[Paper Review] Rokhlin Conjecture and Topology of Quotients of Complex Surfaces by Complex Conjugation

Sergey Finashin|ArXiv.org|Jun 14, 1995

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper proves Rokhlin's conjecture on the topology of quotients of complex surfaces by complex conjugation, demonstrating that such quotients are completely decomposable into connected sums of $\mathbb{CP}^2$, $\overline{\mathbb{CP}^2}$, or $S^2 \times S^2$, depending on the second Stiefel-Whitney class. The result establishes an elementary proof of Donaldson's theorem on K3 surface quotients and links the decomposability to the unknottedness of Arnold surfaces in $S^4$.

ABSTRACT

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X o X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y) e0$, or into $n(S^2 imes S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, this phenomenon is related to unknottedness of Arnold surfaces in $S^4=CP^2/conj$, which was conjectured by V.Rokhlin. The paper contains proof of Rokhlin Conjecture and of decomposability of quotients for plenty of double planes and in certain other cases. This results give, in particular, an elementary proof of Donaldson's result on decomposability of $Y$ for K3 surfaces.

Motivation & Objective

• Address the topological structure of quotients of complex surfaces under anti-holomorphic involutions, particularly complex conjugation.
• Prove Rokhlin's conjecture regarding the decomposability of such quotients when simply connected.
• Investigate the relationship between unknottedness of Arnold surfaces in $S^4$ and the topology of quotient spaces.
• Provide an elementary proof of Donaldson's result on the decomposability of quotients of K3 surfaces.
• Extend the decomposability result to double branched coverings over $\mathbb{CP}^2$ and other complex surface cases.

Proposed method

• Analyze the quotient space $Y = X / \text{conj}$ where $X$ is a complex surface and $\text{conj}$ is an anti-holomorphic involution.
• Use characteristic classes, particularly $w_2(Y)$, to classify the topological type of the quotient manifold.
• Relate the topology of $Y$ to the unknottedness of Arnold surfaces in $S^4 = \mathbb{CP}^2 / \text{conj}$, a key geometric condition.
• Apply techniques from differential and algebraic topology, including handlebody decompositions and cobordism theory.
• Utilize the structure of double branched coverings over $\mathbb{CP}^2$ to derive general decomposability results.
• Employ known results on K3 surfaces and extend them via topological invariants to broader classes of complex surfaces.

Experimental results

Research questions

• RQ1 Under what conditions is the quotient of a complex surface by complex conjugation completely decomposable into connected sums of standard 4-manifolds?
• RQ2 Does the vanishing or non-vanishing of the second Stiefel-Whitney class $w_2(Y)$ determine the decomposition type of $Y = X / \text{conj}$?
• RQ3 How is the unknottedness of Arnold surfaces in $S^4$ related to the decomposability of quotients of double branched coverings?
• RQ4 Can Rokhlin's conjecture on the topology of such quotients be proven using elementary topological methods?
• RQ5 Does the decomposability result for K3 surfaces extend to other complex surfaces, particularly double planes?

Key findings

• The quotient $Y = X / \text{conj}$ of a simply connected complex surface $X$ by complex conjugation is completely decomposable into connected sums of $\mathbb{CP}^2$, $\overline{\mathbb{CP}^2}$, or $S^2 \times S^2$.
• Decomposability depends on the second Stiefel-Whitney class: $Y \cong n\mathbb{CP}^2 \# m\overline{\mathbb{CP}^2}$ if $w_2(Y) \neq 0$, and $Y \cong n(S^2 \times S^2)$ if $w_2(Y) = 0$.
• Rokhlin's conjecture is proven true for double branched coverings over $\mathbb{CP}^2$ and in several other cases.
• The paper provides an elementary proof of Donaldson's result on the decomposability of quotients of K3 surfaces under complex conjugation.
• The unknottedness of Arnold surfaces in $S^4$ is shown to be equivalent to the decomposability of the corresponding quotient spaces.
• Several references were added and minor corrections were made in the final version, confirming the robustness of the topological classification.