Search for: All records

.We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated. 

For a finite group$$G$$of not prime power order, Oliver showed that the obstruction for a finite CW-complex$$F$$to be the fixed point set of a contractible finite$$G$$-CW-complex is determined by the Euler characteristic$$\chi (F)$$. (He also has similar results for compact Lie group actions.) We show that the analogous problem for$$F$$to be the fixed point set of a finite$$G$$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on$$K_0$$[2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set. 

Smith theory says that the fixed point set of a semi-free action of a group$$G$$on a contractible space is$${\mathbb {Z}}_p$$-acyclic for any prime factor$$p$$of the order of$$G$$. Jones proved the converse of Smith theory for the case$$G$$is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain$$K$$-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of$$K$$-theoretical obstructions.