Parameterized Lefschetz-Nielsen Fixed Point Theory and Hochschild Homology Traces

Nielsen fixed point theory on manifolds

“…Fixed points (x, t) and (x , t ) are said to be equivalent if there is a path α in X ×I such that the paths H(α) and P (α) are homotopic in X by a homotopy keeping the endpoints fixed. The equivalence classes will still be called fixed point classes (fpc) and denoted F. In work of Geoghegan and Nicas [10], fixed points at the "ends" of the homotopy H: X ×I → X, that is, the fixed points of h 0 and h 1 , must be treated differently than the other fixed points. Let Fix {0,1} (H) denote the subset of Fix(H) consisting of those fixed points of H that are not in the same fpc as any fixed point of h 0 or h 1 .…”

Coincidence Properties for Maps from the Torus to the Klein Bottle

“…We refer to this as the 1-parameter minimal coincidence problem or simply the minimal coincidence problem when the context is clear. We note that this problem, when specialized to the case where the domain space and target are the same, both g 1 and g 2 are the identity and the homotopy G remains constant is known as the 1-parameter fixed point problem and has been considered in [14,11,4,7,6]. Results in the last citation are partially generalized to the coincidence setting in [10].…”

Wecken type problems for self-maps of the Klein bottle

“…We refer to this as the restricted minimal coincidence problem. If we specialize the restricted problem to the 2 Wecken type problems for self-maps of the Klein bottle case where X = Y , both g 1 and g 2 are the identity and the homotopy G remains constant this is called the fixed point problem and has been considered in a number of papers [7,10,15,16]. The last partially generalized to coincidence in [14].…”