Ba`i Review cu?a TS Nguyen Xuan Tuyen

[tle@msri.org]

Mathematical Reviews, =A9 Copyright American Mathematical Society 1995, 1=
997 =


95j:20050 20J06 55R40 =

Nguen Suan Tuen
On the cohomology of alternating groups and its application in topology. =
(Russian) =

Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzii 97 (1992), 18--47. =


The paper contains four sections. The first two are devoted to the comput=
ation of ${\rm
Im\,Res}(E,A\sb m)$, the image of the restriction ${\rm Res}(E,A\sb m)\co=
lon H\sp *(A\sb
m;{\bf Z}/p)\to H\sp *(E;{\bf Z}/p)$, where $A\sb m$ denotes the alternat=
ing group on $m$
letters, $E$ is a maximal elementary abelian $p$-subgroup of $A\sb m$ and=
 $p$ is a prime. The
main result of these two sections, Theorem 2.5, is an explicit computatio=
n of this image for the
special case where $m=3Dp\sp n$, $A\sb {p\sp n}$ is viewed as the alterna=
ting group on (the point set
of) $({\bf Z}/p)\sp n$ and $E=3DE\sp n$ is the subgroup of all translatio=
ns on $({\bf Z}/p)\sp n$.
Unfortunately, this result is due to Huynh Mui, at least for $p>2$ [Math.=
 Z. 193 (1986), no. 1,
151--163; MR 88e:55015 (see Theorem 3.10)]. Formula (9) in Section 1, the=
 key point in the proof
of Theorem 1.8, is also due to Huynh Mui [J. Fac. Sci. Univ. Tokyo Sect. =
1A Math. 22 (1975), no.
3, 319--369; MR 54 #10440 (see Theorem 5.2)]. =


In Section 3 the author naturally equips $H\sb *A\sb \infty\coloneq \lim =
H\sb *A\sb m$ with an
algebra structure. Let $E\subset A\sb m\subset S\sb m$, where $E$ and $A\=
sb m$ are as above and
$S\sb m$ is the symmetric group on $m$ letters. The author also shows ${\=
rm Im\,Res}(E,A\sb
m)=3D{\rm Im\,Res}(E,S\sb m)$ for some special cases of $E$ and $m$. (In =
the general case, it is
false; for example, if $p>2$, $m=3Dp\sp n$, $E=3DE\sp n$. See Huynh Mui's=
 two papers mentioned
above.) Formula (11) and its proof on pages 39--40 are again due to Huynh=
 Mui [op. cit., 1986 (see
p. 156)]. =


Section 4 is devoted to describing the homology and cohomology algebras o=
f $F({\bf R}\sp
q,\infty)/A\sb \infty$. Here $F({\bf R}\sp q,m)$ denotes the configuratio=
n space of $m$-tuples of
distinct points in ${\bf R}\sp q$ with the usual free action of $A\sb m\s=
ubset S\sb m$, and $F({\bf
R}\sp q,\infty)/A\sb \infty\coloneq \lim F({\bf R}\sp q,m)/A\sb m$. The s=
ection is reproduced
from some parts of three papers by the reviewer [C. R. Acad. Sci. Paris S=
er. I Math. 297 (1983),
no. 12, 611--614; MR 85b:20071; C. R. Acad. Sci. Paris Ser. I Math. 307 (=
1988), no. 18,
911--914; MR 90c:57037; Pacific J. Math. 143 (1990), no. 2, 251--286; MR =
91k:55017].
Actually, the reviewer studied $F({\bf R}\sp q,\infty)/S\sb \infty$. Ther=
efore, the reviewer thinks
what the author should have proved in the paper is that $$H\sp *(F({\bf R=
}\sp q,\infty)/A\sb
\infty;{\bf Z}/p)\cong H\sp *(F({\bf }\sp q,\infty)/S\sb \infty;{\bf Z}/p=
)$$ for $p>2$. However, he
does not do that, but claims it only for $q=3D\infty$ by asserting $H\sb =
*(A\sb \infty; {\bf
Z}/p)\cong H\sb *(S\sb \infty;{\bf Z}/p)$ on page 39. So, there are some =
unclear things in the
section. For instance, one might ask why the hypothesis that $q$ is odd i=
n Theorem 4.1 (and so, in
Theorems 4.4 and 4.5) is not used in the proof. The reader is urged to fo=
llow an explanation for

this hypothesis in the reviewer's paper [op. cit., 1988]. Also, the eleme=
nts $\overline L\sb
n,\overline M\sb {n,s},\overline Q\sb {n,s}$ on page 44, $m\sb {H,R}$ in =
Lemma 4.3 and the sum
$(K,S)+(L,T)$ in Theorem 4.4 are used but not defined. =


              Reviewed by Nguyen H. V. Hung


 =A9 Copyright American Mathematical Society 1995, 1997 =