On the homology of the space of singular knots (Q1945811)

Using ideas derived from the construction of the loop product in string topology, H. Abbaspour and D. Chataur introduce various associative products on the homology of the space of knots and singular knots in \(S^n\). More precisely denote by \(\text{Emb}(S^1, S^n)\) the space of embeddings of \(S^1\) in \(S^n\) and \(\text{Imm}'(S^1, S^n)\) (respectively \(\text{Imm}'_k(S^1, S^n)\)) the space of singular knots with only transverse double points (respectively with exactly \(k\) double points) away from a marked point, and no other singular point. Then the graded groups \(H_{*+2n-1}(\text{Emb}(S^1, S^n))\) and \(H_{*+2n-1}(\text{Imm}'(S^1, S^n))\) are equipped with a graded commutative and associative product. Denote by \(ev : \text{Emb}(S^1, S^n)\to US^n\) the map in the sphere tangent bundle defined by \[ ev(\gamma) = (\gamma (0), \gamma'(0)/|| \gamma'(0)||)\,, \] and equip \(H_{*+2n-1}(US^n)\) with the intersection product. Then the evaluation maps in homology, \[ H_{*+2n-1}(ev) : H_{*+2n-1}(\text{Emb}(S^1, S^n))\to H_{*+2n-1}(US^n) \] and \[ H_{*+2n-1}(ev) : H_{*+2n-1}(\text{Imm}'(S^1,S^n))\to H_{*+2n-1}(US^n) \] are multiplicative maps. The authors introduce also a multiplicative desingularization morphism \[ H_*(\text{Imm}'_k(S^1, S^n))\to H_{*+k(n-3)}(\text{Emb}(S^1, S^n)) \] obtained by resolving the singularity in all possible ways, that is compatible with the multiplicative structures.