# Regular homotopy group of immersions (Ex)

The goal of this exercise is to get more feeling for the regular homotopy group of -immersions in , and the intersection/self-intersection form on it. Below are two definitions of it. Both are used in [Wall1999].

**Definition 0.1.**
In [Lück2001] the group is defined as follows. Elements of are represented by pointed -immersions, i.e pairs with is an immersion which does not necessarily map the basepoint to the basepoint and is a path from to . Two pairs are considered equivalent if they are pointed homotopic, i.e. if there exists a regular homotopy between and such that and are homotopic relative endpoints. The sum of and is defined by forming the (class of the) connected sum immersion along with the (class of the) path . The action of is given by mapping to where is a loop at representing a .

**Definition 0.2.**
In [Ranicki2002] the group is defined as follows. Elements of are represented by with a -immersion and a lift of to the universal cover of . Two pairs are considered equivalent if they are regular homotopy equivalent. The sum is given by connected sum. The action of is via deck transformations on the lift.

To determine the equivariant intersection of and choose and to be transverse. For every doublepoint with there exists an element such that . Define the equivariant index of and at to be where is determined by comparing orientations again.

**1)** Show that the above definitions of -modules are equivalent.

**2)** Show that the descriptions of the equivariant intersections of (regular homotopy classes of) -immersions coincide.

**3)** Show that the corresponding descriptions of Wall's -form (the self-intersection form) coincide up to possible conjugation by a fixed element .