A mapping cone formula for Involutive Heegaard Floer homology Ian Zemke joint w/ K. Hendricks, J. Hom and M. Stoffregen September 17, 2020 If Y is a 3-manifold, and s 2 Spinc(Y ), Ozsv´athand Szab´o − + construct three F[U]-modules HF (Y; s), HF (Y; s) and HF 1(Y; s). 3 If Y is a ZHS and K ⊆ Y , Ozsv´athand Szab´o constructed a graded chain complex, CFK 1(Y; K), which is filtered by Z ⊕ Z. Heegaard Floer homology 3 If Y is a ZHS and K ⊆ Y , Ozsv´athand Szab´o constructed a graded chain complex, CFK 1(Y; K), which is filtered by Z ⊕ Z. Heegaard Floer homology If Y is a 3-manifold, and s 2 Spinc(Y ), Ozsv´athand Szab´o − + construct three F[U]-modules HF (Y; s), HF (Y; s) and HF 1(Y; s). Heegaard Floer homology If Y is a 3-manifold, and s 2 Spinc(Y ), Ozsv´athand Szab´o − + construct three F[U]-modules HF (Y; s), HF (Y; s) and HF 1(Y; s). 3 If Y is a ZHS and K ⊆ Y , Ozsv´athand Szab´o constructed a graded chain complex, CFK 1(Y; K), which is filtered by Z ⊕ Z. Heegaard Floer homology If Y is a 3-manifold, and s 2 Spinc(Y ), Ozsv´athand Szab´o − + construct three F[U]-modules HF (Y; s), HF (Y; s) and HF 1(Y; s). 3 If Y is a ZHS and K ⊆ Y , Ozsv´athand Szab´o constructed a graded chain complex, CFK 1(Y; K), which is filtered by Z ⊕ Z. − Bold HF indicates coefficients in the power series F[[U]]. 3 When Y = S , the map Dn is explicitly computable from just CFK 1(S3;K). Ozsv´athand Szab´o'smapping cone formula Theorem (Ozsv´ath–Szab´o) 3 If K is a knot in Y , a ZHS , then − ∼ Dn HF (Yn(K)) = H∗ Cone A −−! B ; where A and B are chain complexes obtained from subcomplexes 1 of CFK (Y; K), and Dn is a chain map. 3 When Y = S , the map Dn is explicitly computable from just CFK 1(S3;K). Ozsv´athand Szab´o'smapping cone formula Theorem (Ozsv´ath–Szab´o) 3 If K is a knot in Y , a ZHS , then − ∼ Dn HF (Yn(K)) = H∗ Cone A −−! B ; where A and B are chain complexes obtained from subcomplexes 1 of CFK (Y; K), and Dn is a chain map. − Bold HF indicates coefficients in the power series F[[U]]. Ozsv´athand Szab´o'smapping cone formula Theorem (Ozsv´ath–Szab´o) 3 If K is a knot in Y , a ZHS , then − ∼ Dn HF (Yn(K)) = H∗ Cone A −−! B ; where A and B are chain complexes obtained from subcomplexes 1 of CFK (Y; K), and Dn is a chain map. − Bold HF indicates coefficients in the power series F[[U]]. 3 When Y = S , the map Dn is explicitly computable from just CFK 1(S3;K). Y Y A = As and B = Bs; s2Z s2Z 1 where As and Bs are subcomplexes of CFK (K), completed over F[[U]]. − Bs ' CF (Y ) for all s. (Actually, Bs is independent of s). As is a subcomplex of Bs for all s. Ozsv´athand Szab´o'smapping cone formula − Bs ' CF (Y ) for all s. (Actually, Bs is independent of s). As is a subcomplex of Bs for all s. Ozsv´athand Szab´o'smapping cone formula Y Y A = As and B = Bs; s2Z s2Z 1 where As and Bs are subcomplexes of CFK (K), completed over F[[U]]. As is a subcomplex of Bs for all s. Ozsv´athand Szab´o'smapping cone formula Y Y A = As and B = Bs; s2Z s2Z 1 where As and Bs are subcomplexes of CFK (K), completed over F[[U]]. − Bs ' CF (Y ) for all s. (Actually, Bs is independent of s). Ozsv´athand Szab´o'smapping cone formula Y Y A = As and B = Bs; s2Z s2Z 1 where As and Bs are subcomplexes of CFK (K), completed over F[[U]]. − Bs ' CF (Y ) for all s. (Actually, Bs is independent of s). As is a subcomplex of Bs for all s. Dn = v + h. v sends As to Bs, while h sends As to Bs+n. v is the inclusion of As into Bs. h is the inclusion of As into Bes, composed with a homotopy equivalence F : Bes ! Bs, where Bes is as follows: j i j = s As h v v e Bs F Bs+n Bes Ozsv´athand Szab´o'smapping cone formula v sends As to Bs, while h sends As to Bs+n. v is the inclusion of As into Bs. h is the inclusion of As into Bes, composed with a homotopy equivalence F : Bes ! Bs, where Bes is as follows: j i j = s As h v v e Bs F Bs+n Bes Ozsv´athand Szab´o'smapping cone formula Dn = v + h. v is the inclusion of As into Bs. h is the inclusion of As into Bes, composed with a homotopy equivalence F : Bes ! Bs, where Bes is as follows: j i j = s As h v v e Bs F Bs+n Bes Ozsv´athand Szab´o'smapping cone formula Dn = v + h. v sends As to Bs, while h sends As to Bs+n. h is the inclusion of As into Bes, composed with a homotopy equivalence F : Bes ! Bs, where Bes is as follows: j i j = s As h v v e Bs F Bs+n Bes Ozsv´athand Szab´o'smapping cone formula Dn = v + h. v sends As to Bs, while h sends As to Bs+n. v is the inclusion of As into Bs. j i j = s As h v v e Bs F Bs+n Bes Ozsv´athand Szab´o'smapping cone formula Dn = v + h. v sends As to Bs, while h sends As to Bs+n. v is the inclusion of As into Bs. h is the inclusion of As into Bes, composed with a homotopy equivalence F : Bes ! Bs, where Bes is as follows: Ozsv´athand Szab´o'smapping cone formula Dn = v + h. v sends As to Bs, while h sends As to Bs+n. v is the inclusion of As into Bs. h is the inclusion of As into Bes, composed with a homotopy equivalence F : Bes ! Bs, where Bes is as follows: j i j = s As h v v e Bs F Bs+n Bes For knots in S3, the homotopy type of the mapping cone is determined by just CFK 1(K). Indeed, the only ambiguity is the homotopy equivalence F : Bes ! Bs+n, but Bes and Bs+n are both homotopy − 3 equivalent to CF (S ) ' F[U], so there is a unique homotopy equivalence, up to chain homotopy. Ozsv´athand Szab´o'smapping cone formula Indeed, the only ambiguity is the homotopy equivalence F : Bes ! Bs+n, but Bes and Bs+n are both homotopy − 3 equivalent to CF (S ) ' F[U], so there is a unique homotopy equivalence, up to chain homotopy. Ozsv´athand Szab´o'smapping cone formula For knots in S3, the homotopy type of the mapping cone is determined by just CFK 1(K). Ozsv´athand Szab´o'smapping cone formula For knots in S3, the homotopy type of the mapping cone is determined by just CFK 1(K). Indeed, the only ambiguity is the homotopy equivalence F : Bes ! Bs+n, but Bes and Bs+n are both homotopy − 3 equivalent to CF (S ) ' F[U], so there is a unique homotopy equivalence, up to chain homotopy. Suppose Y a 3-manifold, s 2 Spinc(Y ) and s = s. Hendricks and Manolescu study a homotopy involution ι: CF −(Y; s) ! CF −(Y; s): If H = (Σ; α; β) is a Heegaard diagram, there is a canonical isomorphism η : CF − (H; s) ! CF − H; s where H = (Σ; β; α). ι := η ◦ ΨH!H; where ΨH!H is the map from naturality. Involutive Heegaard Floer homology (Hendricks{Manolescu) Hendricks and Manolescu study a homotopy involution ι: CF −(Y; s) ! CF −(Y; s): If H = (Σ; α; β) is a Heegaard diagram, there is a canonical isomorphism η : CF − (H; s) ! CF − H; s where H = (Σ; β; α). ι := η ◦ ΨH!H; where ΨH!H is the map from naturality. Involutive Heegaard Floer homology (Hendricks{Manolescu) Suppose Y a 3-manifold, s 2 Spinc(Y ) and s = s. If H = (Σ; α; β) is a Heegaard diagram, there is a canonical isomorphism η : CF − (H; s) ! CF − H; s where H = (Σ; β; α). ι := η ◦ ΨH!H; where ΨH!H is the map from naturality. Involutive Heegaard Floer homology (Hendricks{Manolescu) Suppose Y a 3-manifold, s 2 Spinc(Y ) and s = s. Hendricks and Manolescu study a homotopy involution ι: CF −(Y; s) ! CF −(Y; s): ι := η ◦ ΨH!H; where ΨH!H is the map from naturality. Involutive Heegaard Floer homology (Hendricks{Manolescu) Suppose Y a 3-manifold, s 2 Spinc(Y ) and s = s. Hendricks and Manolescu study a homotopy involution ι: CF −(Y; s) ! CF −(Y; s): If H = (Σ; α; β) is a Heegaard diagram, there is a canonical isomorphism η : CF − (H; s) ! CF − H; s where H = (Σ; β; α). Involutive Heegaard Floer homology (Hendricks{Manolescu) Suppose Y a 3-manifold, s 2 Spinc(Y ) and s = s. Hendricks and Manolescu study a homotopy involution ι: CF −(Y; s) ! CF −(Y; s): If H = (Σ; α; β) is a Heegaard diagram, there is a canonical isomorphism η : CF − (H; s) ! CF − H; s where H = (Σ; β; α).