Equivariant cohomology of even-dimensional complex quadrics from a combinatorial point of view

TL;DR

The paper gives a complete, combinatorial presentation of the graph equivariant cohomology for the GKM graph associated with the even‑dimensional complex quadric $Q_{2n}$. It introduces two families of generators, degree‑2s $M_{v}$ and higher Thom classes $\Delta_{K}$, and imposes four explicit relations to obtain an isomorphism $\mathbb{Z}[\mathcal{GQ}_{2n}] \cong H^{*}(\mathcal{GQ}_{2n})$, thereby computing $H_T^{*}(Q_{2n})$ via $H^{*}(\mathcal{GQ}_{2n})$. A key result is a multiplicative formula for $\Delta_{K}\Delta_{H}$ that encodes the interaction of Schubert‑type data in this combinatorial model. The work also explains, in purely combinatorial terms, why the ordinary cohomology ring $H^{*}(Q_{2n})$ differs depending on whether $n$ is even or odd, and provides an alternative route to Lai’s classical calculations of $H^{*}(Q_{4n})$ and $H^{*}(Q_{4n+2})$.

Abstract

The purpose of this paper is to determine the ring structure of the graph equivariant cohomology of the GKM graph induced from the even-dimensional complex quadrics. We show that the graph equivariant cohomology is generated by two types of subgraphs in the GKM graph, which are subject to four different types of relations. By utilizing this ring structure, we establish the multiplicative relation for the generators of degree 2n and provide an alternative computation of the ordinary cohomology ring of 4n-dimensional complex quadrics, as previously computed by H. Lai. Additionally, we provide a combinatorial explanation for why the square of the 2n degree generator x vanishes when n is odd and is non-vanishing when n is even.

Figures (14)

• Figure 1: The left graph is $\Gamma_{4}$ ($n=2$) induced from the $T^{3}$-action on $Q_{4}$, and the right graph is $\Gamma_{6}$ ($n=3$) induced from the $T^{4}$-action on $Q_{6}$.
• Figure 2: The axial function $\widetilde{\alpha}$ around the vertex $1$ in $\Gamma_{4}$. This corresponds to the GKM graph induced from the $T^{3}$-action on $Q_{4}$ defined by \ref{['def-noneff-action']}. Note that $\overline{6}=1$, $\overline{5}=2$, $\overline{4}=3$.
• Figure 3: The GKM graph $\mathcal{GQ}_{2n}$ when $n=2$ (also see the left graph in Figure \ref{['2-examples']}). The right figure shows that the axial function $\alpha:E_{4}\to H^{2}(BT^{3})$ of $\mathcal{GQ}_{4}$ around the vertex $1$. The left figure shows its $0$-cochain presentation $f:V_{4}\to H^{2}(BT^{3})$.
• Figure 4: The element $M_{6}\in H^{2}(\mathcal{GQ}_{4})$.
• Figure 5: $\Delta_{K}$ for $K=\{1,2,3\}$, where $\Delta_{K}(2)=(x_{3}-x_{1})(x_{2}-x_{1}+x_{3})$.

Theorems & Definitions (51)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Definition 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof