[Paper Review] The integral cohomology groups of configuration spaces of pairs of points in real projective spaces

Jesús González, Peter S. Landweber|arXiv (Cornell University)|Apr 6, 2010

Homotopy and Cohomology in Algebraic Topology37 references3 citations

TL;DR

This paper computes the integral cohomology groups of configuration spaces of two distinct points in real projective spaces, both ordered and unordered, using spectral sequences and group cohomology. The key result is a complete description of these groups in terms of elementary abelian 2-groups and extensions, resolving a conjecture on 4-torsion and extending prior results on symmetric topological complexity.

ABSTRACT

We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2.

Motivation & Objective

• To compute the integral cohomology and homology groups of the ordered and unordered configuration spaces $F(\mathbb{P}^m, 2)$ and $B(\mathbb{P}^m, 2)$.
• To resolve a conjecture by Fred Cohen regarding the torsion in $H^*(B(\mathbb{P}^m, 2))$, specifically that it contains 2-torsion for $m > 1$ and 4-torsion for $m > 2$, but no 8-torsion.
• To complete the computation of the symmetric topological complexity of $\mathbb{P}^{2^i + \delta}$ for $i \geq 0$ and $0 \leq \delta \leq 2$.
• To relate the cohomology of these configuration spaces to the integral cohomology with twisted coefficients of the dihedral group of order 8 (unordered case) and the elementary abelian 2-group of rank 2 (ordered case).

Proposed method

• The paper employs the Cartan-Leray and Serre spectral sequences to analyze the cohomology of the configuration spaces via the action of the symmetric group on the ordered configuration space.
• It uses the Bockstein spectral sequence to analyze torsion in cohomology, particularly to confirm the presence of 4-torsion and absence of 8-torsion in $H^*(B(\mathbb{P}^m, 2))$.
• The authors apply the Universal Coefficient Theorem to derive integral homology groups from the computed cohomology groups.
• They leverage Poincaré duality with $w_1$-twisting to relate homology and cohomology in non-orientable settings.
• The analysis involves tracking permanent cycles and differentials in spectral sequences, particularly identifying which elements survive to $E_\infty$ via convergence and multiplicative structure.
• The results are derived by relating the configuration space cohomology to group cohomology of $D_4$ and $\mathbb{Z}_2^2$, using known structures and invariants like $\rho^*$-invariants.

Experimental results

Research questions

• RQ1 What are the integral cohomology groups of $F(\mathbb{P}^m, 2)$ and $B(\mathbb{P}^m, 2)$ for real projective spaces?
• RQ2 Does the cohomology of $B(\mathbb{P}^m, 2)$ contain 4-torsion, and if so, in which dimensions?
• RQ3 Can the symmetric topological complexity of $\mathbb{P}^{2^i + \delta}$ be fully computed for $i \geq 0$ and $0 \leq \delta \leq 2$?
• RQ4 How do the cohomology groups of these configuration spaces relate to the integral cohomology of the dihedral group $D_4$ and $\mathbb{Z}_2^2$ with twisted coefficients?
• RQ5 What is the precise structure of torsion in the cohomology of $B(\mathbb{P}^m, 2)$, and is 8-torsion present?

Key findings

• The integral cohomology of $F(\mathbb{P}^{2n}, 2)$ is computed explicitly: it is $\mathbb{Z}$ in degrees 0 and $4n-1$, and torsion groups $\langle k \rangle$ or $\{k\}$ in intermediate degrees depending on parity and range.
• For $F(\mathbb{P}^{2n+1}, 2)$, the cohomology includes an additional $\mathbb{Z} \oplus \langle n \rangle$ summand in degree $2n+1$, reflecting the odd-dimensional case.
• The cohomology of $B(\mathbb{P}^{2n}, 2)$ is described in terms of $\{k\} = \langle k \rangle \oplus \mathbb{Z}_4$ and $\langle k \rangle$, with dimensions grouped modulo 4, showing 4-torsion in degrees $4\ell$ for $0 < \ell < m/2$.
• The cohomology of $B(\mathbb{P}^{2n+1}, 2)$ includes a $\mathbb{Z} \oplus \langle n \rangle$ summand in degree $2n+1$, and the 4-torsion structure is confirmed in the same range as in the even case.
• The paper confirms Fred Cohen's conjecture: $H^*(B(\mathbb{P}^m, 2))$ has 2-torsion for $m > 1$ and 4-torsion for $m > 2$, but no 8-torsion.
• After inverting 2, both $F(\mathbb{P}^m, 2)$ and $B(\mathbb{P}^m, 2)$ become homology spheres, generalizing the known circle case for $m=1$.