*_{m} and the inclusion *P*_{m+1}(*W*_{t}) *⊂* *X*.
Deﬁne a self-map *Ψ* : *X* *→* *X* by *Ψ|*_{P}_{m+1}_{(W}_{t}_{)} = *Ψ*e and *Ψ|*_{C}_{(φ)} = *Λ*.

*S*_{(p)}^{2t} *ΣW*_{t} *−−−→*^{ε}^{m}^{−}^{1} *P*_{m}(*W*_{t}) *−−→*^{ι}^{m} *P*_{m+1}(*W*_{t}) *−→*^{⊂} *X* *←−*^{⊃} *C*(*φ*)

[*λ*]

y ^{ΣΦ}^{Wt}_{λ} y y^{P}^{m}^{(Φ}^{Wt}_{λ} ^{)} y^{Ψ} y^{Λ}
*S*^{2t}

(*p*) *ΣW*_{t} *−−−→*

*ε*_{m}_{−}_{1} *P*_{m}(*W*_{t}) *−−→*

*ι*_{m} *P*_{m+1}(*W*_{t}) *−→*

*⊂* *X* *←−*

*⊃* *C*(*φ*)*,*
From the deﬁnition,

*H*^{∗}(*X*; Z_{(p)}) = Z_{(p)}[*x*]*/*(*x*^{m+2}) *⊕* Z_{(p)}*{y}* as a Z_{(p)}-algebra

with deg *x* = 2*t* and deg *y* = 2*t* + 2(*p* *−* 1)*.*

Since *Ψ|*_{C}_{(φ)} = *Λ*, the induced homomorphism

*Ψ*^{∗} : *H*^{∗}(*X*; Z_{(p)}) *→* *H*^{∗}(*X*; Z_{(p)})

is given by *Ψ*^{∗}(*x*) = *λx* and *Ψ*^{∗}(*y*) = *λy* + *ηx*^{m+1} for some *η* *∈* Z_{(p)}.
Lemma.

If *λ*(*λ*^{m} *−* 1) *≡* 0 mod *p*, then *η* *≡* 0 mod *p*.

Proof.

*H*^{∗}(*P*_{m+1}(*W*_{t}); F*p*) *←−* *H*^{∗}(*X*; F*p*) *−→* *H*^{∗}(*C*(*φ*); F*p*)
Write *P*^{1}(* x*) =

*ξ*

**x**^{m+1}+

*ζ*with

**y***ξ, ζ*

*̸≡*0 mod

*p*. Since

*P*^{1}*Ψ*^{∗}(*x*) = *λξ x*

^{m+1}+

*λζ*and

**y***Ψ*^{∗}*P*^{1}(*x*) = *λ*^{m+1}*ξ x*

^{m+1}+

*λζ*+

**y***ηζ*

**x**^{m+1}

*,*

we have *ξλ*(*λ*^{m} *−* 1) + *ηζ* *≡* 0 mod *p*. Then *η* *≡* 0 mod *p*.

Let **a**,**b***∈* *H*_{2t+2(p}_{−}_{1)}(*X*; Z_{(p)}) denote the Kronecker duals of *x*^{m+1}*, y* *∈*
*H*^{2t+2(p}^{−}^{1)}(*X*; Z_{(p)}), respectively. Using the duality, we can show that

*Ψ*_{∗}(* a*) =

*λ*

^{m+1}

*+*

**a***η*and

**b***Ψ*_{∗}(* b*) =

*λ*

**b**.Consider the homomorphism

*E* : *H*_{2t+2(p}_{−}_{1)}(*X*; Z_{(p)}) *→* *π*_{2t+2(p}_{−}_{1)}_{−}_{1}(*P*_{m}(*W*_{t}))
deﬁned by the following composition:

*H*_{2t+2(p}_{−}_{1)}(*X*; Z_{(p)}) *−→* *H*_{2t+2(p}_{−}_{1)}(*X, P*_{m}(*W*_{t}); Z_{(p)})

*H* ^{−}^{1}

*−−−→*_{∼}

= *π*_{2t+2(p}_{−}_{1)}(*X, P*_{m}(*W*_{t})) *−→*^{∂} *π*_{2t+2(p}_{−}_{1)}_{−}_{1}(*P*_{m}(*W*_{t}))*,*
where *H* denotes the Hurewicz isomorphism. Then *P*_{m}(*Φ*^{W}_{λ} ^{t})_{#}*E* = *E* *Ψ*_{∗}.

Since *E* (* a*) =

*γ*

_{m}and

*E*(

*) =*

**b***φ*, we have thate

*P*_{m}(*Φ*^{W}_{λ} ^{t})_{#}(*γ*_{m}) = *λ*^{m+1}*γ*_{m} + *ηφ*e = *λ*^{m+1}*γ*_{m} by Lemma.

This implies that *ι*_{m}*P*_{m}(*Φ*^{W}_{λ} ^{t})*γ*_{m} is null-homotopic, and so there is a self-
map *ψ*: *P*_{m+1}(*W*_{t}) *→* *P*_{m+1}(*W*_{t}) with *ψι*_{m} *≃* *ι*_{m}*P*_{m}(*Φ*^{W}_{λ} ^{t}). Then *Φ*^{W}_{λ} ^{t} is
an *A*_{m+1}-map.

Remark.

Theorem B is proved in a similar way to the proof of Theorem A. In
the proof, we use the Brown-Peterson cohomology instead of the mod *p*
cohomology.