Conjecture D on the degree-one Poitou–Tate sequence for knot groups
Conjecture D on the degree-one Poitou–Tate sequence for knot groups
Let be a knot group, let be a peripheral subgroup, and let be a normal subgroup of finite index. Write for the associated classifying space, and use and for its first cohomology and homology groups. Conjecture D. These groups fit at the ends of the degree-one subsequence in an exact sequence of groups
\xymatrix@C-6pt{ 0\ar[r] & H^1(E_\mathcal{G} G/U,\mathbb{Z})\ar[d]\\ & H^1(U,\mathbb{Z})\ar[r] & \coprod_{G/U\! H} H_1(H\cap U,\mathbb{Z})\ar[r] & H_1(U,\mathbb{Z})\ar[d] & \\ &&&H_1(E_\mathcal{G} G/U,\mathbb{Z})\ar[r] & 0. }This is a knot-theoretic analogue of the Poitou–Tate sequence for algebraic number fields. The conjecture is proved in some special cases but remains open in full generality.
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Primary source
Federico William Pasini, “Classifying spaces for knots: New bridges between knot theory and algebraic number theory”, arXiv:1609.00820 (2016).
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