[Paper Review] Homotopical Intersection Theory, II: equivariance

John R. Klein, Bruce Williams|ArXiv.org|Mar 2, 2008

Homotopy and Cohomology in Algebraic Topology13 references4 citations

TL;DR

This paper develops a homotopy-theoretic obstruction theory for equivariant intersection problems on manifolds with finite group actions, using parametrized equivariant spectra to bypass transversality issues. The key result is a complete obstruction $ e_G(f) \in H^0_G(P; \mathcal{E}(i_Q)) $ that vanishes if and only if an equivariant map $ f: P \to N $ can be deformed off a $ G $-submanifold $ Q $, with sufficiency guaranteed in the equivariant metastable range.

ABSTRACT

This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. This theory has applications to embedding problems, equivariant fixed point theory and the problem of enumerating the periodic points of a self map of a compact smooth manifold.

Motivation & Objective

• To extend homotopical intersection theory to equivariant settings with finite group actions.
• To overcome the failure of equivariant transversality by using homotopy-theoretic tools.
• To construct a complete obstruction in equivariant cohomology for solving equivariant embedding and fixed point problems.
• To apply the theory to periodic point problems and Nielsen number conjectures in equivariant dynamics.

Proposed method

• Construct a naive parametrized $ G $-spectrum $ \mathcal{E}(i_Q) $ over $ N $ from the inclusion $ i_Q: Q \subset N $.
• Define an equivariant obstruction class $ e_G(f) \in H^0_G(P; \mathcal{E}(i_Q)) $ using the map $ f: P \to N $.
• Use equivariant Poincaré duality and the complement formula to relate the obstruction to stable normal data.
• Apply the tom Dieck splitting to decompose equivariant bordism groups into fixed-point components.
• Use the pushforward and pullback of indexing functions to track isotropy data across $ G $-spaces.
• Establish a link between the obstruction and Nielsen numbers via framed bordism of homotopy fiber products.

Experimental results

Research questions

• RQ1 When can an equivariant map $ f: P \to N $ be deformed to avoid a $ G $-submanifold $ Q \subset N $?
• RQ2 What is the complete obstruction to solving such equivariant intersection problems in the metastable range?
• RQ3 How does the obstruction $ e_G(f) $ relate to classical invariants like Nielsen numbers in periodic point theory?
• RQ4 Can the obstruction detect the existence of $ n $-periodic point free maps in terms of framed bordism?
• RQ5 Is the obstruction $ e_G(f) $ equivalent to the vanishing of Nielsen numbers $ N(f^k) $?

Key findings

• The obstruction $ e_G(f) \in H^0_G(P; \mathcal{E}(i_Q)) $ vanishes if and only if $ f $ is equivariantly homotopic to a map disjoint from $ Q $, under the condition $ p^H \leq 2f^*(i_Q)_!\operatorname{cd}_H(i_Q) - 3 $ for all $ (H) \in \mathcal{I}(G;P) $.
• The obstruction $ e_G(f) $ is constructed from a parametrized $ G $-spectrum $ \mathcal{E}(i_Q) $, enabling a homotopy-theoretic approach without relying on transversality.
• The invariant $ \ell_n(f) \in \Omega_0^{\mathbb{Z}_n, \text{fr}}(\text{ho}P_n(f)) $ detects $ n $-periodic point free maps, with $ \ell_n(f) = 0 $ implying such a homotopy exists when $ \dim M \geq 3 $.
• The conjecture that $ \mathcal{N}_k(f) $, the number of non-zero terms in $ \ell_n^k(f) $, equals the Nielsen number $ N(f^k) $ is supported by the isomorphism $ \pi_0(E\mathbb{Z}_k \times_{\mathbb{Z}_k} \text{ho}P_k(f)) \cong \pi_{\rho,k} $.
• The tom Dieck splitting allows decomposition of equivariant bordism groups into components indexed by isotropy subgroups, enabling computation of the obstruction in terms of fixed-point data.
• The proof of Theorem J establishes that $ \pi_0(E\mathbb{Z}_k \times_{\mathbb{Z}_k} \text{ho}P_k(f)) \cong \pi_{\rho,k} $, confirming the link between the obstruction and Nielsen theory.