Persistence modules with operators
in Morse and Floer theory

As explained in Arnol’d’s [4], an important invariant formulation of the equations of motion of classical mechanics involves a manifold $M,$ the phase space, a closed non-degenerate two-form $\omega$ on $M,$ the symplectic form, and a smooth, possibly time-dependent, Hamiltonian function $H:[0,1]\times M\rightarrow\mathbb{R}$ on $M,$ the total energy function of the system. The dynamics on the symplectic manifold $(M,\omega),$ which we assume to be closed throughout this paper, is then described by the Hamiltonian flow of $H$,

$$\{\phi^{t}_{H}:M\to M\}_{t\in[0,1]},$$

obtained by integrating the time-dependent vector field $X^{t}_{H}$ on $M$, given by setting $H_{t}(-)=H(t,-),$ and

$$\omega(X^{t}_{H},\cdot)=-dH_{t}(\cdot).$$

The time-one map $\phi=\phi_{H}^{1}$ of this flow is called a Hamiltonian diffeomorphism and it is clear that 1-periodic orbits of the flow $\{\phi^{t}_{H}\}_{t\in[0,1]}$ correspond to the fixed points of $\phi$. Hamiltonian diffeomorphisms form a group, that we denote $\operatorname{\mathrm{Ham}}(M,\omega).$

In the 1960’s Arnol’d has proposed a famous conjecture [2, 3] that, essentially, the number of fixed points of $\phi\in\operatorname{\mathrm{Ham}}(M,\omega)$ should satisfy the same lower bounds as the number of critical points of a smooth function $f$ on $M.$ The most common interpretation of this conjecture states that under the nondegeneracy^aaaMeaning that $\det({\bf{1}}-D\phi_{1}^{H}(x))\neq 0$ for every fixed point $x$ of $\phi^{1}_{H}$, i.e. the graph of $\phi^{1}_{H}$ intersects the diagonal in $M\times M$ transversely. assumption on $H$, the number of fixed points of $\phi^{1}_{H}$ is bounded from below by the sum of the rational Betti numbers of $M$. This conjecture has been a major driving force for the development of the field of symplectic topology. It was first proven in dimension $2$ by Eliashberg [21], for tori of arbitrary dimension by Conley and Zehnder [16], and on complex projective spaces by Fortune and Weinstein [27, 28]. The decisive breakthrough on this question was achieved by Floer [23, 24, 26], who combined the variational methods of Conley-Zehnder and Gromov’s then-recent discovery of the theory of pseudoholomorphic curves in symplectic manifolds [35], to construct a homology theory on the loop space of $M$ which parallels the more classical Morse homology (in turn originating in Witten’s interpretation of Morse theory [60]; cf. [50]).

Let us briefly describe Floer’s work in the simplest setting (cf. [5]). Assuming that $M$ is symplectically aspherical, that is $\omega|_{\pi_{2}(M)}=0,\;c_{1}(M,\omega)|_{\pi_{2}(M)}=0,$ one can define the action functional $\mathcal{A}_{H}$ on the space $\mathcal{L}_{c}M$ of contractible loops on $M$ by setting

$$\mathcal{A}_{H}(z)=\int_{0}^{1}H_{t}(z(t))~dt-\int_{D^{2}}{\overline{z}}^{*}\omega,$$

where $z:[0,1]\rightarrow M$, $z(0)=z(1)$ and $\overline{z}:D^{2}\rightarrow M$, $\overline{z}(e^{2\pi it})=z(t)$. Indeed by the asphericity assumption this value depends only on $z,$ and not on $\overline{z}.$

The periodic orbits of the Hamiltonian flow coincide with the critical points of the action functional $\mathcal{A}_{H}$, which serves as a Morse function in the construction of Floer homology. Since critical points give generators of Morse chain complexes, if $H$ is non-degenerate, the Floer chain complex $CF_{*}(H)$ will be generated by contractible periodic orbits of $H$. Floer homology will be isomorphic to singular homology of $M$ with rational coefficients, and hence there must be at least

$$\dim CF_{*}(H)\geq\dim HF_{*}(H)=\sum_{k}\dim H_{k}(M,\mathbb{Q}),$$

such periodic orbits in $M.$ This solves the rational homological version of the Arnol’d conjecture. The detailed construction of Floer homology is rather involved and it has been developed in increasing generality over the years by the combined work of many people (cf. [32, 33, 45, 41, 48, 44]), in particular proving the above statement for general $M$ (without assumptions on $\pi_{2}(M)$). However, various other interpretations of the conjecture are still open in general, with only partial results currently achieved (see e.g. [26, 49, 25, 36, 19, 43, 7]).

In both Morse and Floer theory, the differential is given by counting certain trajectories of the negative gradient vector field connecting pairs of critical points. Since the Morse function, or the action functional in the Floer case, decreases along these trajectories, for each $s\in{\mathbb{R}}$ the generators whose critical values are $<s,$ form a subcomplex. In the case of a Morse function $f$ on a closed manifold $X$, the homology $V^{s}_{*}(f)$ of this subcomplex with coefficients in a field ${\mathbb{K}}$ is isomorphic to the homology $H_{*}(\{f<s\},{\mathbb{K}})$ of the sublevel set $\{f<s\}$ with coefficients in ${\mathbb{K}}$ (which is a vector space of finite dimension!). Inclusions $\{f<s\}\subset\{f<t\}$ for $s\leq t$ yield maps $\pi_{s,t}:V^{s}_{*}(f)\to V^{t}_{*}(f)$ such that $\pi_{s,t}\circ\pi_{r,s}=\pi_{r,t}$ for $r\leq s\leq t.$ This family of vector spaces parametrized by a real parameter forms an algebraic structure called a persistence module which was introduced and studied extensively since the early 2000s in the data analysis community (see e.g. [15, 14, 20, 8, 9, 12, 17, 34, 62]). Quite recently, persistence modules found applications in symplectic topology, see [1, 47, 56, 61, 29, 52], with preludes in [54, 55, 31, 6].

In the case of Floer theory on a symplectically aspherical manifold, the exact same procedure applies. In the so-called monotone case, when $\omega|_{\pi_{2}(M)}=\kappa\cdot c_{1}(TM)|_{\pi_{2}(M)}$ for a constant $\kappa>0,$ one can define a persistence module by looking at $V^{s}_{m}(H)$ for a fixed degree $m,$ which is the approach we choose later in the paper. The structure theorem of persistence modules allows us to associate to this situation a barcode: a multi-set $\mathcal{I}$ of intervals of the form $(a,b]$ or $(a,\infty),$ such that

$$V^{t}_{m}(H)\cong\bigoplus_{I\in\mathcal{I}}Q^{t}(I),$$

where $Q^{t}(I)$ is a persistence modules equal to $\mathbb{K}$ for $t\in I$ and 0 otherwise. The number of the infinite intervals in the barcode is equal to the dimension of the ambient (quantum) homology in a given degree, while the left ends of these intervals correspond to the well know spectral invariants. Spectral invariant can be defined in complete generality (without assumptions on $\pi_{2}(M)$), and have been used extensively in symplectic topology in the last few decades starting with the foundational papers [42, 51, 57] (see [40, 53] for recent developments and numerous further references), before persistence modules entered the field. In terms of barcodes, in complete generality one can only expect bars with endpoints in ${\mathbb{R}}/\mathcal{P}(\omega),$ where $\mathcal{P}(\omega)=\operatorname{im}\,(\int\omega:\pi_{2}(M)\to{\mathbb{R}})$ is the period group of $\omega.$ In [56] such a persistence module is constructed, with, remarkably, the lengths of intervals being well-defined.

Denote the $C^{0}$-distance between two smooth functions $f$ and $g$ on a compact manifold $X$ by

$$|f-g|_{C^{0}}=\max_{x\in X}|f(x)-g(x)|,$$

and Hofer’s $L^{1,\infty}$-distance between Hamiltonians $F_{t}$ and $G_{t}$ on $M$, $t\in[0,1]$ by

$$\mathcal{E}(F-G)=\int_{0}^{1}\bigg{(}\max_{x\in M}(F_{t}(x)-G_{t}(x))-\min_{x\in M}(F_{t}(x)-G_{t}(x))\bigg{)}~dt.$$

The Hofer’s pseudo-metric on the universal cover $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega)$ of $\operatorname{\mathrm{Ham}}(M,\omega)$ is defined as

$$\widetilde{d}(\widetilde{f},\widetilde{g})=\inf\mathcal{E}(F-G),$$

where the infimum runs over all the $F,G$ such that $[\{\phi^{t}_{F}\}]=\widetilde{f},$ $[\{\phi^{t}_{G}\}]=\widetilde{g}$ in $\widetilde{\operatorname{\mathrm{Ham}}}(M,\omega).$ Similarly, Hofer’s metric [37, 39] (cf. [46, 38]) on $\operatorname{\mathrm{Ham}}(M,\omega)$ is

$$d(f,g)=\inf\mathcal{E}(F-G),$$

where the infimum runs over all the $F,G$ such that $\phi^{1}_{F}=f,$ $\phi^{1}_{G}=g$.

One crucial feature of the barcodes is that if the $C^{0}$-distance between Morse functions, or the Hofer’s $L^{1,\infty}$-distance between two Hamiltonians is at most $c,$ then one barcode can be obtained from the other, by a procedure that allows to move each endpoint of a bar by distance at most $c$ (this allows erasing intervals of length $\leq 2c,$ as well as creating such intervals). This follows from the celebrated isometry theorem of persistence modules. Such a procedure is encoded by the formal notion of a $c$-matching of barcodes, whose algebraic counterpart is the notion of a $c$-interleaving. More precisely, we say that two persistence module morphisms $f_{t}:V^{t}\rightarrow W^{t+c}$ and $g_{t}:W^{t}\rightarrow V^{t+c}$ induce a $c$-interleaving between persistence modules $V$ and $W$ if $g_{t+c}\circ f_{t}=\pi^{V}_{t,t+2c}$ and $f_{t+c}\circ g_{t}=\pi^{W}_{t,t+2c}$ for every $t\in\mathbb{R}$. The main point is that Morse and Floer continuation maps with respect to linear interpolations of functions $f$ and $g$ or Hamiltonians $F$ and $G$ (or small perturbations thereof) yield the required $c$-interleavings, where $c=|f-g|_{C^{0}}$ or $c=\mathcal{E}(F-G)$. This allows us to bound the $C^{0}$-distance between Morse functions as well as Hofer’s distance between Hamiltonians (and consequently Hamiltonian diffeomorphisms) from below by the minimal $c$ needed to match the corresponding barcodes.

In this paper we primarily investigate an additional structure on Morse and Floer persistence modules coming from the ambient homology. Our main observation is that the ambient homology acts on the persistence module by intersecting cycles in the sublevel sets of functions (and a similar picture holds in the Floer case). We consider this action as a particular case of the notion of a persistence module with an operator. Namely, we consider pairs $(V,A)$ where $A:V^{t}\rightarrow V^{t+c_{A}}$ is a persistence module morphism as main objects of interest and define morphisms between these objects to be usual persistence module morphisms which commute with the corresponding operators. We may now define operator interleaving as an interleaving in this new category, i.e. an interleaving which commutes with the operators. The fact that $(V,A)$ and $(W,B)$ are $c$-operator interleaved will immediately imply that $\operatorname{im}A$ and $\operatorname{im}B$ (as well as $\ker A$ and $\ker B$) are $c$-interleaved (see Section 2.3 for a discussion of persitence modules with operators).

In the Morse and the Floer case, fixing a (quantum) homology class $a$, we obtain an operator $a*$ induced by intersection (or quantum) product. Continuation maps commute with this operator, hence constitute morphisms of persistence modules with operators and induce operator interleavings. Finally, they provide both $\operatorname{im}(a*)$ and $\ker(a*)$ for two functions $f$ and $g$ or two Hamiltonians $F$ and $G$, with $c$-interleavings, for $c=|f-g|_{C^{0}}$ or $c=\mathcal{E}(F-G)$ respectively. This means that we may bound these values from below by using barcodes associated to $\operatorname{im}(a*)$ or $\ker(a*)$. Following this line of reasoning, we show that there exists a pair $f,g$ of Morse function on a manifold (even of dimension $2$) such that all their spectral invariants, as well as their barcodes coincide, and yet the corresponding $\operatorname{im}(a*)$ modules are at a positive (computable) interleaving distance $c$. We conclude that the two functions must be at $C^{0}$-distance at least $c$ (see Section 2.4 for an example).

Finally, we present an application to Hofer’s geometry, by proving new cases of the conjecture that on any closed symplectic manifold and for any integer $k\geq 2,$ there exist Hamiltonian diffeomorphisms which are arbitrarily far away, in Hofer’s metric, from having a root of order $k.$ First results of this kind were obtained in [47], and were then extended to certain other cases in [61] (for $k$ sufficiently large). In our situation, the multiplication with classes in ambient homology allows to adjust multiplicities of certain long bars, the number theoretic properties of which are crucial to the argument, and allow to reduce the $k$ for which the result holds, yielding Theorem 1.2 (see Section 1.3).

In this section we shall assume that the ground field ${\mathbb{K}}$ has characteristic $\mathrm{char}({\mathbb{K}})\neq p,$ contains all $p$-th roots of unity^bbbThat is the polynomial $x^{p}-1\in{\mathbb{K}}[x],$ which is separable by the assumption $\mathrm{char}({\mathbb{K}})\neq p,$ splits over ${\mathbb{K}}.$, and fixing a primitive $p$-th root of unity $\zeta_{p},$ the equation $x^{p}-(\zeta_{p})^{q}=0,$ for each integer $q$ coprime to $p,$ has no solutions in ${\mathbb{K}}.$ An example of such a field is the splitting field ${\mathbb{Q}}_{p}$ over ${\mathbb{Q}}$ of $x^{p}-1\in{\mathbb{Q}}[x].$

Let $(M,\omega)$ be a closed symplectic manifold, and put $\operatorname{\mathrm{Powers}}_{k}(M,\omega)\subset\operatorname{\mathrm{Ham}}(M,\omega),$ where $k$ is an integer, for the set of all diffeomorphisms in $\operatorname{\mathrm{Ham}}(M,\omega),$ admitting a root of order $k$ (in the same group). The following result was proven in [47].

Now, assume that $N$ is a monotone symplectic manifold, fix a prime number $p$ and denote by $\Lambda_{{\mathbb{K}}}$ the field of Laurent power series in variable $q^{-1}$ with coefficients in ${\mathbb{K}}$,

$$\Lambda_{{\mathbb{K}}}=\bigg{\{}\sum\limits_{n\in\mathbb{Z}}a_{n}q^{n}~\bigg{|}~a_{n}\in{\mathbb{K}},~(\exists n_{0}\in\mathbb{N})~a_{n}=0~\text{for}~n\geq n_{0}\bigg{\}}.$$

The quantum homology of $N$ with ${\mathbb{K}}$ coefficients is the vector space $H_{*}(N,{\mathbb{K}})\otimes_{{\mathbb{K}}}\Lambda_{{\mathbb{K}}}$ over $\Lambda_{{\mathbb{K}}}$ which we denote by $QH(N)$. Assuming that $\deg q=2c_{N}$ where $c_{N}$ is the minimal Chern number of $N$, $QH(N)$ has a natural $\mathbb{Z}$-grading, that is we can define $QH_{r}(N)$ for $r\in\mathbb{Z}$, which will be vector spaces over the base field ${\mathbb{K}}$. We also have that $QH_{r+2c_{N}}(N)\cong QH_{r}(N)$ for every $r\in\mathbb{Z}$, where the isomorphism is given by multiplication by $q$. Let $e\in QH(N)$ be a homogeneous element and define a map

$$e*:QH(N)\rightarrow QH(N),~(e*)a=e*a,$$

where $*$ denotes quantum product. This map is a linear morphism between vector spaces over $\Lambda_{{\mathbb{K}}}$ which restricts to a linear morphism between vector spaces over ${\mathbb{K}}$ after fixing the grading:

$$e*:QH_{r}(N)\rightarrow QH_{r-2n+\deg e}(N),~\text{for}~r\in\mathbb{Z}.$$

Now $E:=e*(QH(N))\subset QH(N)$ is a vector space over $\Lambda_{{\mathbb{K}}}$ and

$$E_{r}:=e*(QH_{r}(N))\subset QH_{r-2n+\deg e}(N),$$

are vector spaces over ${\mathbb{K}}$ which satisfy $E_{r}\cong E_{r+2c_{N}}$, the isomorphism being induced by multiplication by $q$. These spaces give us $2c_{N}$ Betti numbers associated to a homogeneous element $e\in QH(N)$, which we define as:

$$b_{r}(e)=\dim_{{\mathbb{K}}}E_{r},~r=0,\ldots,2c_{N}-1.$$

Now, we can state the result regarding Hofer’s geometry. Denote by $\operatorname{\mathrm{powers}}_{p}(M)$ the supremum of the Hofer distance to $p$-th powers in $\operatorname{\mathrm{Ham}}(M)$. That is for each $\phi\in\operatorname{\mathrm{Ham}}(M)$ define $d(\phi,\operatorname{\mathrm{Powers}}_{p}(M))=\displaystyle\inf_{\theta\in\operatorname{\mathrm{Powers}}_{p}(M)}d(\phi,\theta),$ and define $\operatorname{\mathrm{powers}}_{p}(M):=\sup_{\phi\in\operatorname{\mathrm{Ham}}(M)}d(\phi,\operatorname{\mathrm{Powers}}_{p}(M)).$

To prove this result we describe the Floer theoretical setup that fits into our algebraic framework of equivariant persistence modules with operators, and then make a concrete computation in the case of the egg-beater flow which yields the result.

We recall briefly the category $\operatorname{{\mathbf{pmod}}}$ of persistence modules that we work with, together with their relevant properties. For detailed treatment of these topics see [8, 9, 12, 17, 34, 62].

Let $\mathbb{K}$ be a field. A persistence module over $\mathbb{K}$ is a pair $(V,\pi)$ where, $\{V^{t}\}_{t\in\mathbb{R}}$ is a family of finite dimensional vector spaces over $\mathbb{K}$ and $\pi_{s,t}:V^{s}\rightarrow V^{t}$ for $s<t,~s,t\in\mathbb{R}$ is a family of linear maps, called structure maps, which satisfy:

1. 1)

   $V^{t}=0$ for $t\ll 0$ and $\pi_{s,t}$ are isomorphisms for all $s,t$ sufficiently large;

2. 2)

   $\pi_{t,r}\circ\pi_{s,t}=\pi_{s,r}$ for all $s<t<r$;

3. 3)

   For every $r\in\mathbb{R}$ there exists $\varepsilon>0$ such that $\pi_{s,t}$ are isomorphisms for all $r-\varepsilon<s<t\leq r$;

4. 4)

   For all but a finite number of points $r\in\mathbb{R}$, there is a neighbourhood $U\ni r$ such that $\pi_{s,t}$ are isomorphisms for all $s<t$ with $s,t\in U$.

The set of the exceptional points in 4), i.e. the set of all points $r\in\mathbb{R}$ for which there does not exist a neighbourhood $U\ni r$ such that $\pi_{s,t}$ are isomorphisms for all $s,t\in U$, is called the spectrum of the persistence module $(V,\pi)$ and is denoted by $\mathcal{S}(V)$. One easily checks that for two consecutive points $a<b$ of the spectrum and $a<s<t\leq b$, $\pi_{s,t}$ is an isomorphism. This means that $V^{t}$ only changes when $t$ ”passes through points in the spectrum”.

We define a morphism between two persistence modules $A:(V,\pi)\rightarrow(V^{\prime},\pi^{\prime})$ as a family of linear maps $A_{t}:V^{t}\rightarrow(V^{\prime})^{t}$ for every $t\in\mathbb{R}$ which satisfies

$$A_{t}\pi_{s,t}=\pi_{s,t}^{\prime}A_{s}~~\text{for}~s<t.$$

Note that the kernel $\ker A$ and an image $\operatorname{im}A$ are naturally persistence modules whose families of vector spaces are $\{\ker A_{t}\subset V^{t}\}_{t\in{\mathbb{R}}},$ $\{\operatorname{im}A_{t}\subset(V^{\prime})^{t}\}_{t\in{\mathbb{R}}},$ since the structure maps $\pi_{s,t}$ restrict to these systems of subspaces. In fact, it is not difficult to prove that $\operatorname{{\mathbf{pmod}}}$ forms an abelian category, with the direct sum of two persistence modules $(V,\pi)$ and $(V^{\prime},\pi^{\prime})$ given by

$$(V,\pi)\oplus(V^{\prime},\pi^{\prime})=(V\oplus V^{\prime},\pi\oplus\pi^{\prime}).$$

An important object in our story is the barcode associated to a persistence module. It arises from the structure theorem for persistence modules, which we now recall. Let $I$ be an interval of the form $(a,b]$ or $(a,+\infty)$, $a,b\in\mathbb{R}$ and denote by $Q(I)=(Q(I),\pi)$ the persistence module which satisfies $Q^{t}(I)=\mathbb{K}$ for $t\in I$ and $Q^{t}(I)=0$ otherwise and $\pi_{s,t}=id$ for $s,t\in I$ and $\pi_{s,t}=0$ otherwise.

The multi-set which contains $m_{i}$ copies of each $I_{i}$ appearing in the structure theorem is called the barcode associated to $V$ and is denoted by $\mathcal{B}(V)$. Intervals $I_{i}$ are called bars.

For an interval $I=(a,b]$ or $I=(a,+\infty)$, let $I^{-c}=(a-c,b+c]$ or $I^{-c}=(a-c,+\infty)$, and similarly $I^{c}=(a+c,b-c]$ or $I^{c}=(a+c,+\infty),$ when $b-a>2c$. We say that barcodes $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ admit a $\delta$-matching if it is possible to delete some of the bars of length $\leq 2\delta$ from $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ (and thus obtain $\bar{\mathcal{B}}_{1}$ and $\bar{\mathcal{B}}_{2}$) such that there exists a bijection $\mu:\bar{\mathcal{B}}_{1}\rightarrow\bar{\mathcal{B}}_{2}$ which satisfies

$$\mu(I)=J\Rightarrow I\subset J^{-\delta},~J\subset I^{-\delta}.$$

We define the bottleneck distance $d_{bottle}(\mathcal{B}_{1},\mathcal{B}_{2})$ between barcodes $\mathcal{B}_{1},\mathcal{B}_{2}$ as infimum over $\delta>0$ such that there exists a $\delta$-matching between them. One readily checks that the following lemma holds.

For a persistence module $V=(V,\pi)$ denote by $V[\delta]=(V[\delta],\pi[\delta])$ a shifted persistence module given by $V[\delta]^{t}=V^{t+\delta},\pi_{s,t}=\pi_{s+\delta,t+\delta}$ and by $sh(\delta)_{V}:V\rightarrow V[\delta]$ a canonical shift morphism given by $(sh(\delta)_{V})_{t}=\pi_{t,t+\delta}:V^{t}\rightarrow V^{t+\delta}$. Note also that a morphism $f:V\rightarrow W$ induces a morphism of $f[\delta]:V[\delta]\rightarrow W[\delta]$. We say that a pair of morphisms $f:V\rightarrow W[\delta]$ and $g:W\rightarrow V[\delta]$ is a $\delta$-interleaving between $V$ and $W$ if

$$g[\delta]\circ f=sh(2\delta)_{V}\text{ and }f[\delta]\circ g=sh(2\delta)_{W}.$$

Now we can define the interleaving distance $d_{inter}(V,W)$ between $V$ and $W$ as infimum over all $\delta>0$ such that $V$ and $W$ admit a $\delta$-interleaving. The isometry theorem for persistence modules states that $d_{inter}(V,W)=d_{bottle}(\mathcal{B}(V),\mathcal{B}(W))$ (see [8]).

As we mentioned before, $\operatorname{{\mathbf{pmod}}}$ is an abelian category, and we wish to define a monoidal structure $\otimes$ and its derived functors in this category in a similar fashion to the situation which we have for $\mathbb{Z}$ modules (similar constructions, yet with different aims and applications, appeared in [10, 11, 18, 58, 59]). Let $(V^{s},\pi^{V})$ and $(W^{t},\pi^{W})$ be two persistence modules and define vector spaces

$$X^{r}=\bigoplus_{t+s=r}V^{s}\otimes W^{t},~\text{and}~Y^{r}\subset X^{r}~\text{for every}~r\in\mathbb{R},$$

given by

$$Y^{r}=\bigg{\langle}\bigg{\{}(\pi^{V}_{\alpha,s_{1}}v_{\alpha})\otimes(\pi^{W}_{\beta,t_{1}}w_{\beta})-(\pi^{V}_{\alpha,s_{2}}v_{\alpha})\otimes(\pi^{W}_{\beta,t_{2}}w_{\beta})\bigg{\}}\bigg{\rangle},$$

where $\langle S\rangle$ stands for vector space over $\mathbb{K}$ generated by the set $S$ and indices $s_{1},s_{2},t_{1},t_{2},\alpha$ and $\beta$ satisfy $s_{1}+t_{1}=s_{2}+t_{2}=r,\alpha\leq\min\{s_{1},s_{2}\},~\beta\leq\min\{t_{1},t_{2}\}$. We may now define $(V\otimes W)^{r}=X^{r}/Y^{r}$ and maps $\pi^{V}\otimes\pi^{W}$ on $X^{r}$ induce maps $\pi^{V\otimes W}$ on $V\otimes W$ which give this space the structure of persistence module. We call this module the tensor product of persistence modules $(V,\pi^{V})$ and $(W,\pi^{W})$. Another way to think of $V\otimes W$ is that $(V\otimes W)^{r}$ is the colimit in the category of (finite-dimensional, as is easy to see) vector spaces over our ground field of the diagram with objects $\{V^{s}\otimes W^{t}\}_{s+t\leq r}$ and maps $\pi_{s_{1},s_{2}}\otimes\pi_{t_{1},t_{2}}:V^{s_{1}}\otimes W^{t_{1}}\to V^{s_{2}}\otimes W^{t_{2}}$ for $s_{1}\leq s_{2}$ and $t_{1}\leq t_{2}$ (we use the convention that $\pi_{t,t}={\bf{1}}_{V^{t}}$).

It is easy to see that we can also define the tensor product $f\otimes g:V\otimes W\rightarrow V^{\prime}\otimes W^{\prime}$ of persistence morphisms $f:V\rightarrow V^{\prime}$ and $g:W\rightarrow W^{\prime}$ by setting $f\otimes g([v_{\alpha}\otimes w_{\beta}])=[f(v_{\alpha})\otimes g(w_{\beta})]$.

Fixing a persistence module $W$ we get a functor $\otimes W:\operatorname{{\mathbf{pmod}}}\rightarrow\operatorname{{\mathbf{pmod}}}$ which acts on objects and morphisms by

$$\otimes W(V)=V\otimes W,~\otimes W(f)=f\otimes{\bf{1}}_{W}.$$

One can check that $\otimes W$ is a right exact functor and in order to define its derived functors we need to construct a projective resolution of every persistence module $V$. In the simplest case when $V$ is an interval module $Q((a,b])$ we have the following projective resolution of $V$ of length two:

$$0\rightarrow Q((b,+\infty))\rightarrow Q((a,+\infty))\rightarrow Q((a,b])\rightarrow 0,$$

where arrows denote obvious maps. Note that we used the fact that $Q((a,+\infty))$ is projective object for every $a\in\mathbb{R}$. One may also check that in fact a persistence module $V$ is projective if and only if its barcode contains no finite bars. Using this fact together with Theorem 2.2 we may construct a projective resolution of length two of every persistence module $V$ in the same manner as we did for the interval module. Recall that (classical) derived functors of $\otimes W$ applied to $V$ are computed as homologies of the sequence

$$\ldots\rightarrow P_{2}\otimes W\xrightarrow{f_{2}\otimes{\bf{1}}}P_{1}\otimes W\xrightarrow{f_{1}\otimes{\bf{1}}}P_{0}\otimes W\rightarrow 0,$$

where $\ldots\rightarrow P_{2}\xrightarrow{f_{2}}P_{1}\xrightarrow{f_{1}}P_{0}\xrightarrow{f_{0}}V\rightarrow 0$ is a projective resolution of $V$. Since every persistence module has a projective resolution of length two, there is only one non-trivial derived functor of $\otimes W$ which we denote by $Tor(\cdot,W)$. Both $\otimes$ and $Tor$ are symmetric in the sense that $V\otimes W\cong W\otimes V$ and $Tor(V,W)\cong Tor(W,V)$ and it immediately follows that if either $V$ or $W$ is projective $Tor(V,W)=0$.

Our goal is to establish a Künneth type formula for filtered homology groups using $\otimes$ and $Tor$. Let us first recall the following definition.