A finite ℚ–bad space

Ivanov, Sergei¹ ; Mikhailov, Roman²

¹ Laboratory of Modern Algebra and Applications, St. Petersburg State University, Saint Petersburg, Russia
² Laboratory of Modern Algebra and Applications, St. Petersburg State University, Saint Petersburg, Russia, St. Petersburg Department of Steklov Mathematical Institute, Saint Petersburg, Russia

Geometry & topology, Tome 23 (2019) no. 3, pp. 1237-1249.

Résumé

We prove that, for a free noncyclic group $F$, the second homology group $H_2\left({\widehat{F}}_{\mathbb{Q}},\mathbb{Q}\right)$ is an uncountable $\mathbb{Q}$–vector space, where ${\widehat{F}}_{\mathbb{Q}}$ denotes the $\mathbb{Q}$–completion of $F$. This solves a problem of A K Bousfield for the case of rational coefficients. As a direct consequence of this result, it follows that a wedge of two or more circles is $\mathbb{Q}$–bad in the sense of Bousfield–Kan. The same methods as used in the proof of the above result serve to show that $H_2\left({\widehat{F}}_{\mathbb{Z}},\mathbb{Z}\right)$ is not a divisible group, where ${\widehat{F}}_{\mathbb{Z}}$ is the integral pronilpotent completion of $F$.

Détail

DOI : 10.2140/gt.2019.23.1237

Classification : 14F35, 16W60, 55P60
Keywords: homology, nilpotent completion, Bousfield–Kan completion, R–good space, R–bad space

Bibliographie

[1] A K Bousfield, The localization of spaces with respect to homology, Topology 14 (1975) 133 | DOI

[2] A K Bousfield, Homological localization towers for groups and Π–modules, 186, Amer. Math. Soc. (1977) | DOI

[3] A K Bousfield, On the p–adic completions of nonnilpotent spaces, Trans. Amer. Math. Soc. 331 (1992) 335 | DOI

[4] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972) | DOI

[5] R Brown, J L Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311 | DOI

[6] S O Ivanov, R Mikhailov, On a problem of Bousfield for metabelian groups, Adv. Math. 290 (2016) 552 | DOI

[7] S O Ivanov, R Mikhailov, On discrete homology of a free pro-p–group, Compos. Math. 154 (2018) 2195 | DOI

[8] C Miller, The second homology group of a group; relations among commutators, Proc. Amer. Math. Soc. 3 (1952) 588 | DOI