A Mapping Cone Formula for Involutive Heegaard Floer Homology

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 ∈ Spinc(Y ), Ozsváthand Szabó − + construct three F[U]-modules HF (Y, s), HF (Y, s) and HF ∞(Y, s). 3 If Y is a ZHS and K ⊆ Y , Ozsváthand Szabó constructed a graded chain complex, CFK ∞(Y,K), which is filtered by Z ⊕ Z.

Heegaard Floer homology 3 If Y is a ZHS and K ⊆ Y , Ozsváthand Szabó constructed a graded chain complex, CFK ∞(Y,K), which is filtered by Z ⊕ Z.

Heegaard Floer homology

If Y is a 3-manifold, and s ∈ Spinc(Y ), Ozsv´athand Szab´o − + construct three F[U]-modules HF (Y, s), HF (Y, s) and HF ∞(Y, s). Heegaard Floer homology

If Y is a 3-manifold, and s ∈ Spinc(Y ), Ozsváthand Szabó − + construct three F[U]-modules HF (Y, s), HF (Y, s) and HF ∞(Y, s). 3 If Y is a ZHS and K ⊆ Y , Ozsváthand Szabó constructed a graded chain complex, CFK ∞(Y,K), which is filtered by Z ⊕ Z. Heegaard Floer homology

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 ∞ of CFK (Y,K), and Dn is a chain map. 3 When Y = S , the map Dn is explicitly computable from just CFK ∞(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 ∞ of CFK (Y,K), and Dn is a chain map.

− Bold HF indicates coefficients in the power series F[[U]]. Ozsváthand Szabó’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 ∞ 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 ∞(S3,K). Y Y A = As and B = Bs, s∈Z s∈Z ∞ 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, s∈Z s∈Z ∞ 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, s∈Z s∈Z ∞ 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, s∈Z s∈Z ∞ 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

Bes For knots in S3, the homotopy type of the mapping cone is determined by just CFK ∞(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 ∞(K). Ozsv´athand Szab´o’smapping cone formula

For knots in S3, the homotopy type of the mapping cone is determined by just CFK ∞(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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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. Hendricks and Manolescu deﬁne

Q(id +ι) CFI −(Y, s) := Cone CF −(Y, s) −−−−−→ Q · CF −(Y, s) .

2 Module over F[U, Q]/Q . Applications to the homology cobordism group. E.g. ∃ ∞ summand of Θ3 (Dai, Hom, Stoﬀregen, Truong). Z Z

Involutive Heegaard Floer homology 2 Module over F[U, Q]/Q . Applications to the homology cobordism group. E.g. ∃ ∞ summand of Θ3 (Dai, Hom, Stoﬀregen, Truong). Z Z

Involutive Heegaard Floer homology

Hendricks and Manolescu deﬁne

Q(id +ι) CFI −(Y, s) := Cone CF −(Y, s) −−−−−→ Q · CF −(Y, s) . Applications to the homology cobordism group. E.g. ∃ ∞ summand of Θ3 (Dai, Hom, Stoﬀregen, Truong). Z Z

Involutive Heegaard Floer homology

Hendricks and Manolescu deﬁne

Q(id +ι) CFI −(Y, s) := Cone CF −(Y, s) −−−−−→ Q · CF −(Y, s) .

2 Module over F[U, Q]/Q . E.g. ∃ ∞ summand of Θ3 (Dai, Hom, Stoﬀregen, Truong). Z Z

Involutive Heegaard Floer homology

Hendricks and Manolescu deﬁne

Q(id +ι) CFI −(Y, s) := Cone CF −(Y, s) −−−−−→ Q · CF −(Y, s) .

2 Module over F[U, Q]/Q . Applications to the homology cobordism group. Involutive Heegaard Floer homology

Hendricks and Manolescu deﬁne

Q(id +ι) CFI −(Y, s) := Cone CF −(Y, s) −−−−−→ Q · CF −(Y, s) .

2 Module over F[U, Q]/Q . Applications to the homology cobordism group. E.g. ∃ ∞ summand of Θ3 (Dai, Hom, Stoﬀregen, Truong). Z Z For example, is there an analog of the mapping cone formula?

Computing involutive Heegaard Floer homology

Question How can we compute HFI −(Y )? Computing involutive Heegaard Floer homology

Question How can we compute HFI −(Y )?

For example, is there an analog of the mapping cone formula? Hence

− ∼ Q(id +ιK ) HFI (Yn(K), [0]) = H∗ Cone A0 −−−−−−→ Q · A0 .

Hendricks and Manolescu also deﬁned a knot involution ∞ ∞ ιK : CFK (K) → CFK (K).

Theorem (Hendricks–Manolescu) 3 If n is large, and K ⊆ Y is a knot in a ZHS , Y , then

− (CF (Yn(K), [0]), ι) ' (A0, ιK ),

where ' denotes homotopy equivalence of ι-complexes.

Here, [0] denotes the Spinc structure identiﬁed with 0 under c ∼ Spin (Yn(K)) = Zn.

Previous results Hence

− ∼ Q(id +ιK ) HFI (Yn(K), [0]) = H∗ Cone A0 −−−−−−→ Q · A0 .

Theorem (Hendricks–Manolescu) 3 If n is large, and K ⊆ Y is a knot in a ZHS , Y , then

− (CF (Yn(K), [0]), ι) ' (A0, ιK ),

where ' denotes homotopy equivalence of ι-complexes.

Here, [0] denotes the Spinc structure identiﬁed with 0 under c ∼ Spin (Yn(K)) = Zn.

Previous results

Hendricks and Manolescu also deﬁned a knot involution ∞ ∞ ιK : CFK (K) → CFK (K). Hence

− ∼ Q(id +ιK ) HFI (Yn(K), [0]) = H∗ Cone A0 −−−−−−→ Q · A0 .

Here, [0] denotes the Spinc structure identiﬁed with 0 under c ∼ Spin (Yn(K)) = Zn.

Previous results

Hendricks and Manolescu also deﬁned a knot involution ∞ ∞ ιK : CFK (K) → CFK (K).

Theorem (Hendricks–Manolescu) 3 If n is large, and K ⊆ Y is a knot in a ZHS , Y , then

− (CF (Yn(K), [0]), ι) ' (A0, ιK ),

where ' denotes homotopy equivalence of ι-complexes. Here, [0] denotes the Spinc structure identiﬁed with 0 under c ∼ Spin (Yn(K)) = Zn.

Previous results

Hendricks and Manolescu also deﬁned a knot involution ∞ ∞ ιK : CFK (K) → CFK (K).

Theorem (Hendricks–Manolescu) 3 If n is large, and K ⊆ Y is a knot in a ZHS , Y , then

− (CF (Yn(K), [0]), ι) ' (A0, ιK ),

where ' denotes homotopy equivalence of ι-complexes. Hence

− ∼ Q(id +ιK ) HFI (Yn(K), [0]) = H∗ Cone A0 −−−−−−→ Q · A0 . Previous results

Hendricks and Manolescu also deﬁned a knot involution ∞ ∞ ιK : CFK (K) → CFK (K).

Theorem (Hendricks–Manolescu) 3 If n is large, and K ⊆ Y is a knot in a ZHS , Y , then

− (CF (Yn(K), [0]), ι) ' (A0, ιK ),

where ' denotes homotopy equivalence of ι-complexes. Hence

− ∼ Q(id +ιK ) HFI (Yn(K), [0]) = H∗ Cone A0 −−−−−−→ Q · A0 .

Here, [0] denotes the Spinc structure identiﬁed with 0 under c ∼ Spin (Yn(K)) = Zn. If K ⊆ Y is a framed knot there is an exact sequence

··· HFId (Y ) → HFId (Y0) → HFId (Y1) → HFId (Y ) ··· .

Computing involutive Heegaard Floer homology

Theorem (Hendricks–Lipshitz) Using bordered Floer homology, HFId (Y ) may be computed combinatorially. Computing involutive Heegaard Floer homology

Theorem (Hendricks–Lipshitz) Using bordered Floer homology, HFId (Y ) may be computed combinatorially. If K ⊆ Y is a framed knot there is an exact sequence

··· HFId (Y ) → HFId (Y0) → HFId (Y1) → HFId (Y ) ··· . More computational tools

Theorem (Dai–Manolescu) Involutive Heegaard Floer homology is computable for three manifolds obtained by plumbing along almost rational graphs. (This includes all Seifert ﬁbered homology 3-spheres). Theorem (Hendricks–Manolescu–Z.)

If Y1 and Y2 are homology spheres, then under the equivalence − − − CF (Y1#Y2) ' CF (Y1) ⊗ CF (Y2), the involution ιY1#Y2 is

equivalent to ιY1 ⊗ ιY2 .

Theorem (Z.) ∞ ∞ Given (CFK (K1), ιK1 ) and (CFK (K2), ιK2 ), there is a ∞ formula for (CFK (K1#K2), ιK1#K2 ).

More computational tools Theorem (Z.) ∞ ∞ Given (CFK (K1), ιK1 ) and (CFK (K2), ιK2 ), there is a ∞ formula for (CFK (K1#K2), ιK1#K2 ).

More computational tools

Theorem (Hendricks–Manolescu–Z.)

If Y1 and Y2 are homology spheres, then under the equivalence − − − CF (Y1#Y2) ' CF (Y1) ⊗ CF (Y2), the involution ιY1#Y2 is

equivalent to ιY1 ⊗ ιY2 . More computational tools

Theorem (Hendricks–Manolescu–Z.)

If Y1 and Y2 are homology spheres, then under the equivalence − − − CF (Y1#Y2) ' CF (Y1) ⊗ CF (Y2), the involution ιY1#Y2 is

equivalent to ιY1 ⊗ ιY2 .

Theorem (Z.) ∞ ∞ Given (CFK (K1), ιK1 ) and (CFK (K2), ιK2 ), there is a ∞ formula for (CFK (K1#K2), ιK1#K2 ). 3 If K is a knot in a ZHS Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Yn) → HFI (Yn+m) → HFI (Y ) → · · · .

Theorem (In prep. Hendricks–Hom–Stoﬀregen–Z.) If K is a framed knot in Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Y0) → HFI (Y1) → HFI (Y ) ··· .

Bold denotes coeﬃcients in F[[U]]. Underline denotes twisted coeﬃcients.

New developments: exact sequences 3 If K is a knot in a ZHS Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Yn) → HFI (Yn+m) → HFI (Y ) → · · · .

Bold denotes coeﬃcients in F[[U]]. Underline denotes twisted coeﬃcients.

New developments: exact sequences

Theorem (In prep. Hendricks–Hom–Stoﬀregen–Z.) If K is a framed knot in Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Y0) → HFI (Y1) → HFI (Y ) ··· . Bold denotes coeﬃcients in F[[U]]. Underline denotes twisted coeﬃcients.

New developments: exact sequences

Theorem (In prep. Hendricks–Hom–Stoﬀregen–Z.) If K is a framed knot in Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Y0) → HFI (Y1) → HFI (Y ) ··· .

3 If K is a knot in a ZHS Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Yn) → HFI (Yn+m) → HFI (Y ) → · · · . New developments: exact sequences

Theorem (In prep. Hendricks–Hom–Stoﬀregen–Z.) If K is a framed knot in Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Y0) → HFI (Y1) → HFI (Y ) ··· .

3 If K is a knot in a ZHS Y , then there is an exact sequence

− − − − ··· HFI (Y ) → HFI (Yn) → HFI (Yn+m) → HFI (Y ) → · · · .

Bold denotes coefficients in F[[U]]. Underline denotes twisted coefficients. Theorem (In prep., Hendricks–Hom–Stoffregen–Z.) 3 (Weak version) If K is a knot in a ZHS Y , then there is a homotopy equivalence

Dn A B CFI −(Y (K)) ' QHn n Q(id +ιA) Q(id +ιB)

Dn Q · A Q · B

Not amenable for computations, since changing Hn could change the homotopy type.

The involutive mapping cone formula (weak form) Not amenable for computations, since changing Hn could change the homotopy type.

The involutive mapping cone formula (weak form)

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) 3 (Weak version) If K is a knot in a ZHS Y , then there is a homotopy equivalence

Dn A B CFI −(Y (K)) ' QHn n Q(id +ιA) Q(id +ιB)

Dn Q · A Q · B The involutive mapping cone formula (weak form)

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) 3 (Weak version) If K is a knot in a ZHS Y , then there is a homotopy equivalence

Dn A B CFI −(Y (K)) ' QHn n Q(id +ιA) Q(id +ιB)

Dn Q · A Q · B

Not amenable for computations, since changing Hn could change the homotopy type. (Strong version) If K is a knot in a S3, then we may choose the maps to satisfy the following: Dn = v + h, where h factors through ve: As ,→ Bes. Hn factors as ve, followed by a map from Bes to B−s. s ιA is U ιK , and sends As to A−s. s ιB is the composition of U ιK , which sends Bs to Be−s, followed by a homotopy equivalence from Be−s to B−s+n. Most importantly, these conditions completely determine the homotopy type of the mapping cone.

The involutive mapping cone formula for knots in S3:

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Dn = v + h, where h factors through ve: As ,→ Bes. Hn factors as ve, followed by a map from Bes to B−s. s ιA is U ιK , and sends As to A−s. s ιB is the composition of U ιK , which sends Bs to Be−s, followed by a homotopy equivalence from Be−s to B−s+n. Most importantly, these conditions completely determine the homotopy type of the mapping cone.

The involutive mapping cone formula for knots in S3:

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) (Strong version) If K is a knot in a S3, then we may choose the maps to satisfy the following: Hn factors as ve, followed by a map from Bes to B−s. s ιA is U ιK , and sends As to A−s. s ιB is the composition of U ιK , which sends Bs to Be−s, followed by a homotopy equivalence from Be−s to B−s+n. Most importantly, these conditions completely determine the homotopy type of the mapping cone.

The involutive mapping cone formula for knots in S3:

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) (Strong version) If K is a knot in a S3, then we may choose the maps to satisfy the following: Dn = v + h, where h factors through ve: As ,→ Bes. s ιA is U ιK , and sends As to A−s. s ιB is the composition of U ιK , which sends Bs to Be−s, followed by a homotopy equivalence from Be−s to B−s+n. Most importantly, these conditions completely determine the homotopy type of the mapping cone.

The involutive mapping cone formula for knots in S3:

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) (Strong version) If K is a knot in a S3, then we may choose the maps to satisfy the following: Dn = v + h, where h factors through ve: As ,→ Bes. Hn factors as ve, followed by a map from Bes to B−s. s ιA is U ιK , and sends As to A−s. s ιB is the composition of U ιK , which sends Bs to Be−s, followed by a homotopy equivalence from Be−s to B−s+n. Most importantly, these conditions completely determine the homotopy type of the mapping cone. − 3 In particular, the homotopy type of CFI (Sn(K)) is completely determined by, and is easily computed from ∞ (CFK (K), ιK ). We prove similar mapping cone formulas for rational surgeries and 0-surgeries (and prove a similar computability result for knots in S3).

On the strong version of the mapping cone formula We prove similar mapping cone formulas for rational surgeries and 0-surgeries (and prove a similar computability result for knots in S3).

On the strong version of the mapping cone formula

− 3 In particular, the homotopy type of CFI (Sn(K)) is completely determined by, and is easily computed from ∞ (CFK (K), ιK ). On the strong version of the mapping cone formula

− 3 In particular, the homotopy type of CFI (Sn(K)) is completely determined by, and is easily computed from ∞ (CFK (K), ιK ). We prove similar mapping cone formulas for rational surgeries and 0-surgeries (and prove a similar computability result for knots in S3). Here is D1 on X1:

··· A−2 A−1 A0 A1 A2 ···

v h v h v h v h v h

··· B−1 B0 B1 B2 ···

Diagrams when n = 1 Diagrams when n = 1

Here is D1 on X1:

··· A−2 A−1 A0 A1 A2 ···

v h v h v h v h v h

··· B−1 B0 B1 B2 ··· Here is ιX on X1:

ιA

··· A−2 A−1 A0 A1 A2 ···

H H ιA H H H

··· B2 B1 B0 B−1 ··· ιB

ιB

(Note, Bs are shown in reverse order).

Diagrams when n = 1 Diagrams when n = 1

Here is ιX on X1:

ιA

··· A−2 A−1 A0 A1 A2 ···

H H ιA H H H

··· B2 B1 B0 B−1 ··· ιB

ιB

(Note, Bs are shown in reverse order). It’s often useful to consider the algebraic categories of ι-complexes and ιK -complexes. These are the categories consisting of F[U]-chain complexes, equipped with involutions, and an extra ﬁltration structure for ιK -complexes. Deﬁnition

We say an algebraic ιK -complex is of L-space type if ∼ H∗(Bs) = F[U]. ∼ − For complexes arising from a knot K in Y , H∗(Bs) = HF (Y ).

On the algebra of the involutive mapping cone These are the categories consisting of F[U]-chain complexes, equipped with involutions, and an extra ﬁltration structure for ιK -complexes. Deﬁnition

We say an algebraic ιK -complex is of L-space type if ∼ H∗(Bs) = F[U]. ∼ − For complexes arising from a knot K in Y , H∗(Bs) = HF (Y ).

On the algebra of the involutive mapping cone

It’s often useful to consider the algebraic categories of ι-complexes and ιK -complexes. Deﬁnition

We say an algebraic ιK -complex is of L-space type if ∼ H∗(Bs) = F[U]. ∼ − For complexes arising from a knot K in Y , H∗(Bs) = HF (Y ).

On the algebra of the involutive mapping cone

It’s often useful to consider the algebraic categories of ι-complexes and ιK -complexes. These are the categories consisting of F[U]-chain complexes, equipped with involutions, and an extra ﬁltration structure for ιK -complexes. ∼ − For complexes arising from a knot K in Y , H∗(Bs) = HF (Y ).

On the algebra of the involutive mapping cone

We say an algebraic ιK -complex is of L-space type if ∼ H∗(Bs) = F[U]. On the algebra of the involutive mapping cone

We say an algebraic ιK -complex is of L-space type if ∼ H∗(Bs) = F[U]. ∼ − For complexes arising from a knot K in Y , H∗(Bs) = HF (Y ). Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) There is a well-deﬁned (algebraic) map

{ι -complexes of L-space type} {ι-complexes} alg : K −→ , XIn ' '

sending an algebraic ιK -complex to a model of the involutive mapping cone with the above factorization properties.

On the algebra of the involutive mapping cone On the algebra of the involutive mapping cone

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) There is a well-deﬁned (algebraic) map

{ι -complexes of L-space type} {ι-complexes} alg : K −→ , XIn ' '

sending an algebraic ιK -complex to a model of the involutive mapping cone with the above factorization properties. Expanding on the work of Dai, Hom, Stoffregen and Truong, we prove the following: Theorem (In prep., Hendricks–Hom–Stoffregen–Z.) The homology cobordism group is not generated by Seifert fibered homology 3-spheres.

Previous work of Frøyshov (unpublished), F. Lin (2017) and Stoffregen (2020) construct classes in Θ3 which are not Z represented by Seifert fibered spaces. However none of these proofs imply that the classes are not connected sums of such classes. The standard complexes approach of Dai, Hom, Stoffregen and Truong give an algebraic obstruction to being in the span of Seifert fibered spaces, and we use the cone formula to find an example.

An application to the homology cobordism group Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) The homology cobordism group is not generated by Seifert ﬁbered homology 3-spheres.

An application to the homology cobordism group

Expanding on the work of Dai, Hom, Stoffregen and Truong, we prove the following: Previous work of Frøyshov (unpublished), F. Lin (2017) and Stoffregen (2020) construct classes in Θ3 which are not Z represented by Seifert fibered spaces. However none of these proofs imply that the classes are not connected sums of such classes. The standard complexes approach of Dai, Hom, Stoffregen and Truong give an algebraic obstruction to being in the span of Seifert fibered spaces, and we use the cone formula to find an example.

An application to the homology cobordism group

Expanding on the work of Dai, Hom, Stoffregen and Truong, we prove the following: Theorem (In prep., Hendricks–Hom–Stoffregen–Z.) The homology cobordism group is not generated by Seifert fibered homology 3-spheres. The standard complexes approach of Dai, Hom, Stoffregen and Truong give an algebraic obstruction to being in the span of Seifert fibered spaces, and we use the cone formula to find an example.

An application to the homology cobordism group

Two ι-complexes (C1, ι1) and (C2, ι2) are locally equivalent if there are grading preserving chain maps

F : C1 → C2 and G: C2 → C1

such that F ι1 + ι2F ' 0 and Gι2 + ι1G ' 0, such that F and G become isomorphisms on homology after inverting U.

The local class of (CF −(Y ), ι) contains all the algebraic obstructions to homology cobordism coming from HFI .

More applications: local equivalence classes The local class of (CF −(Y ), ι) contains all the algebraic obstructions to homology cobordism coming from HFI .

More applications: local equivalence classes

Deﬁnition

Two ι-complexes (C1, ι1) and (C2, ι2) are locally equivalent if there are grading preserving chain maps

F : C1 → C2 and G: C2 → C1

such that F ι1 + ι2F ' 0 and Gι2 + ι1G ' 0, such that F and G become isomorphisms on homology after inverting U. More applications: local equivalence classes

Deﬁnition

Two ι-complexes (C1, ι1) and (C2, ι2) are locally equivalent if there are grading preserving chain maps

F : C1 → C2 and G: C2 → C1

such that F ι1 + ι2F ' 0 and Gι2 + ι1G ' 0, such that F and G become isomorphisms on homology after inverting U.

The local class of (CF −(Y ), ι) contains all the algebraic obstructions to homology cobordism coming from HFI . 3 c 3 ∼ If K ⊆ S , then Spin (Sn(K)) = Z/n. If n is odd, then [0] is the only self-conjugate Spinc structure. If n is even, then [0] and [n/2] are the only self-conjugate Spinc structures.

More applications: local equivalence classes

Recall: If n is odd, then [0] is the only self-conjugate Spinc structure. If n is even, then [0] and [n/2] are the only self-conjugate Spinc structures.

More applications: local equivalence classes

Recall: 3 c 3 ∼ If K ⊆ S , then Spin (Sn(K)) = Z/n. If n is even, then [0] and [n/2] are the only self-conjugate Spinc structures.

More applications: local equivalence classes

Recall: 3 c 3 ∼ If K ⊆ S , then Spin (Sn(K)) = Z/n. If n is odd, then [0] is the only self-conjugate Spinc structure. More applications: local equivalence classes

Recall: 3 c 3 ∼ If K ⊆ S , then Spin (Sn(K)) = Z/n. If n is odd, then [0] is the only self-conjugate Spinc structure. If n is even, then [0] and [n/2] are the only self-conjugate Spinc structures. − 3 Then CFI (Sn(K), [0]) is locally equivalent to (A0, ιK ). − 3 Also, CFI (S2n(K), [n]) is locally equivalent to

An An v v

with the involution which swaps the two copies of An via the identity map.

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0.

A similar story holds for rational surgeries.

More applications: local equivalence classes − 3 Then CFI (Sn(K), [0]) is locally equivalent to (A0, ιK ). − 3 Also, CFI (S2n(K), [n]) is locally equivalent to

An An v v

with the involution which swaps the two copies of An via the identity map.

A similar story holds for rational surgeries.

More applications: local equivalence classes

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0. − 3 Also, CFI (S2n(K), [n]) is locally equivalent to

An An v v

with the involution which swaps the two copies of An via the identity map.

A similar story holds for rational surgeries.

More applications: local equivalence classes

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0. − 3 Then CFI (Sn(K), [0]) is locally equivalent to (A0, ιK ). A similar story holds for rational surgeries.

More applications: local equivalence classes

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0. − 3 Then CFI (Sn(K), [0]) is locally equivalent to (A0, ιK ). − 3 Also, CFI (S2n(K), [n]) is locally equivalent to

An An v v

with the involution which swaps the two copies of An via the identity map. More applications: local equivalence classes

An An v v

with the involution which swaps the two copies of An via the identity map.

A similar story holds for rational surgeries. 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 3 d(S2n(K), [n]) = d(S2n(K), [n]). 3 d(S2n(K), [n]) = d(L(2n, 1), [n]).

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0.

This is an analog of a result by Ni and Wu, concerning the ordinary d-invariants of surgeries.

More applications: correction terms 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 3 d(S2n(K), [n]) = d(S2n(K), [n]). 3 d(S2n(K), [n]) = d(L(2n, 1), [n]).

This is an analog of a result by Ni and Wu, concerning the ordinary d-invariants of surgeries.

More applications: correction terms

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0. 3 3 d(S2n(K), [n]) = d(S2n(K), [n]). 3 d(S2n(K), [n]) = d(L(2n, 1), [n]).

More applications: correction terms

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0. 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K).

This is an analog of a result by Ni and Wu, concerning the ordinary d-invariants of surgeries. More applications: correction terms

Theorem (In prep., Hendricks–Hom–Stoﬀregen–Z.) Suppose K ⊆ S3, and n > 0. 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 d(Sn(K), [0]) = d(L(n, 1), [0]) − 2V 0(K). 3 3 d(S2n(K), [n]) = d(S2n(K), [n]). 3 d(S2n(K), [n]) = d(L(2n, 1), [n]).

This is an analog of a result by Ni and Wu, concerning the ordinary d-invariants of surgeries. Recall the main steps in Ozsv´athand Szab´o’sproof of the ordinary mapping cone formula: A large surgeries formula, which states that

− ∼ HF (Yn(K), [i]) = H∗(Ai(K)),

if n is suﬃciently large. A surgery exact sequence

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+m) → HF (Y ) ··· ,

where HF −(Y ) = Lm HF −(Y ).

Idea of proof A large surgeries formula, which states that

− ∼ HF (Yn(K), [i]) = H∗(Ai(K)),

if n is suﬃciently large. A surgery exact sequence

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+m) → HF (Y ) ··· ,

where HF −(Y ) = Lm HF −(Y ).

Idea of proof

Recall the main steps in Ozsv´athand Szab´o’sproof of the ordinary mapping cone formula: A surgery exact sequence

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+m) → HF (Y ) ··· ,

where HF −(Y ) = Lm HF −(Y ).

Idea of proof

Recall the main steps in Ozsv´athand Szab´o’sproof of the ordinary mapping cone formula: A large surgeries formula, which states that

− ∼ HF (Yn(K), [i]) = H∗(Ai(K)),

if n is suﬃciently large. Idea of proof

Recall the main steps in Ozsv´athand Szab´o’sproof of the ordinary mapping cone formula: A large surgeries formula, which states that

− ∼ HF (Yn(K), [i]) = H∗(Ai(K)),

if n is suﬃciently large. A surgery exact sequence

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+m) → HF (Y ) ··· ,

where HF −(Y ) = Lm HF −(Y ). Consider the m = 1 case

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+1) → HF (Y ) ··· ,

Deﬁne a “cobordism” map

− − CFI (Yn+1) → CFI (Y )

as well as a quasi-isomorphism − − − Φ: CFI (Yn) → Cone CFI (Yn+1) → CFI (Y ) .

The exact sequence for mapping cones from homological algebra gives the surgery exact sequence.

Idea of proof: exact sequence, m = 1 Deﬁne a “cobordism” map

− − CFI (Yn+1) → CFI (Y )

as well as a quasi-isomorphism − − − Φ: CFI (Yn) → Cone CFI (Yn+1) → CFI (Y ) .

The exact sequence for mapping cones from homological algebra gives the surgery exact sequence.

Idea of proof: exact sequence, m = 1

Consider the m = 1 case

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+1) → HF (Y ) ··· , The exact sequence for mapping cones from homological algebra gives the surgery exact sequence.

Idea of proof: exact sequence, m = 1

Consider the m = 1 case

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+1) → HF (Y ) ··· ,

Deﬁne a “cobordism” map

− − CFI (Yn+1) → CFI (Y )

as well as a quasi-isomorphism − − − Φ: CFI (Yn) → Cone CFI (Yn+1) → CFI (Y ) . Idea of proof: exact sequence, m = 1

Consider the m = 1 case

− − − − ··· HF (Y ) → HF (Yn) → HF (Yn+1) → HF (Y ) ··· ,

Deﬁne a “cobordism” map

− − CFI (Yn+1) → CFI (Y )

as well as a quasi-isomorphism − − − Φ: CFI (Yn) → Cone CFI (Yn+1) → CFI (Y ) .

The exact sequence for mapping cones from homological algebra gives the surgery exact sequence. To build

− − − Φ: CFI (Yn) → Cone(CFI (Yn+1) → CFI (Y )) we start by building a hypercube (i.e. a cubical diagram whose total complex is a chain complex)

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y ) Furthermore, the maps along top coincide with the maps along the bottom.

Idea of proof: exact sequence, m = 1 Furthermore, the maps along top coincide with the maps along the bottom.

Idea of proof: exact sequence, m = 1

To build

− − − Φ: CFI (Yn) → Cone(CFI (Yn+1) → CFI (Y )) we start by building a hypercube (i.e. a cubical diagram whose total complex is a chain complex)

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y ) Idea of proof: exact sequence, m = 1

To build

− − − Φ: CFI (Yn) → Cone(CFI (Yn+1) → CFI (Y )) we start by building a hypercube (i.e. a cubical diagram whose total complex is a chain complex)

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y ) Furthermore, the maps along top coincide with the maps along the bottom. The maps along the top and bottom were constructed by Ozsv´athand Szab´o. The maps along the left and front face were constructed by Hendricks and Manolescu. The challenging part which is new to our work is the length 3 dotted arrow.

Idea of proof: exact sequence, m = 1

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y ) The maps along the left and front face were constructed by Hendricks and Manolescu. The challenging part which is new to our work is the length 3 dotted arrow.

Idea of proof: exact sequence, m = 1

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y )

The maps along the top and bottom were constructed by Ozsv´athand Szab´o. The challenging part which is new to our work is the length 3 dotted arrow.

Idea of proof: exact sequence, m = 1

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y )

The maps along the top and bottom were constructed by Ozsv´athand Szab´o. The maps along the left and front face were constructed by Hendricks and Manolescu. Idea of proof: exact sequence, m = 1

− CF (Yn)

− − ι CF (Yn+1) CF (Y )

− CF (Yn) ι ι

− − CF (Yn+1) CF (Y )

The maps along the top and bottom were constructed by Ozsv´athand Szab´o. The maps along the left and front face were constructed by Hendricks and Manolescu. The challenging part which is new to our work is the length 3 dotted arrow. Since the maps along the top and bottom, agree, we can add id to each vertical map, and total complex will still be a chain complex:

− CF (Yn)

− − Q(id +ι) CF (Yn+1) CF (Y )

− QCF (Yn) Q(id +ι) Q(id +ι)

− − QCF (Yn+1) QCF (Y )

− This is the same as a chain map from CFI (Yn) to − − Cone(CFI (Yn+1) → CFI (Y )).

Idea of proof: exact sequence, m = 1 − CF (Yn)

− − Q(id +ι) CF (Yn+1) CF (Y )

− QCF (Yn) Q(id +ι) Q(id +ι)

− − QCF (Yn+1) QCF (Y )

− This is the same as a chain map from CFI (Yn) to − − Cone(CFI (Yn+1) → CFI (Y )).

Idea of proof: exact sequence, m = 1

Since the maps along the top and bottom, agree, we can add id to each vertical map, and total complex will still be a chain complex: − This is the same as a chain map from CFI (Yn) to − − Cone(CFI (Yn+1) → CFI (Y )).

Idea of proof: exact sequence, m = 1

Since the maps along the top and bottom, agree, we can add id to each vertical map, and total complex will still be a chain complex:

− CF (Yn)

− − Q(id +ι) CF (Yn+1) CF (Y )

− QCF (Yn) Q(id +ι) Q(id +ι)

− − QCF (Yn+1) QCF (Y ) Idea of proof: exact sequence, m = 1

Since the maps along the top and bottom, agree, we can add id to each vertical map, and total complex will still be a chain complex:

− CF (Yn)

− − Q(id +ι) CF (Yn+1) CF (Y )

− QCF (Yn) Q(id +ι) Q(id +ι)

− − QCF (Yn+1) QCF (Y )

− This is the same as a chain map from CFI (Yn) to − − Cone(CFI (Yn+1) → CFI (Y )). Thanks for listening!