Flatlander - the collected "Gil the ARM" stories of Larry Niven. In his "Known Space" chronology, these are contemporary with A World of Ptavvs. This book can be considered an expansion of the 1976 collection The Long ARM of Gil Hamilton: with The Patchwork Girl (1980 - without the illustrations that bulked that short novel into a book) plus an original novelet "The Woman in Del Rey Crater" inserted after the 3 novelets, and before an updated version of the afterword.

Gilbert Hamilton was a flatlander who gave up his UN citizenship to become a Belter. For years he worked as part of a 3-man crew, mining all around the Solar System. Then an accident cost one crew member his life, and cost Gil his right arm. That's when he found he had a limited psi-power: an imaginary arm, but only with the reach of his lost limb. After he could raise the price of a return-ticket to Earth, Gil re-applied for citizenship, for access to the organ-banks - to get back a real arm. When he found that the arm he received was from an organlegger's banks, he joined the UN police - the part that tracked down organleggers. At the start of the first story, Gil finds out that his friend - the Belter who saved his life - has died, and the local police think it suicide by "wireheading." Gil thinks it was murder. Niven crafts a perfect mix of detective story and SF extrapolation, mixing wireheads (direct stimulation of the brain), organ-banks and their effect on the criminal justice system, matter-of-fact dealing with overcrowding, and inventive details which make this world clear to us. "Death By Ecstasy" is my absolute favorite Niven story, and any book that contains it will get my recommendation.

Death By Ecstasy
The Defenseless Dead
ARM
PATCHWORK GIRL
The Woman in Del Rey Crater
Science/Mystery Fiction (Afterword)

"The Woman in Del Rey Crater" is original to this 1995 paperback; the rest first appeared from 1969 to 1980: "Death By Ecstasy" in Galaxy January 1969 (as "The Organleggers"), "The Defenseless Dead" in Ten Tomorrows (1973), "ARM" in Epoch (1975), The Patchwork Girl (1980.) The afterword is revised from "The Last Word about SF/Detectives" in The Long ARM of Gil Hamilton (1976.)