An explicit peripheral basis for nonparabolic adjoint torus homology

The fixed line and a rational complementary plane. The direct $2\times 2$ determinant identity gives $M^2-tM+I=0$, and therefore $v^2=(\mathrm{\Delta }\mathrm{/}4)I$, $\mathrm{tr}v=0$, and $\mathrm{tr}(v^2)=\mathrm{\Delta }\mathrm{/}2\mathrm{\ne }0$. In particular $v\mathrm{\ne }0$. The map $\pi :V\to Fv$ defined by $\pi (X)=2\mathrm{tr}(vX)v\mathrm{/}\mathrm{\Delta }$ is a projection: $\pi (v)=v$, its image is $Fv$ and its kernel is $W$. Thus $V=Fv\oplus W$. Commutation of $L,M$ implies $P(v)=Q(v)=v$. For either $A=L$ or $A=M$, the direct identity $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ gives $\mathrm{tr}(vAX{A}^{-1})=\mathrm{tr}(A^{-1}vAX)=\mathrm{tr}(vX)$. Hence $P,Q$ preserve $W$ and commute with $\pi$.

An actual inverse on the complementary plane. On $W$ the inverse of $Q-I$ is the explicit operator $R=(Q+(3-t^2)I)\mathrm{/}\mathrm{\Delta }$. To check this without a splitting-field assumption, the polynomial $z^2-tz+1$ either has a root in $F$ or is irreducible, in which case its quotient gives a quadratic extension. In that field its two distinct roots are $r,r^{-1}$, and a change of basis diagonalizes $M$, since the two kernels of $M-rI$ and $M-r^{-1}I$ are distinct one-dimensional subspaces. In these coordinates $v=\mathrm{diag}((r-r^{-1})\mathrm{/}2,-(r-r^{-1})\mathrm{/}2)$, $W$ is exactly the off-diagonal plane, and $Q$ has eigenvalues $r^2,r^{-2}$ on it. Consequently $Q^2-(t^2-2)Q+I=0$ on $W$, and multiplication gives $(Q-I)(Q+(3-t^2)I)=\mathrm{\Delta }I$ there. This identity descends to $F$, because scalar extension of a finite matrix over a field is injective. The two factors commute, so $R$ is a two-sided inverse. It commutes with $P$ because $P$ commutes with $Q$. This finite calculation uses no chosen eigenvectors in the final formulas.

The actual torus chain complex. Use one vertex, the two oriented edges and the square whose attaching path is $\lambda \mu {\lambda }^{-1}{\mu }^{-1}$. Lift its lower left corner and transport a coefficient from that corner along its four sides. The bottom and oppositely oriented top edges give coefficients $z$ and $-Qz$ in the $\lambda$ coordinate; the right and oppositely oriented left edges give $Pz$ and $-z$ in the $\mu$ coordinate. Thus the square boundary is $((I-Q)z,(P-I)z)$. An edge with monodromy $S$ has boundary $(S-I)$ on its coefficient, which gives $d_1$ as stated. The two composites cancel because $PQ=QP$. This is the supplied-lift cellular model in [F1], so it computes the actual intrinsic homology, rather than an independently assumed complex. The direct-sum decomposition of step1.1 splits this entire complex into its $Fv$ and $W$ parts.

A chain contraction, including both ends of the complex. On the $W$ part define $h_0:W\to W\oplus W$ by $h_0(z)=(0,Rz)$ and $h_1:W\oplus W\to W$ by $h_1(x,y)=-Rx$, and put all other homotopy maps zero. Then $d_1{h}_0=I$. Also $h_1{d}_2=R(Q-I)=I$. Finally $d_2{h}_1(x,y)=(x,-(P-I)Rx)$ and $h_0{d}_1(x,y)=(0,R(P-I)x+y)$; their sum is $(x,y)$ because $R$ commutes with $P$. Thus $dh+hd=I$ in each of degrees $0,1,2$. The $W$ summand contributes no homology. On $Fv$ both differentials vanish, leaving one, two and one copies of $Fv$ in those degrees. Their cycles are precisely the vertex, the two supplied oriented edges, and the supplied oriented square with coefficient $v$. This proves the stated basis and not merely a dimension count.

Compatibility and exact scope. For a field extension, the same formulas for $v,\pi ,R,d,h$ remain defined because $2$ and $\mathrm{\Delta }$ remain nonzero; the split contractible and zero-differential summands persist, so their classes identify the actual extended homology. For simultaneous conjugation by an invertible matrix $A$, the map $X\mapsto AX{A}^{-1}$ sends $v$ to the new $v$, intertwines $P,Q$, and preserves all traces in $\pi$; it therefore carries the splitting and contraction to the displayed new ones and takes each named peripheral cycle to its corresponding cycle. The formulas use only addition, multiplication and division by $2$ and $\mathrm{\Delta }$ in the entries of the original matrices (the inverse of an $S{L}_2$ matrix is its polynomial adjugate). The eigenvalue extension in step2.1 verified an identity, and supplies no extra scalar choice to these formulas. The hypothesis $\mathrm{\Delta }\mathrm{\ne }0$ excludes the parabolic and scalar cases; neither their homology nor any exterior inclusion rank is asserted. All conclusions, including the fixed order of the two peripheral classes, follow.