Is The Ordinary Cup Square
Abstract.
We use equivariant methods and product structures to derive a relation between the fixed point Floer cohomology of an exact symplectic automorphism and that of its square.
This paper concerns the Floer cohomology of symplectic automorphisms, and its behaviour under iterations: more specifically, when passing to the square of a given automorphism (one expects parallel results for odd prime powers, but they are beyond our scope here). The concrete situation is as follows. Let Ï•italic-Ï•\phiitalic_Ï• be an exact symplectic automorphism of a Liouville domain Mݑ€Mitalic_M (there are some additional conditions on Ï•italic-Ï•\phiitalic_Ï•, see Setup 2.12 for details). The Floer cohomology ݻݹ*(Ï•)superscriptݻݹitalic-Ï•\mathitHF^{*}(\phi)italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) (defined in [20], generalizing the Hamiltonian case [24]) is a ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded ݕ‚ݕ‚\mathbbKblackboard_K-vector space. Here and throughout the paper, ݕ‚=ݔ½2ݕ‚subscriptݔ½2\mathbbK=\mathbbF_2blackboard_K = blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is the field with two elements. The Floer cohomology of Ï•2superscriptitalic-Ï•2\phi^2italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT carries additional structure, page_seo_titlely an action of ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2. Denote the invariant part by ݻݹ*(Ï•2)ℤ/2superscriptݻݹsuperscriptsuperscriptitalic-Ï•2ℤ2\mathitHF^{*}(\phi^2)^\mathbbZ/2italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT blackboard_Z / 2 end_POSTSUPERSCRIPT. From the viewpoint of applications, our most significant result is the following Smith-type inequality (the page_seo_title refers to a topological result reproduced as (2.20) below, see [8, Chapter III, 4.3]):
There is an inequality of total dimensions,
(1.1) dimݻݹ*(Ï•2)ℤ/2≥dimݻݹ*(Ï•).dimsuperscriptݻݹsuperscriptsuperscriptitalic-Ï•2ℤ2dimsuperscriptݻݹitalic-Ï•\mathrmdim\,\mathitHF^{*}(\phi^2)^\mathbbZ/2\geq\mathrmdim\,% \mathitHF^{*}(\phi).roman_dim italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT blackboard_Z / 2 end_POSTSUPERSCRIPT ≥ roman_dim italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) .
This is not entirely new: under additional topological restrictions (stated below as Assumption 2.21), it has been previously proved by Hendricks [31]. As in [31], the proof involves an equivariant form of Floer cohomology, written as ݻݹݑ’ݑž*(Ï•2)subscriptsuperscriptݻݹݑ’ݑžsuperscriptitalic-Ï•2\mathitHF^{*}_\mathiteq(\phi^2)italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_eq end_POSTSUBSCRIPT ( italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). This is a finitely generated ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded module over ݕ‚[[h]]ݕ‚delimited-[]delimited-[]â„Ž\mathbbK[[h]]blackboard_K [ [ italic_h ] ], the ring of formal power series in one variable hâ„Žhitalic_h (the variable has degree 1111). The information encoded in this equivariant theory can be viewed as a refinement of the previously mentioned ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-action. What we obtain is a page_content of equivariant Floer cohomology after inverting hâ„Žhitalic_h, which means after tensoring with the ring ݕ‚((h))ݕ‚ℎ\mathbbK((h))blackboard_K ( ( italic_h ) ) of Laurent series:
There is an isomorphism of ungraded ݕ‚((h))ݕ‚ℎ\mathbbK((h))blackboard_K ( ( italic_h ) )-modules,
(1.2) ݻݹ*(Ï•)((h))=ݻݹ*(Ï•)⊗ݕ‚((h))≅ݻݹݑ’ݑž*(Ï•2)⊗ݕ‚[[h]]ݕ‚((h)).superscriptݻݹitalic-Ï•â„Žtensor-productsuperscriptݻݹitalic-ϕݕ‚ℎsubscripttensor-productݕ‚delimited-[]delimited-[]â„Žsubscriptsuperscriptݻݹݑ’ݑžsuperscriptitalic-Ï•2ݕ‚ℎ\mathitHF^{*}(\phi)((h))=\mathitHF^{*}(\phi)\otimes\mathbbK((h))\cong% \mathitHF^{*}_\mathiteq(\phi^2)\otimes_\mathbbK[[h]]\mathbbK% ((h)).italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) ( ( italic_h ) ) = italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) ⊗ blackboard_K ( ( italic_h ) ) ≅ italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_eq end_POSTSUBSCRIPT ( italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ⊗ start_POSTSUBSCRIPT blackboard_K [ [ italic_h ] ] end_POSTSUBSCRIPT blackboard_K ( ( italic_h ) ) .
Corollary 1.1 follows from this by purely algebraic arguments (the same step appears in [59, 31], as well as in ordinary equivariant cohomology [8, Chapter IV.4]).
Naively, (1.1) may not be surprising: if one thinks of Floer cohomology as a measure of fixed points, ϕ2superscriptitalic-ϕ2\phi^2italic_ϕ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT clearly has more of them than ϕitalic-ϕ\phiitalic_ϕ. In the same intuitive spirit (and with the localization theorem for equivariant cohomology in mind, which we will recall as Theorem 2.9 below), one can think of tensoring with ݕ‚((h))ݕ‚ℎ\mathbbK((h))blackboard_K ( ( italic_h ) ) as throwing away the fixed points of ϕ2superscriptitalic-ϕ2\phi^2italic_ϕ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT which are not fixed points of ϕitalic-ϕ\phiitalic_ϕ, leading to (1.2). Indeed, in a sense, the proofs ultimately reduce to such very basic considerations. Before one can get to that point, however, a map has to be defined which allows one to compare the two sides of (1.2). It is at this point that our approach diverges from that in [31]. We construct an equivariant refinement of the pair-of-pants product [55, 52], which is a homomorphism of ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded ݕ‚[[h]]ݕ‚delimited-[]delimited-[]ℎ\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-modules,
(1.3) H*(ℤ/2;ݶݹ*(Ï•)⊗ݶݹ*(Ï•))⟶ݻݹݑ’ݑž*(Ï•2).⟶superscriptÝ»ℤ2tensor-productsuperscriptݶݹitalic-Ï•superscriptݶݹitalic-Ï•subscriptsuperscriptݻݹݑ’ݑžsuperscriptitalic-Ï•2H^{*}(\mathbbZ/2;\mathitCF^{*}(\phi)\otimes\mathitCF^{*}(\phi))% \longrightarrow\mathitHF^{*}_\mathiteq(\phi^2).italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_CF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) ⊗ italic_CF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) ) ⟶ italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_eq end_POSTSUBSCRIPT ( italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .
Here ݶݹ*(Ï•)superscriptݶݹitalic-Ï•\mathitCF^{*}(\phi)italic_CF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) is the chain complex underlying ݻݹ*(Ï•)superscriptݻݹitalic-Ï•\mathitHF^{*}(\phi)italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ). We take its tensor product with itself (as a chain complex), equip it with the involution that exchanges the two factors, and consider the associated group cohomology H*(ℤ/2;ݶݹ*(Ï•)⊗ݶݹ*(Ï•))superscriptÝ»ℤ2tensor-productsuperscriptݶݹitalic-Ï•superscriptݶݹitalic-Ï•H^{*}(\mathbbZ/2;\mathitCF^{*}(\phi)\otimes\mathitCF^{*}(\phi))italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_CF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) ⊗ italic_CF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ) ). We will see, as part of the elementary formalism of group cohomology, that this depends only on ݻݹ*(Ï•)superscriptݻݹitalic-Ï•\mathitHF^{*}(\phi)italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_Ï• ). Our main theorem is:
The equivariant pair-of-pants product (1.3) becomes an isomorphism after tensoring with ݕ‚((h))ݕ‚ℎ\mathbbK((h))blackboard_K ( ( italic_h ) ) on both sides.
Corollary 1.2 is a purely algebraic consequence of this statement. Note that in principle, the map (1.3) contains additional information, which is lost when taking the tensor product with ݕ‚((h))ݕ‚ℎ\mathbbK((h))blackboard_K ( ( italic_h ) ).
Addendum 1.4.
The construction of ݻݹ*(ϕ)superscriptݻݹitalic-ϕ\mathitHF^{*}(\phi)italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ϕ ) assumes nondegeneracy of fixed points, and involves additional choices of almost complex structures. Ultimately, one uses continuation maps [51] to show that Floer cohomology is independent of those choices up to canonical isomorphism, and also to extend the definition to the degenerate case.
Similarly, the construction of ݻݹݑ’ݑž*(Ï•2)subscriptsuperscriptݻݹݑ’ݑžsuperscriptitalic-Ï•2\mathitHF^{*}_\mathiteq(\phi^2)italic_HF start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_eq end_POSTSUBSCRIPT ( italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and of (1.3) requires nondegeneracy of the fixed points of Ï•2superscriptitalic-Ï•2\phi^2italic_Ï• start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, and involves further auxiliary choices (of almost complex structures and, in the case of the product, Hamiltonian functions which serve as inhomogeneous terms for the ∂¯normal-¯\bar\partialover¯ start_ARG ∂ end_ARG-equations). Even though this should not affect the outcome, in the same sense as before, we will not prove that statement here.
Now, the proof of Theorem 1.3 makes some specific requirements: in addition to the nondegeneracy of fixed points of ϕ2superscriptitalic-ϕ2\phi^2italic_ϕ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, there is an additional condition on the action functional (see Setup 6.8; this can be achieved by a small perturbation). One then needs to choose the auxiliary data (specifically, the inhomogeneous terms) that define the equivariant pair-of-pants product to be sufficiently small. The precise statement should therefore be that, for this particular class of ϕitalic-ϕ\phiitalic_ϕ, one can define (1.3) in such a way that it becomes an isomorphism after tensoring with ݕ‚((h))ݕ‚ℎ\mathbbK((h))blackboard_K ( ( italic_h ) ). The same applies to Corollary 1.2. However, Corollary 1.1 does not require any such additional language (because the statement only concerns ordinary Floer cohomology groups).
The structure of the paper is as follows. Section 2, a kind of extended introduction, provides background and context for our constructions. In particular, it describes the algebraic arguments that tie together the statements made above; explains the motivation from classical equivariant cohomology; and discusses some applications. Section 3 constructs certain auxiliary Morse-theoretic moduli spaces. Using those plus rather standard Floer-theoretic machinery, we construct equivariant Floer cohomology and (1.3), in Section 4. Section 5 contains further background material, this time from symplectic linear algebra. This is used in Section 6 to prove Theorem 1.3. Finally, Section 7 takes a brief look at some of the new phenomena that one can expect if the exactness assumptions are dropped.
Acknowledgments. I am indebted to Kristen Hendricks, Graeme Segal, David Treumann and Jingyu Zhao for helpful explanations. This work was partially supported by NSF through grant DMS-1005288; by the Simons Foundation, through a Simons Investigator grant; and by a Fellowship at the Radcliffe Institute for Advanced Study. I would also like to thank the IBS Center for Geometry and Physics (Pohang), where part of the paper was written, for its hospitality.
2. Context
Since the constructions in this paper are modelled on ones in equivariant cohomology, we include a review of that theory (specialized to the group ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2), emphasizing its algebraic aspects. After that, we outline the structure of the Floer-theoretic analogue, and in particular, explain how one goes from Theorem 1.3 to Corollaries 1.1 and 1.2. We will then discuss some sample application. Finally, returning to the general picture, we consider how our approach to relating the Floer cohomology of ϕitalic-ϕ\phiitalic_ϕ and ϕ2superscriptitalic-ϕ2\phi^2italic_ϕ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT compares to that in [31], as well as to the purely algebraic theory in [41]. Surprisingly, the attempt to combine the picture here with that in [31] naturally seems to involve another theory, page_seo_titlely, the Floer homotopy type proposed in [15].
2.1. Algebra background
Let Vݑ‰Vitalic_V be a vector space over ݕ‚=ݔ½2ݕ‚subscriptݔ½2\mathbbK=\mathbbF_2blackboard_K = blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, with a linear action of the group ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2, or in other words, an involution ι:V→V:ݜ„→ݑ‰ݑ‰\iota:V\rightarrow Vitalic_ι : italic_V → italic_V. The associated group cochain complex is
(2.1) C*(ℤ/2;V)=V[[h]],dC=h(ݑ–ݑ‘+ι),formulae-sequencesuperscriptݶℤ2ݑ‰ݑ‰delimited-[]delimited-[]â„Žsubscriptݑ‘ݶℎݑ–ݑ‘ݜ„C^{*}(\mathbbZ/2;V)=V[[h]],\quad d_C=h(\mathitid+\iota),italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = italic_V [ [ italic_h ] ] , italic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = italic_h ( italic_id + italic_ι ) ,
where hâ„Žhitalic_h is a formal variable of degree 1111. Its cohomology, called group cohomology with coefficients in Vݑ‰Vitalic_V and denoted by H*(ℤ/2;V)superscriptÝ»ℤ2ݑ‰H^{*}(\mathbbZ/2;V)italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ), is a ℤℤ\mathbbZblackboard_Z-graded module over ݕ‚[[h]]ݕ‚delimited-[]delimited-[]â„Ž\mathbbK[[h]]blackboard_K [ [ italic_h ] ]. There is also a version where one inverts hâ„Žhitalic_h, whose cohomology is called Tate cohomology:
(2.2) C^*(ℤ/2;V)=C*(ℤ/2;V)⊗ݕ‚[[h]]ݕ‚((h))=V((h)),superscript^ݶℤ2ݑ‰subscripttensor-productݕ‚delimited-[]delimited-[]â„Žsuperscriptݶℤ2ݑ‰ݕ‚ℎݑ‰ℎ\displaystyle\hatC^{*}(\mathbbZ/2;V)=C^{*}(\mathbbZ/2;V)\otimes_% \mathbbK[[h]]\mathbbK((h))=V((h)),over^ start_ARG italic_C end_ARG start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) ⊗ start_POSTSUBSCRIPT blackboard_K [ [ italic_h ] ] end_POSTSUBSCRIPT blackboard_K ( ( italic_h ) ) = italic_V ( ( italic_h ) ) ,
(2.3) H^*(ℤ/2;V)=H*(C^*(ℤ/2;V))≅H*(ℤ/2;V)⊗ݕ‚[[h]]ݕ‚((h)).superscript^Ý»ℤ2ݑ‰superscriptÝ»superscript^ݶℤ2ݑ‰subscripttensor-productݕ‚delimited-[]delimited-[]â„ŽsuperscriptÝ»ℤ2ݑ‰ݕ‚ℎ\displaystyle\hatH^{*}(\mathbbZ/2;V)=H^{*}(\hatC^{*}(\mathbbZ/2;V)% )\cong H^{*}(\mathbbZ/2;V)\otimes_\mathbbK[[h]]\mathbbK((h)).over^ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( over^ start_ARG italic_C end_ARG start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) ) ≅ italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) ⊗ start_POSTSUBSCRIPT blackboard_K [ [ italic_h ] ] end_POSTSUBSCRIPT blackboard_K ( ( italic_h ) ) .
Both versions are functorial in Vݑ‰Vitalic_V (under ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-equivariant linear maps).
Let Vݑ‰Vitalic_V be a vector space with ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-action, which is equivariantly isomorphic to a direct sum of copies of the standard representation ݕ‚[ℤ/2]ݕ‚delimited-[]ℤ2\mathbbK[\mathbbZ/2]blackboard_K [ blackboard_Z / 2 ]. In simpler terms, this means that Vݑ‰Vitalic_V has a basis freely acted on by ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2. Direct computation shows that then, H^*(ℤ/2;V)=0superscriptnormal-^Ý»ℤ2ݑ‰0\hatH^{*}(\mathbbZ/2;V)=0over^ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = 0.
Group cohomology, which applies to representations of arbitrary groups, was defined in [21]. The Tate version, for finite groups, was introduced in [63]. However, the general relation between the two theories takes on a more complicated form than (2.3). Example 2.1 is a special case of the vanishing of Tate cohomology with coefficients in a free module (see e.g. [10, p. 136]).
The definitions made above generalize to the situation where Vݑ‰Vitalic_V is a (ℤℤ\mathbbZblackboard_Z-graded or ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded) chain complex of vector spaces acted on by ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2, in which case the differential on C*(ℤ/2;V)superscriptݶℤ2ݑ‰C^{*}(\mathbbZ/2;V)italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) becomes dC=dV+h(ݑ–ݑ‘+ι)subscriptݑ‘ݶsubscriptݑ‘ݑ‰ℎݑ–ݑ‘ݜ„d_C=d_V+h(\mathitid+\iota)italic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT + italic_h ( italic_id + italic_ι ). Its cohomology H*(ℤ/2;V)superscriptÝ»ℤ2ݑ‰H^{*}(\mathbbZ/2;V)italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) is again a (ℤℤ\mathbbZblackboard_Z-graded or ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded) ݕ‚[[h]]ݕ‚delimited-[]delimited-[]â„Ž\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-module. We summarize some of its basic properties:
Lemma 2.3.
(i) If H*(V)=0superscriptÝ»ݑ‰0H^{*}(V)=0italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_V ) = 0, then H*(ℤ/2;V)=0superscriptÝ»ℤ2ݑ‰0H^{*}(\mathbbZ/2;V)=0italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = 0.
(ii) If H*(V)superscriptÝ»ݑ‰H^{*}(V)italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_V ) is of finite (total) dimension, then H*(ℤ/2;V)superscriptÝ»ℤ2ݑ‰H^{*}(\mathbbZ/2;V)italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) is a finitely generated ݕ‚[[h]]ݕ‚delimited-[]delimited-[]â„Ž\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-module.
(iii) Suppose that V1subscriptݑ‰1V_1italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and V2subscriptݑ‰2V_2italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are chain complexes with ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-actions, and that we have a chain map V1→V2normal-→subscriptݑ‰1subscriptݑ‰2V_1\rightarrow V_2italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT which is ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-equivariant, and which induces an isomorphism H*(V1)→H*(V2)normal-→superscriptÝ»subscriptݑ‰1superscriptÝ»subscriptݑ‰2H^{*}(V_1)\rightarrow H^{*}(V_2)italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). Then the associated map H*(ℤ/2;V1)→H*(ℤ/2;V2)normal-→superscriptÝ»ℤ2subscriptݑ‰1superscriptÝ»ℤ2subscriptݑ‰2H^{*}(\mathbbZ/2;V_1)\rightarrow H^{*}(\mathbbZ/2;V_2)italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) is also an isomorphism.
(iv) Suppose that we have three chain complexes with ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-actions, and equivariant chain maps between them, which form a short exact sequence
(2.4) 0→V1⟶V2⟶V3→0.→0subscriptݑ‰1⟶subscriptݑ‰2⟶subscriptݑ‰3→00\rightarrow V_1\longrightarrow V_2\longrightarrow V_3\rightarrow 0.0 → italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟶ italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟶ italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT → 0 .
Then, the associated maps on group cohomology fit into a long exact sequence
(2.5) ⋯→H*(ℤ/2;V1)⟶H*(ℤ/2;V2)⟶H*(ℤ/2;V3)⟶H*+1(ℤ/2;V1)→⋯→⋯superscriptÝ»ℤ2subscriptݑ‰1⟶superscriptÝ»ℤ2subscriptݑ‰2⟶superscriptÝ»ℤ2subscriptݑ‰3⟶superscriptÝ»absent1ℤ2subscriptݑ‰1→⋯\cdots\rightarrow H^{*}(\mathbbZ/2;V_1)\longrightarrow H^{*}(\mathbbZ% /2;V_2)\longrightarrow H^{*}(\mathbbZ/2;V_3)\longrightarrow H^*+1(% \mathbbZ/2;V_1)\rightarrow\cdots⋯ → italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⟶ italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⟶ italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ⟶ italic_H start_POSTSUPERSCRIPT * + 1 end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) → ⋯
(i) Take a cocycle v∈C*(ℤ/2;V)=V[[h]]ݑ£superscriptݶℤ2ݑ‰ݑ‰delimited-[]delimited-[]â„Žv\in C^{*}(\mathbbZ/2;V)=V[[h]]italic_v ∈ italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = italic_V [ [ italic_h ] ], and write it as v=v0+O(h)ݑ£superscriptݑ£0ݑ‚ℎv=v^0+O(h)italic_v = italic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_O ( italic_h ), where v0∈Vsuperscriptݑ£0ݑ‰v^0\in Vitalic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ∈ italic_V (the notation O(h)ݑ‚ℎO(h)italic_O ( italic_h ) means a multiple of hâ„Žhitalic_h, or in other words, an element of hV[[h]]ℎݑ‰delimited-[]delimited-[]â„ŽhV[[h]]italic_h italic_V [ [ italic_h ] ]). Then, dVv0=0subscriptݑ‘ݑ‰superscriptݑ£00d_Vv^0=0italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = 0. By assumption, there is a w0∈Vsuperscriptݑ¤0ݑ‰w^0\in Vitalic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ∈ italic_V such that dVw0=v0subscriptݑ‘ݑ‰superscriptݑ¤0superscriptݑ£0d_Vw^0=v^0italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT. One can therefore write v-dCw0=hv1+O(h2)ݑ£subscriptݑ‘ݶsuperscriptݑ¤0â„Žsuperscriptݑ£1ݑ‚superscriptâ„Ž2v-d_Cw^0=hv^1+O(h^2)italic_v - italic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_h italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT + italic_O ( italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) for some v1∈Vsuperscriptݑ£1ݑ‰v^1\in Vitalic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ italic_V, and then repeat the previous argument to find a w1∈Vsuperscriptݑ¤1ݑ‰w^1\in Vitalic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ italic_V such that v-dC(w0+hw1)=O(h2)ݑ£subscriptݑ‘ݶsuperscriptݑ¤0â„Žsuperscriptݑ¤1ݑ‚superscriptâ„Ž2v-d_C(w^0+hw^1)=O(h^2)italic_v - italic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_h italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) = italic_O ( italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). This iteratively constructs w=w0+hw1+⋯∈V[[h]]ݑ¤superscriptݑ¤0â„Žsuperscriptݑ¤1⋯ݑ‰delimited-[]delimited-[]â„Žw=w^0+hw^1+\cdots\in V[[h]]italic_w = italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_h italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT + ⋯ ∈ italic_V [ [ italic_h ] ] which satisfies dCw=vsubscriptݑ‘ݶݑ¤ݑ£d_Cw=vitalic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_w = italic_v.
(ii) The quotient map C*(ℤ/2;V)=V[[h]]→V[[h]]/hV[[h]]=Vsuperscriptݶℤ2ݑ‰ݑ‰delimited-[]delimited-[]ℎ→ݑ‰delimited-[]delimited-[]ℎℎݑ‰delimited-[]delimited-[]ℎݑ‰C^{*}(\mathbbZ/2;V)=V[[h]]\rightarrow V[[h]]/hV[[h]]=Vitalic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) = italic_V [ [ italic_h ] ] → italic_V [ [ italic_h ] ] / italic_h italic_V [ [ italic_h ] ] = italic_V induces a map
(2.6) H*(ℤ/2;V)⟶H*(V).⟶superscriptÝ»ℤ2ݑ‰superscriptÝ»ݑ‰H^{*}(\mathbbZ/2;V)\longrightarrow H^{*}(V).italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) ⟶ italic_H start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_V ) .
Take cocycles u1,…,ur∈C*(ℤ/2;V)subscriptݑ¢1…subscriptݑ¢ݑŸsuperscriptݶℤ2ݑ‰u_1,\dots,u_r\in C^{*}(\mathbbZ/2;V)italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ∈ italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) whose images in Vݑ‰Vitalic_V yield cohomology classes which span the image of (2.6). Write them as uk=uk0+O(h)subscriptݑ¢ݑ˜superscriptsubscriptݑ¢ݑ˜0ݑ‚ℎu_k=u_k^0+O(h)italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_O ( italic_h ). Given any cocycle v∈C*(ℤ/2;V)ݑ£superscriptݶℤ2ݑ‰v\in C^{*}(\mathbbZ/2;V)italic_v ∈ italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ), write it as v=v0+O(h)ݑ£superscriptݑ£0ݑ‚ℎv=v^0+O(h)italic_v = italic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_O ( italic_h ) as well. By assumption, one can find γ10,…,γr0∈ݕ‚superscriptsubscriptݛ¾10…superscriptsubscriptݛ¾ݑŸ0ݕ‚\gamma_1^0,\dots,\gamma_r^0\in\mathbbKitalic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ∈ blackboard_K and a w0∈Vsuperscriptݑ¤0ݑ‰w^0\in Vitalic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ∈ italic_V such that v0=γ10u10+⋯+γr0ur0+dVw0superscriptݑ£0superscriptsubscriptݛ¾10superscriptsubscriptݑ¢10⋯superscriptsubscriptݛ¾ݑŸ0superscriptsubscriptݑ¢ݑŸ0subscriptݑ‘ݑ‰superscriptݑ¤0v^0=\gamma_1^0u_1^0+\cdots+\gamma_r^0u_r^0+d_Vw^0italic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + ⋯ + italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT. One can therefore write
(2.7) v-γ10u1-⋯-γr0ur-dCw0=hv1+O(h2)ݑ£superscriptsubscriptݛ¾10subscriptݑ¢1⋯superscriptsubscriptݛ¾ݑŸ0subscriptݑ¢ݑŸsubscriptݑ‘ݶsuperscriptݑ¤0â„Žsuperscriptݑ£1ݑ‚superscriptâ„Ž2v-\gamma_1^0u_1-\cdots-\gamma_r^0u_r-d_Cw^0=hv^1+O(h^2)italic_v - italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - ⋯ - italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_h italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT + italic_O ( italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )
for some v1∈Vsuperscriptݑ£1ݑ‰v^1\in Vitalic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ italic_V. The expression on either side of (2.7) is hâ„Žhitalic_h times some cocycle in C*(ℤ/2;V)superscriptݶℤ2ݑ‰C^{*}(\mathbbZ/2;V)italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ). We can apply the same argument to that cocycle, and then proceed iteratively, which constructs γ1,…,γr∈ݕ‚[[h]]subscriptݛ¾1…subscriptݛ¾ݑŸݕ‚delimited-[]delimited-[]â„Ž\gamma_1,\dots,\gamma_r\in\mathbbK[[h]]italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ∈ blackboard_K [ [ italic_h ] ] and a w∈C*(ℤ/2;V)ݑ¤superscriptݶℤ2ݑ‰w\in C^{*}(\mathbbZ/2;V)italic_w ∈ italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V ) such that v=γ1u1+⋯+γrur+dCwݑ£subscriptݛ¾1subscriptݑ¢1⋯subscriptݛ¾ݑŸsubscriptݑ¢ݑŸsubscriptݑ‘ݶݑ¤v=\gamma_1u_1+\cdots+\gamma_ru_r+d_Cwitalic_v = italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT + italic_d start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_w.
(iii) can be proved by a similar order-by-order argument, whose details we omit.
(iv) is obvious, since the complexes C*(ℤ/2;Vk)superscriptݶℤ2subscriptݑ‰ݑ˜C^{*}(\mathbbZ/2;V_k)italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( blackboard_Z / 2 ; italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) themselves form a short exact sequence (inspection of the standard argument shows that the boundary operator is a ݕ‚[[h]]ݕ‚delimited-[]delimited-[]â„Ž\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-linear map). ∎
The acyclicity result (i) is an instance of a much more general principle. page_seo_titlely, take any (ℤℤ\mathbbZblackboard_Z-graded or ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded) chain complex of vector spaces (V,dV)ݑ‰subscriptݑ‘ݑ‰(V,d_V)( italic_V , italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ). Suppose that on V[[h]]ݑ‰delimited-[]delimited-[]â„ŽV[[h]]italic_V [ [ italic_h ] ], we have a ݕ‚[[h]]ݕ‚delimited-[]delimited-[]â„Ž\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-linear differential of the form dv=dVv+O(h)ݑ‘ݑ£subscriptݑ‘ݑ‰ݑ£ݑ‚ℎdv=d_Vv+O(h)italic_d italic_v = italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT italic_v + italic_O ( italic_h ). Then, if (V,dV)ݑ‰subscriptݑ‘ݑ‰(V,d_V)( italic_V , italic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ) is acyclic, the same holds for (V[[h]],d)ݑ‰delimited-[]delimited-[]ℎݑ‘(V[[h]],d)( italic_V [ [ italic_h ] ] , italic_d ). The proof is the same as in the previously considered special case. Alternatively, one can think in terms of spectral sequences: (V[[h]],d)ݑ‰delimited-[]delimited-[]ℎݑ‘(V[[h]],d)( italic_V [ [ italic_h ] ] , italic_d ) carries a complete decreasing filtration (by powers of hâ„Žhitalic_h), and the differential on the associated graded space is given by dVsubscriptݑ‘ݑ‰d_Vitalic_d start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT (at each level of the filtration). Under our assumption, the E1subscriptݸ1E_1italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT page of the spectral sequence is zero, which implies the acyclicity of (V[[h]],d)ݑ‰delimited-[]delimited-[]ℎݑ‘(V[[h]],d)( italic_V [ [ italic_h ] ] , italic_d ).
There is a similar generalization of (ii). Abstractly, one should be able think of it as a vanishing result parallel to (i), by working modulo the Serre subcategory of finitely generated ݕ‚[[h]]ݕ‚delimited-[]delimited-[]ℎ\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-modules [60] (but we have not checked the details of this approach; in any case, the proof we have given also works in this more general context).
A similar observation applies to part (iii). Take chain complexes Vksubscriptݑ‰ݑ˜V_kitalic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT (k=1,2ݑ˜12k=1,2italic_k = 1 , 2; with no group actions). Suppose that we have differentials dk=dVk+O(h)subscriptݑ‘ݑ˜subscriptݑ‘subscriptݑ‰ݑ˜ݑ‚ℎd_k=d_V_k+O(h)italic_d start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_O ( italic_h ) on Vk[[h]]subscriptݑ‰ݑ˜delimited-[]delimited-[]ℎV_k[[h]]italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ [ italic_h ] ]. Consider a ݕ‚[[h]]ݕ‚delimited-[]delimited-[]ℎ\mathbbK[[h]]blackboard_K [ [ italic_h ] ]-linear chain map V1[[h]]→V2[[h]]normal-→subscriptݑ‰1delimited-[]delimited-[]ℎsubscriptݑ‰2delimited-[]delimited-[]ℎV_1[[h]]\rightarrow V_2[[h]]italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT [ [ italic_h
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