Regular homotopy group of immersions (Ex)

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The goal of this exercise is to get more feeling for the regular homotopy group of k-immersions in M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vjOEtO, I_k(M) 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 I_k(M) are represented by pointed k-immersions, i.e pairs (f,w) with f:S^k\looparrowright M is an immersion which does not necessarily map the basepoint 1\in S^k to the basepoint b\in M and w:I\rightarrow M is a path from b to g(1). Two pairs (f,w), (f',w') are considered equivalent if they are pointed homotopic, i.e. if there exists a regular homotopy H:S^k\times I\rightarrow M between f and f' such that w\ast H(1,-) and w' are homotopic relative endpoints. The sum of [(f_0,w_0)] and [(f_1,w_1)] is defined by forming the (class of the) connected sum immersion f_1\# f_2 along with the (class of the) path w_1\ast w_0^{-1}. The action of \pi_1(M,b) is given by mapping [(f,w)] to [(f,w\ast\omega)] where \omega is a loop at b representing a g\in\pi_1(M).

The equivariant intersection of (g,w), (g',w') is described as follows. Choose representavives with g_0 and g_1 transverse. For every double point (x_0,x_1) with g_0(x_0)=g_1(x_1)=d determine the sign \epsilon(d) in the usual way, i.e. by comparing orientations of T_{x_0}S^k\oplus T_{x_1}S^k and T_dM. The element g(d) is given by the class of the loop
\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}
where u_i is a path in S^k from 1 to x_i.

Definition 0.2. In [Ranicki2002] the group is defined as follows. Elements of I_k(M) are represented by (f,\widetilde{f}) with f:S^k\looparrowright M a k-immersion and \widetilde{g}:S^k\looparrowright \widetilde{M} a lift of f to the universal cover of M. Two pairs (f,\widetilde{f}), (f',\widetilde{f'}) are considered equivalent if they are regular homotopy equivalent. The sum is given by connected sum. The action of \pi_1(M,b) is via deck transformations on the lift.

To determine the equivariant intersection of (f_0,\widetilde{f_0}) and (f_1,\widetilde{f_1}) choose f_0 and f_1 to be transverse. For every doublepoint (x_0,x_1) with f_0(x_0)=f_1(x_1)=d there exists an element g(d) such that g(d)f_0(x_0)=f_1(x_1). Define the equivariant index of f_0 and f_1 at d to be \epsilon(d)g(d)\in\pm\pi_1(M) where \epsilon(d) is determined by comparing orientations again.

1) Show that the above definitions of \mathbb{Z}[\pi_1(M)]-modules are equivalent.

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

3) Show that the corresponding descriptions of Wall's \mu-form (the self-intersection form) coincide up to possible conjugation by a fixed element \alpha\in\pi_1(M).

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