On Integral Cohomology Ring of Symmetric Products

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Publication date: 6 February 2015

Abstract: We prove that the integral cohomology ring modulo torsion

H^{*} (m a t h r m {S y m}^{n} X; m a t h b b Z) / m a t h r m T o r

for the symmetric product of a connected CW-complex

X

of finite homology type is a functor of

H^{*} (X; m a t h b b Z) / m a t h r m T o r

(see Theorem 1). Moreover, we give an explicit description of this functor. We also consider the important particular case when

X

is a compact Riemann surface

M_{g}^{2}

of genus

g

. There is a famous theorem of Macdonald of 1962, which gives an explicit description of the integral cohomology ring

H^{*} (m a t h r m {S y m}^{n} M_{g}^{2}; m a t h b b Z)

. The analysis of the original proof by Macdonald shows that it contains three gaps. All these gaps were filled in by Seroul in 1972, and, therefore, he obtained a complete proof of Macdonald's theorem. Nevertheless, in the unstable case

2 l e n l e 2 g - 2

Macdonald's theorem has a subsection, that needs a slight correction even over

m a t h b b Q

(see Theorem 2).

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