From: "Pedro Real" Subject: To toplist:Question about cup-i products Date: Thu, 13 May 1999 12:09:22 +0200 In [``A combinatorial method for computing Steenrod squares'' . Roc=EDo Gonz=E1lez-D=EDaz and Pedro Real. To appear in JPAA.(We have sent it to Hopf)] explicit combinatorial formulae for the cup-i products are established. I admit that in this work we only rediscover the old (1947) description given by Steenrod and we clarify it in a general combinatorial framework. This paper has been written for a wide audience (a non-expert in Algebraic Topology can read it without problems). The organization of the face operators in the formula for the cup-i product is simple and it is already outlined by Steenrod but the signs involved are complicated (no problems, of course, if you work over Z_2). Nevertheless, our approach to Steenrod cohomology operations is new in the sense: a) we use Homological perturbation techniques in the combinatorial fiber bundle (twisted cartesian product) $(X\times X) \times_{\tau} B(Z_2)$, being $B( )$ the classifying functor of a simplicial group, $\tau: B(Z_2) --->Z_2$ the canonical twisting function and the action of $Z_2$ over $(X \times X)$ consists in interchanging the factors. b) If $X$ is a simplicial set and $C_*( X)$ denotes the normalized chain complex associated to $X$, we make use of an explicit simplicial description of the homotopy operator $SHI: C_*(X\times X) ---> C_{*+1}(X \times X) of the Eilenberg-Zilber Strong Deformation Retract from C_*(X\times X) to C_*(X) \otimes C_*(X). The work of Hess [K. Hess. ``Generic Perturbation and transfer''. Contemporary Math. Vol 227, 1999, 103-143, Section 6.2] is near to us. Pedro. Pedro Real Dpto. de Matematica Aplicada I Fac. de Informatica y Estadistica Univ. de Sevilla Avda. Reina Mercedes, s/n 41012 Sevilla Tfno: 95-34-4556921 Fax: 95-34-4557878 e-mail: real@cica.es ------=_NextPart_000_0004_01BE9D39.6FEAAD20 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

In=20 [``A combinatorial method for = computing Steenrod=20 squares'' . Rocío González-Díaz and

Pedro Real. To appear in JPAA.(We have = sent it to=20 Hopf)] explicit combinatorial formulae for the cup-i products are = established. I admit that in this work we = only =20 rediscover the old (1947) description given by=20 Steenrod and we clarify it in a general combinatorial framework. This = paper has=20 been written

for a wide audience (a non-expert in Algebraic Topology can read it without = problems). The=20 organization of the face operators in the=20 formula

for the cup-i product is simple and it = is already=20 outlined by Steenrod but the signs involved are complicated (no problems, of course, if you work over = Z_2).

Nevertheless, our approach = to=20 Steenrod cohomology=20 operations is new in the sense:

a) we use Homological perturbation = techniques in the combinatorial fiber bundle = (twisted=20 cartesian product) $(X\times X) = \times_{\tau} B(Z_2)$, being $B( )$ the classifying functor of a = simplicial=20 group,

$\tau: B(Z_2) --->Z_2$ the canonical = twisting=20 function and the action of $Z_2$ over $(X \times X)$

consists in interchanging the=20 factors.

b) If $X$ is a = simplicial set and=20 $C_*( X)$ denotes the normalized chain complex associated to = $X$,

we make use of an = explicit simplicial=20 description of the homotopy operator $SHI: C_*(X\times X) = --->

C_{*+1}(X \times X) of the Eilenberg-Zilber Strong Deformation = Retract=20 from C_*(X\times X) to

C_*(X) \otimes C_*(X). =

&nbs= p; The=20 work of Hess [K. Hess. ``Generic Perturbation and = transfer''.Contemporary Math.=20 Vol 227,

1999,=20 103-143, Section 6.2] is near to us.

Pedro.

Pedro Real

Dpto.=20 de Matematica Aplicada I

Fac. de Informatica y=20 Estadistica

Univ. de Sevilla

Avda. Reina Mercedes, = s/n

41012 Sevilla

Tfno: 95-34-4556921

Fax: = 95-34-4557878

e-mail: real@cica.es

=20

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