String topology initiated by M. Chas and D. Sullivan [Abel Symp. 4, 33–37 (2009; Zbl 1185.55013)] studies plentiful structures of the homology of the free loop space \(M^{S^1}=\text{map}(S^1, M)\) of an oriented manifold \(M\). In particular, the general properties and explicit calculations of the loop product and coproduct on \(H_*(M^{S^1})\) are investigated. The product can be generalized naturally to one on the homology of the mapping space \(M^{S^n}\) whose domain is the \(n\)-dimensional sphere. This is called the brane product. In fact, for an oriented manifold \(M\), the intersection product \(\Delta^! : H_*(M\times M) \to H_{*-\dim M}(M)\) can be lifted to the homology of the pullback \(M^{S^n}\times_M M^{S^n}\) for the diagram \[ M^{S^n}\times M^{S^n} \stackrel{ev\times ev}{\longrightarrow}M \times M \stackrel{\Delta}{\longleftarrow} M, \] where \(ev\) and \(\Delta\) are the evaluation map and the diagonal map, respectively. In consequence, one has the shriek map \(incl^! : H_*(M^{S^n}\times M^{S^n}) \to H_{*-\dim M}(M^{S^n}\times_M M^{S^n})\). By definition, the composite \((comp)_*\circ incl^!\) with the map \((comp)_*\) induced by the composition \(comp : M^{S^n}\times_M M^{S^n} \to M^{S^n}\) is the brane product [R. L. Cohen et al., String topology and cyclic homology. Basel: Birkhäuser (2006; Zbl 1089.57002)]. If we consider naturally the coproduct in the setting of brane topology, it is necessary to deal with the pullback \(M^{S^n}\times_M M^{S^n}\) for the diagram \[ M^{S^n} \stackrel{res}{\longrightarrow} M^{S^{n-1}} \stackrel{c}{\longleftarrow} M, \] in which the map \(c\) is defined by the assignment to constant maps and the map \(res\) is induced by the inclusion \(S^{n-1} \to S^n\) into the equator. However, since the map \(c : M \to M^{S^{n-1}}\) is not regarded as an embedding of finite codimension in general, the same construction as that of the intersection product does not enable us to obtain a “wrong way map” \(c^! : H_*(M^{S^{n-1}}) \to H_*(M)\). In order to define a relevant brane coproduct, the author overcomes the problem appealing to rational homotopy theory.
We call a path-connected space \(M\) a \({\mathbb K}\)-Gorenstein space of dimension \(m\) if \[ \dim \text{Ext}_{C^*(M)}^l({\mathbb K}, C^*(M)) = \begin{cases} 1&\text{if } l= m, \\ 0&\text{otherwise}, \end{cases} \] where \({\mathbb K}\) is a field and \(C^*(M)\) denotes the singular cochain algebra of \(M\) with coefficients in \({\mathbb K}\); see [Y. Félix et al., Adv. Math. 71, No. 1, 92–112 (1988; Zbl 0659.57011)]. In what follows, we assume that \({\mathbb K}\) is a field of characteristic zero. Let \(S\) and \(T\) be manifolds of dimension \(k\) and \(M\) a \(k\)-connected space whose iterated based loop space \(\Omega^{k-1}M\) is a \({\mathbb K}\)-Gorenstein space of dimension \(\overline{m}\). Thanks to the Gorensteinness of \(\Omega^{k-1}M\), one has a wrong way map \(c^! : H_*(M^{S^{k-1}}) \to H_{*-\overline{m}}(M)\). In fact, Theorem 3.1 enables us to deduce that \[ \text{Ext}_{C^*(M^{S^{k-1}})}^*(C^*(M), C^*(M^{S^{k-1}})) \cong H^{*-\overline{m}}(M). \] As a consequence, the generator in \(H^0(M)\cong {\mathbb K}\) gives rise to the dual to the map \(c^!\).
Then the author defines the \((S, T)\)-brane coproduct \(\delta_{ST} : H_*(M^{S\#T}) \to H_{*-\overline{m}}(M^S\times M^T)\) by considering the pullback \(M^S\times_MM^T\) for the diagram \[ M^{S\#T} \stackrel{res}{\longrightarrow} M^{S^{n-1}} \stackrel{c}{\longleftarrow} M, \] where \(res\) is the map induced by the inclusion of \(S^{n-1}\) into the disk which is used when constructing the connected sum \(S\#T\). It is worth mentioning that the spectral sequence due to Félix and Thomas in the proof of [Y. Félix and J.-C. Thomas, Math. Ann. 345, No. 2, 417–452 (2009; Zbl 1204.55008), Theorem 2.3] plays a crucial role in proving Theorem 3.1 mentioned above. One of the main theorems (Theorem 1.5) asserts that, under the same assumption as above, the brane product and coproduct for manifolds of dimension \(k\) are associative, commutative and enjoy the Frobenius compatibility.
The explicit calculations (Theorem 1.6) of the brane product and coproduct on \(H_*((S^{2n+1})^{S^2})\) for \(n \geq 1\) allow us to conclude that \((\delta_{S^2S^2}\otimes 1)\circ \delta_{S^2S^2}\) is non trivial. We observe that the composite \((\delta \otimes 1)\circ \delta\) for the loop coproduct \(\delta=\delta_{S^1S^1}\) is trivial in general [H. Tamanoi, J. Pure Appl. Algebra 214, No. 5, 605–615 (2010; Zbl 1201.55003), Theorem A].