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Re: Ba`i Review cu?a TS Nguyen Xuan Tuyen
Hi ca'c ba'c
Review negative dde^'n va^.y trong toa'n qua? la\ hie^'m hoi.
Ba'c Tuye^n na\y tui ddoa'n chi? co^'t ba?o ve^. cho co' ba(`ng
dde^? la\m administration, dda(ng ba\i + ba?o ve^. o+? ta^.n
xu' "khi? ho" Tbilisi cho qua chuye^.n, kho^ng ngo+\
le^n dde^'n Math. Review la.i ro+i va\o tay "ddao phu?" NHV Hu+ng
(chu+' ne^'u he^n ho+n mo^.t chu't co' the^? ro+i va\o tay kho^'i bo.n
reviewers vo+' va^? vie^'t chung chung kho^ng khen kho^ng che^)
O^i ddau thu+o+ng!
Ba'c DV An va\ ba'c Tha('ng co\n co' ta\i lie^.u gi\ the^m ve^\ vu. na\y
xin dda(ng no^'t le^n ddi. Chuye^.n ba'c Tuye^n ddi.nh u+'ng cu+?
Hie^.u Tru+o+?ng Dda.i Ho.c Quo^'c Gia o+? Hue^' co' pha?i nguye^n
nha^n chi'nh kho^ng?
A\ ma\ ba'c Tuye^n na\y cha('c o+? Tbilisi qua~ng cuo^'i '80.
Vu~ (Mainz) co' nho+' ba'c Tuye^n na`y kho^ng?
Cheers, Z
----------
> De : tle@msri.org
> A : Multiple recipients of list <vnsa-l@csd.uwm.edu>
> Objet : Ba`i Review cu?a TS Nguyen Xuan Tuyen
> Date : vendredi 13 juin 1997 20:53
>
>
>
>
>
> 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 =
>