Related papers: Simple homotopy invariance of the loop coproduct

Given $f: M \to N$ a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace $[T] \in \pi_1^{st}(\mathcal{L} N, N)$. We realize the Goresky-Hingston coproduct as…

We show that the string coproduct is not homotopy invariant. More precisely, we show that the (reduced) coproducts are different on $L(1,7)$ and $L(2,7)$. Moreover, the coproduct on $L(k,7)$ can be expressed in terms of the Reidemeister…

Let M be a closed Riemannian manifold. We extend the product of Goresky-Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and…

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as…

We describe the Goresky-Hingston coproduct on the free loop space with real coefficients via the quasi-isomorphism $C_*(\Lambda M)\simeq C_*(M,C_*(\Omega M))$. This lets us describe the coproduct on the Leray-Serre spectral sequence as the…

We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as…

The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum $LM^{-TM}$ for a manifold $M$. Using this, they presented a proof of the statement…

The string topology coproduct is often perceived as a counterpart in string topology to the Chas-Sullivan product. However, in certain aspects the string topology coproduct is much harder to understand than the Chas-Sullivan product. In…

We construct an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct in string topology. The construction takes as its initial input a simplicial complex equipped with a local pairing on its simplicial chains,…

We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which…

We construct and study an algebraic analogue of the loop coproduct in string topology, also known as the Goresky-Hingston coproduct. Our algebraic setup, which under this analogy takes the place of the complex of chains on the free loop…

We study the space of paths in a closed manifold $M$ with endpoints determined by an involution $f\colon M\to M$. If the involution is fixed point free and if $M$ is $2$-connected then this path space is the universal covering space of the…

Given a closed manifold $M$. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincar\'e duality model of…

We construct a lift of the Goresky-Hingston coproduct on the based loop space. Using this lift, we produce formulas for the interaction of the coproduct with the $\pi_1$-action and with maps of manifolds. These results lets us compute the…

We describe two major string topology operations, the Chas-Sullivan product and the Goresky-Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom-Pontrjagin intersection theory while the…

We prove that the string cobracket is not a homotopy invariant. Adapting Naef's method arXiv:2106.11307 for computing the string coproduct, we show that the string cobrackets on the three-dimensional lens spaces $L(9;1)$ and $L(9;4)$…

The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into…

The Goresky-Hingston coproduct was first introduced by D. Sullivan and later extended by M. Goresky and N. Hingston. In this article we give a Morse theoretic description of the coproduct. Using the description we prove homotopy invariance…

Let $M$ be any simply-connected Gorenstein space over any field. F\'elix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We…

The primary goal of this paper is to study topological invariants in two dimensional twofold rotation and time-reversal symmetric spinful systems. In this paper, firstly we build a new homotopy invariant based on the lifting of the Wilson…