Filtered tower splitting for the Upsilon minimax

Put $\nu ={\nu }_t(C)$ and choose a Maslov-zero representing cycle $\alpha$ with $F_t(\alpha )=\nu$, using [F1]. Let $C_0,C_1$ be the finite-dimensional chain groups in Maslov degrees zero and one. Set $B_0=\mathrm{\partial }{C}_1$ and $L=\{z\in C_0:F_t(z)<\nu \}$, including zero. The maximum-filtration rule makes $L$ a subspace. One has $\alpha \mathrm{\notin }{B}_0+L$: otherwise $\alpha =b+\mathrm{\ell }$ would make $\mathrm{\ell }$ a cycle representing the same nonzero class at a smaller filtration. Finite-dimensional linear algebra therefore gives a functional $\lambda :C_0\to {\mathbb{F}}_2$ with $\lambda (\alpha )=1$ and $\lambda (B_0+L)=0$. Translate each homogeneous Laurent basis vector by the unique power of $U$ putting its Maslov degree in $\{0,1\}$. Define a $\mathrm{\Lambda }$-linear map $P:C\to \mathrm{\Lambda }\alpha$ by $P(x)=\lambda (x)\alpha$ on these degree-zero basis vectors and $P(y)=0$ on the degree-one ones. Boundaries from degree one are annihilated by $\lambda$, and the differential of a degree-zero vector lies in $U{C}_1$, on which $P$ is zero. Thus $P$ is a chain map and $P(\alpha )=\alpha$. If $P(x)\mathrm{\ne }0$, then $F_t(x)\ge \nu$ because $\lambda (L)=0$; the $U$-translation rule gives the same inequality for every translate. Hence $P$ preserves filtration.

Write $\alpha$ in the degree-zero basis and choose a term of largest filtration, which has filtration $\nu$. Replacing that basis vector by $\alpha$ is invertible and preserves filtration in both directions, since all other terms in $\alpha$ have filtration at most $\nu$. For every remaining degree-zero basis vector $x$, replace it by $x+\lambda (x)\alpha$. When the extra term is present, $F_t(x)\ge \nu$, so this and its inverse preserve filtration. The degree-one basis is unchanged. The resulting basis consists of $\alpha$ and a basis of $\ker P$. Since $P$ is a chain map, this is a filtered graded direct-sum decomposition $C=T\oplus A$, with $T=\mathrm{\Lambda }\alpha$ and $A=\ker P$. Its total homology on $T$ already maps isomorphically to $H(C)=\mathrm{\Lambda }$, so $H(A)=0$. This proves the first assertion by an explicit filtered projection, without assuming that an arbitrary unfiltered acyclic complex is filtered contractible.

Return to the original basis $x_r$ at algebraic level zero, and put $m_r=M(x_r)$ and $a_r=A(x_r)$. Its differential has coefficients $c_{rs}{U}^{(m_s-m_r+1)\mathrm{/}2}$ with $c_{rs}\in {\mathbb{F}}_2$. Extend scalars to $\mathcal{K}$ by $U=v^2$, and write $e_r=v^{m_r}{x}_r$. The differential is then $\mathrm{\partial }{e}_r=v{\sum }_s{c}_{rs}{e}_s$. Let $W$ be the finite ${\mathbb{F}}_2$-space on these formal $e_r$, with differential $d_0{e}_r={\sum }_s{c}_{rs}{e}_s$. It squares to zero, since the two-step coefficient in the original differential has the same $U$ exponent for every intermediate index. Separate $W$ by the parity of $m_r$. The even space identifies with $C_0$ by $e_r\mapsto U^{m_r\mathrm{/}2}{x}_r$, and the odd space with $C_1$ by $e_r\mapsto U^{(m_r-1)\mathrm{/}2}{x}_r$. The two matrices of $d_0$ are respectively those of $\mathrm{\partial }:C_1\to C_0$ and $U^{-1}\mathrm{\partial }:C_0\to C_1$. Consequently $H(W{)}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}={\mathbb{F}}_2$ and $H(W{)}_{\mathrm{o}\mathrm{d}\mathrm{d}}=0$, by the given total graded homology. Finite vector-space splittings extend to $\mathcal{K}$; since $v$ is invertible, $\mathrm{\partial }=v{d}_0$ has one-dimensional homology over $\mathcal{K}$, detected by the same even homology functional.

Let $C_t$ be the exact $t$-modified polynomial-basis complex in the definition, and put $g_r=m_r-t{a}_r$. The $\mathcal{K}$-linear map $\mathrm{\Phi }(x_r)=v^{t{a}_r}{x}_r$ from its scalar extension to the scalar extension of $C$ is a grading-preserving chain isomorphism. Indeed, a differential term on the source has coefficient $v^{m_s-m_r+1-t(a_s-a_r)}$, and multiplying its target by $v^{t{a}_s}$ gives exactly $v^{t{a}_r}{v}^{m_s-m_r+1}$, the differential of the image. The weighted filtration on the latter scalar extension is defined by $F_t(v^{\beta }{x}_r)=t{a}_r\mathrm{/}2-\beta \mathrm{/}2$, extended to a series by its least nonzero exponent and to a finite basis expansion by the maximum. Thus the image of $C_t$ is exactly the lattice $F_t\le 0$: its $r$th coefficient has least exponent at least $t{a}_r$. This is the explicit coordinate change, and it uses $U=v^2$.

A cycle in $C_t$ is non-torsion in homology exactly when its class remains nonzero over $\mathcal{K}$. One direction follows by inverting a nonzero annihilating coefficient. Conversely, if the cycle becomes a boundary over $\mathcal{K}$, the bounding chain has finitely many coefficients; a single monomial $v^c$ with $c\ge 0$ shifts all their least exponents into $[0,\mathrm{\infty })$ by [F2]. Multiplying the bounding equation by $v^c$ gives an annihilator of the original class over $\mathcal{R}$. Now a homogeneous cycle of degree $g$ in $C_t$ has the form $z={\sum }_r{ϵ}_r{v}^{g_r-g}{x}_r$, with ${ϵ}_r\in {\mathbb{F}}_2$, all occurring exponents nonnegative. Its image is $v^{-g}{\sum }_r{ϵ}_r{e}_r$, so the vector $\zeta ={\sum }_r{ϵ}_r{e}_r$ is a $d_0$-cycle. If $z$ is non-torsion, step 3.1 shows that the even part of $\zeta$ represents the nonzero even homology class. Hence $z_0={\sum }_{m_r\text{ even}}{ϵ}_r{U}^{m_r\mathrm{/}2}{x}_r$ is an original Maslov-zero cycle representing the normalized generator. For every term in $z_0$, the nonnegative-exponent condition gives $g\le m_r-t{a}_r$, and therefore $F_t(z_0)\le -g\mathrm{/}2$. It follows that every homogeneous non-torsion degree satisfies $g\le -2\nu$.

Conversely write a minimizing original Maslov-zero cycle as $\alpha ={\sum }_{m_r\text{ even}}{ϵ}_r{U}^{m_r\mathrm{/}2}{x}_r$ and put $g=-2\nu$. Each exponent $m_r-t{a}_r-g$ in its support is nonnegative by the definition of $F_t(\alpha )=\nu$. Thus $z_t={\sum }_r{ϵ}_r{v}^{m_r-t{a}_r-g}{x}_r$ is a homogeneous cycle in $C_t$, since its image under step 4.1 is $v^{-g}\alpha$ and $\mathrm{\Phi }$ is injective. Its scalar-extended homology class is nonzero by step 3.1, so it is non-torsion by step 5.1. Its degree is $g=-2\nu$, attaining the upper bound. Finally the filtered projection and direct sum in steps 1.1–2.1 extend over $\mathcal{K}$ and restrict to the lattice $F_t\le 0$. On $T$ this lattice is $\mathcal{R}{v}^{2\nu }\alpha$, with zero differential and generator degree $-2\nu$; the complementary lattice has only torsion homology, since its scalar extension is acyclic. By step 4.1 this is precisely the non-torsion tower in the $t$-modified complex. This proves the complete tower/minimax identification, including irrational $t$ and both endpoints.