Conjecture D on the degree-one Poitou–Tate sequence for knot groups

Let GG be a knot group, let HGH\leq G be a peripheral subgroup, and let UGU\trianglelefteq G be a normal subgroup of finite index. Write EGG/UE_\mathcal{G}G/U for the associated classifying space, and use H1(,Z)H^1(-,\mathbb{Z}) and H1(,Z)H_1(-,\mathbb{Z}) 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.

Sources & referencesView supporting material

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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