  In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional
topology, Furuta proposed a question on the existence of Pin(2)-equivariant
stable maps between certain representation spheres. In this paper, we present a
complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald
invariants. As a geometric application of our result, we prove a
"10/8+4"-Theorem.
  We prove our theorem by analyzing maps between certain finite spectra arising
from BPin(2) and various Thom spectra associated with it. To analyze these
maps, we use the technique of cell diagrams, known results on the stable
homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch spectral
sequence.
