Expression de la differ\'entielle $d_3$ de la suite spectrale de Hochishild-Serre en cohomologie born\'ee r\'eelle

zbMATH Open1280.55012arXiv1211.4451MaRDI QIDQ4914912

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Publication date: 15 April 2013

Abstract: For discrete groups, we construct two bounded cohomology classes with coefficients in the second space of the reduced real

e l l_{1}

-homology. Precisely, we associate to any discrete group

G

a bounded cohomology class of degree two noted

. For

G

and

P i

groups and

h e t a : P i i g h t a r r o w O u t (G)

any homomorphism we associate a bounded cohomology class of degree three noted

. When the outer homomorphism

h e t a : P i i g h t a r r o w O u t (G)

induces an extension of

G

by

P i

we show that the class

f r a k g_{2}

is

P i

-invariant and that the differential

d_{3}

of Hochschild-Serre spectral sequence sends the class

f r a k g_{2}

on the class

[h e t a]

:

d_{3} (f r a k g_{2}) = [h e t a]

. Moreover, we show that for any integer

n g e q 0

the differential

d_{3} : E_{3}^{n, 2} i g h t a r r o w E_{3}^{n + 3, 0}

of Hochschild-Serre spectral sequence in real bounded cohomology is given as a cup-product by the class

[h e t a]

.

Full work available at URL: https://arxiv.org/abs/1211.4451

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