The Additivity Theorem in K

190
H½s; ðP; EÞŠ to interpolate between P and Pip. We first must make an orderpreserving
map hs : D mþnþ1 ! D mþnþ1 which interpolates between 1 and ip.
The map hs is defined on vertices in the following way:
8
< a: a 2½mŠ
hsðaÞ ¼ a: a ¼ m þ 1 þ b; b 2½nŠ; sðbÞ ¼1 :
:
m: a ¼ m þ 1 þ b; b 2½nŠ; sðbÞ ¼0
Note that h0 ¼ ip and h1 ¼ 1, so we can now set P s ¼ Phs. Also, hsi ¼ i, so
P s i ¼ M, which is necessary for P s to be the first component of an element of
MjS:q.
Next, we need to construct the second component, E s . Note that for all
a 2½m þ n þ 1Š, hsðaÞOa. So there is a natural transformation hs ! 1, which
induces a map P s j ¼ Phsj ! Pj ¼ qE. Identifying E with the exact sequence
0 ! sE ! tE ! qE ! 0, we can now form the pullback of that sequence
along the induced map.
Es : 0ƒƒ ƒ!sEs ¼ sEƒƒ ƒ!tEsƒƒ ƒ!Psj ¼ qEsƒƒ ƒ!0
? ? ?
? ? ?
? ? ?
? ? ?
? 1 ? ?
? ?
? ? ?
y y y
E: 0ƒƒƒƒƒƒƒƒƒ!sEƒƒƒƒƒƒ!tEƒƒƒƒƒƒ!qEƒƒƒƒƒƒ!0
As suggested by our notation, E s is defined to be this pullback. By definition,
P s j ¼ qE s , and since we already know that P s i ¼ M, the pair ðP; EÞ is an
element of MjS:q. It is shown in [3] that this is indeed a simplicial homotopy,
and so we can now apply Theorem A^ 0 to WM ¼ S:s p, which shows that the
map S:E !
ðs;qÞ
S:M 2 is a homotopy equivalence. h
Acknowledgements
KEVIN CHARLES JONES ET AL.
Jones and Kim were supported by University of Illinois, Mhoon by NSF
grant DMS 99-83160, Santhanam by NSF grant DMS 03-06429, and
Grayson by NSF grant DMS 03-11378. We thank Randy McCarthy for
useful feedback during the preparation of this paper.
References
1. Gillet, H. and Grayson, D. R.: The loop space of the Q-construction, Illinois J. Math. 31
(4) (1987), 574–597. MR 89h: 18012 1, 6, 7.
2. Goerss, P. G. and Jardine, J. F.: Simplicial Homotopy Theory, Progress in Mathematics,
Vol. 174, Birkhäuser Verlag, Basel, (1999). MR 2001d: 55012 4.
3. Grayson, D. R.: Algebraic K-theory, notes for math 416, available at http://www.math.uiuc.edu/~dan/Courses/2003/Spring/416/,
May (2003) 2, 6, 8, 9.