Vol. 34, No. 3, 1970

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Twisted self-homotopy equivalences

Allan John Sieradski

Vol. 34 (1970), No. 3, 789–802

This paper studies the group G(A × B) of (homotopy classes of) self-homotopy equivalences of a product A × B of two connected CW homotopy associative H-spaces A and B. It establishes the existence of an exact sequence of multiplicative groups

1 → [A ∧ B,A × B ] → G (A × B ) → GL (2,ΛIJ) → 1

provided that i[A×B,A×B] q [AB,A×B] = 0, where q : A×B AB is the cofibration induced by the inclusion i : A B A × B of the sum into the product. The entry GL(2,ΛIJ) is the group of invertible matrices

       (            )
(hIJ) =   hAA  hAB
hBA  hBB

with entries hIJ in the homotopy sets ΛIJ = [I,J] for I,J = A,B, where matrix multiplication is defined by

(hIJ)(kIJ ) = (hIA ∘k∠1J + hIB ∘ kBJ)

in terms of composition and the operation + in the homotopy sets [I,J], and where the multiplicative unit is

       ( 1A  0   )
(δIJ) =  0   1B   .

The homomorphism G(A × B) GL(2,ΛIiI) is given by the correspondence of h : A × B A × B with the matrix

(                       )
iA ∘h ∘pa4  iA ∘ h∘pB
iB ∘h ∘pA   iB ∘ h∘pB

with entries obtained from h by composing with the inclusions iA : A A×B and iB : B A × B and the projections pA : A × B A and pB : A × B B; for a preliminary result states that under the hypothesis above h : A × B A × B is a homotopy equivalence if and only if the matrix (iI h pJ) is invertible.

A homotopy equivalence f ×g : A×B A×B is referred to as untwisted. These determine a subgroup G(A) × G(B) G(A × B) which is isomorphic under the homomorphism G(A×B) GL(2,ΛIJ) to the subgroup of diagonal matrices, and so the nondiagonal matrices give measure of the twisted selfhomotopy equivalences A × B A × B. The extreme case in which all self-homotopy equivalences are untwisted is considered, and it is shown that G(A) ×G(B) = G(A×B) if and only if the homotopy sets [A,B], [B,A], and [A B,A × B] are trivial.

Next, four settings are considered in which

i∘[A × B,A × B]∘ q∘[A ∧ B,A × B] = 0

and the exact sequence is valid. In the last section the dual situation of the group G(M N) of self-homotopy equivalences of a sum M N of two co-H-spaces M and N is briefly sketched.

Mathematical Subject Classification
Primary: 55.40
Received: 12 February 1970
Revised: 30 March 1970
Published: 1 September 1970
Allan John Sieradski