[Paper Review] Cohomology rings of certain seven dimensional manifolds

Christine Escher, Shari K. Ultman|ArXiv.org|Oct 11, 2008

Commutative Algebra and Its Applications13 references3 citations

TL;DR

This paper computes the integral cohomology rings of four families of seven-dimensional simply connected cohomogeneity-one manifolds with $S^3 \times S^3$-action, using double disk bundle structures and long exact sequences in cohomology. The key result is that members of the $N_{(p_-,q_-)(p_+,q_+)}$ and $O_{(p,q:m)}$ families have cohomology rings of type $E_r$, with explicit generators and relations, while the $L$ and $M$ families are shown to have rings isomorphic to those of known manifolds like Eschenburg spaces and $S^3$-bundles over $S^4$. These rings distinguish diffeomorphism types via topological invariants.

ABSTRACT

We calculate the cohomology rings of a collection of seven dimensional manifolds supporting an S^3 x S^3-action with one dimensional orbit space. These manifolds are of interest to differential geometers studying non-negative and positive sectional curvature. From this collection, we identify several families of manifolds for which there exist well-known topological invariants distinguishing homeomorphism and diffeomorphism types.

Motivation & Objective

• To determine the integral cohomology rings of four families of seven-dimensional simply connected cohomogeneity-one manifolds with $S^3 \times S^3$-action.
• To identify topological invariants that distinguish diffeomorphism types within these families.
• To extend prior results on cohomology groups by computing the full ring structure, particularly for the $N$ and $O$ families where the ring structure was previously unknown.
• To establish that certain manifolds have cohomology rings of type $E_r$, enabling the use of topological invariants for diffeomorphism classification.

Proposed method

• Uses the double disk bundle construction $M = D(G/K_-) \cup_{\text{id}} D(G/K_+)$, where $G = S^3 \times S^3$, to model the manifolds topologically.
• Applies long exact sequences in cohomology to the pairs $(M, G/K_\pm)$ to compute cohomology groups and relations between generators.
• Employs Lemma 2.2 to verify conditions under which the cohomology ring is of type $E_r$, based on surjectivity and image conditions of restriction homomorphisms.
• Analyzes isotropy groups $H \subseteq K_- \subseteq G$ and $H \subseteq K_+ \subseteq G$ to determine the structure of $G/K_\pm$ and $G/H$, which are spheres or lens spaces.
• Computes the ring structure via generators in $H^2$, $H^4$, $H^5$, and $H^7$, with products like $xy$ generating $H^7$.
• Distinguishes cases based on parameter constraints (e.g., parity of $p_\pm, q_\pm$, and $m=1$ or $m=2$) to handle different cohomology group structures.

Experimental results

Research questions

• RQ1 What is the integral cohomology ring structure of the $N_{(p_-,q_-)(p_+,q_+)}$ family of seven-dimensional manifolds with $S^3 \times S^3$-action?
• RQ2 How do the cohomology rings of the $O_{(p,q:m)}$ family differ based on whether $m=1$ or $m=2$, and what is their ring type?
• RQ3 Can the cohomology ring of the $L_{(p_-,q_-)(p_+,q_+)}$ family be fully described, especially when $p_+$ is odd or even?
• RQ4 Do the cohomology rings of these families support topological invariants that distinguish diffeomorphism types?
• RQ5 Is the cohomology ring of the $M_{(p_-,q_-)(p_+,q_+)}$ family isomorphic to that of an $S^3$-bundle over $S^4$, as suggested by prior work?

Key findings

• The cohomology ring of $N_{(p_-,q_-)(p_+,q_+)}$ is isomorphic to that of an Eschenburg space, with $H^2(N) = \mathbb{Z} \cdot x$, $H^5(N) = \mathbb{Z} \cdot y$, and $xy$ generating $H^7(N) \cong \mathbb{Z}$, where $r = |p_+^2 q_-^2 - p_-^2 q_+^2|$ is odd.
• For $O_{(p,q:m)}$, the cohomology ring is of type $E_r$ with $r = |p^2 - q^2|$, and $H^2(O) = \mathbb{Z} \cdot x$, $H^5(O) = \mathbb{Z} \cdot y$, and $xy$ generating $H^7(O) \cong \mathbb{Z}$, provided $|p|$ and $|q|$ are not both 1.
• When $p_+$ is odd in the $L$ family, $H^4(L) \cong \mathbb{Z}_r$ with $r = \frac{1}{4}|p_+^2 q_-^2 - p_-^2 q_+^2|$, and the ring is of type $E_r$ with $r$ even.
• When $p_+$ is even in the $L$ family, $H^3(L) \cong \mathbb{Z}_2$, $H^5(L) \cong \mathbb{Z} \oplus \mathbb{Z}_2$, and $H^4(L) \cong \mathbb{Z}_r$ with $r = |p_+^2 q_-^2 - p_-^2 q_+^2|$ odd, and the ring structure is determined by $x^2$ generating $H^4(L)$ and $xy$ generating $H^7(L)$.
• The $M_{(p_-,q_-)(p_+,q_+)}$ family has cohomology ring isomorphic to that of an $S^3$-bundle over $S^4$, with $H^4(M) \cong \mathbb{Z}_r$ for $r = \frac{1}{8}|p_+^2 q_-^2 - p_-^2 q_+^2|$, and $y$ and $z$ generating $H^4(M)$ and $H^7(M)$ respectively.
• For all families, the cohomology ring is of type $E_r$, and when $r$ is odd, the ring structure supports a diffeomorphism invariant computable via topological means, as in [KS].