Regular homotopy group of immersions (Ex)

The goal of this exercise is to get more feeling for the regular homotopy group of $k$$<wikitex>; The goal of this exercise is to get more feeling for the regular homotopy group of $k$-immersions in $M$, $I_k(M)$ and the intersection/self-intersection form on it. Below are two definitions of it. Both are used in {{citeD|Wall1999}}. {{beginthm|Definition}} In {{citeD|Lück2001}} the group is defined as follows. Elements of $I_k(M)$ are represented by pointed $k$-immersions, i.e pairs $(f,w)$ with $f:S^k\looparrowright M$ is an immersion which does not necessarily map the basepoint k$-immersions in $M$$M$, $I_k(M)$$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)$$I_k(M)$ are represented by pointed $k$$k$-immersions, i.e pairs $(f,w)$$(f,w)$ with $f:S^k\looparrowright M$$f:S^k\looparrowright M$ is an immersion which does not necessarily map the basepoint $1\in S^k$$1\in S^k$ to the basepoint $b\in M$$b\in M$ and $w:I\rightarrow M$$w:I\rightarrow M$ is a path from $b$$b$ to $g(1)$$g(1)$. Two pairs $(f,w), (f',w')$$(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$$H:S^k\times I\rightarrow M$ between $f$$f$ and $f'$$f'$ such that $w\ast H(1,-)$$w\ast H(1,-)$ and $w'$$w'$ are homotopic relative endpoints. The sum of $[(f_0,w_0)]$$[(f_0,w_0)]$ and $[(f_1,w_1)]$$[(f_1,w_1)]$ is defined by forming the (class of the) connected sum immersion $f_1\# f_2$$f_1\# f_2$ along with the (class of the) path $w_1\ast w_0^{-1}$$w_1\ast w_0^{-1}$. The action of $\pi_1(M,b)$$\pi_1(M,b)$ is given by mapping $[(f,w)]$$[(f,w)]$ to $[(f,w\ast\omega)]$$[(f,w\ast\omega)]$ where $\omega$$\omega$ is a loop at $b$$b$ representing a $g\in\pi_1(M)$$g\in\pi_1(M)$.

The equivariant intersection of $(g,w), (g',w')$$(g,w), (g',w')$ is described as follows. Choose representavives with $g_0$$g_0$ and $g_1$$g_1$ transverse. For every double point $(x_0,x_1)$$(x_0,x_1)$ with $g_0(x_0)=g_1(x_1)=d$$g_0(x_0)=g_1(x_1)=d$ determine the sign $\epsilon(d)$$\epsilon(d)$ in the usual way, i.e. by comparing orientations of $T_{x_0}S^k\oplus T_{x_1}S^k$$T_{x_0}S^k\oplus T_{x_1}S^k$ and $T_dM$$T_dM$. The element $g(d)$$g(d)$ is given by the class of the loop

$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$ $$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$$

where $u_i$$u_i$ is a path in $S^k$$S^k$ from $1$$1$ to $x_i$$x_i$.

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

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

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

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

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

\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 $$w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$$where $u_i$ is a path in $S^k$ from k-immersions in $M$$M$, $I_k(M)$$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)$$I_k(M)$ are represented by pointed $k$$k$-immersions, i.e pairs $(f,w)$$(f,w)$ with $f:S^k\looparrowright M$$f:S^k\looparrowright M$ is an immersion which does not necessarily map the basepoint $1\in S^k$$1\in S^k$ to the basepoint $b\in M$$b\in M$ and $w:I\rightarrow M$$w:I\rightarrow M$ is a path from $b$$b$ to $g(1)$$g(1)$. Two pairs $(f,w), (f',w')$$(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$$H:S^k\times I\rightarrow M$ between $f$$f$ and $f'$$f'$ such that $w\ast H(1,-)$$w\ast H(1,-)$ and $w'$$w'$ are homotopic relative endpoints. The sum of $[(f_0,w_0)]$$[(f_0,w_0)]$ and $[(f_1,w_1)]$$[(f_1,w_1)]$ is defined by forming the (class of the) connected sum immersion $f_1\# f_2$$f_1\# f_2$ along with the (class of the) path $w_1\ast w_0^{-1}$$w_1\ast w_0^{-1}$. The action of $\pi_1(M,b)$$\pi_1(M,b)$ is given by mapping $[(f,w)]$$[(f,w)]$ to $[(f,w\ast\omega)]$$[(f,w\ast\omega)]$ where $\omega$$\omega$ is a loop at $b$$b$ representing a $g\in\pi_1(M)$$g\in\pi_1(M)$.

The equivariant intersection of $(g,w), (g',w')$$(g,w), (g',w')$ is described as follows. Choose representavives with $g_0$$g_0$ and $g_1$$g_1$ transverse. For every double point $(x_0,x_1)$$(x_0,x_1)$ with $g_0(x_0)=g_1(x_1)=d$$g_0(x_0)=g_1(x_1)=d$ determine the sign $\epsilon(d)$$\epsilon(d)$ in the usual way, i.e. by comparing orientations of $T_{x_0}S^k\oplus T_{x_1}S^k$$T_{x_0}S^k\oplus T_{x_1}S^k$ and $T_dM$$T_dM$. The element $g(d)$$g(d)$ is given by the class of the loop

$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$ $$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$$

where $u_i$$u_i$ is a path in $S^k$$S^k$ from $1$$1$ to $x_i$$x_i$.

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

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

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

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

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

$ to $x_i$. {{endthm|Definition}} {{beginthm|Definition}} In {{citeD|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. {{endthm|Definition}} '''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 $\alpha\in\pi_1(M)$. [[Category:Exercises]] [[Category:Exercises without solution]]k-immersions in $M$$M$, $I_k(M)$$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)$$I_k(M)$ are represented by pointed $k$$k$-immersions, i.e pairs $(f,w)$$(f,w)$ with $f:S^k\looparrowright M$$f:S^k\looparrowright M$ is an immersion which does not necessarily map the basepoint $1\in S^k$$1\in S^k$ to the basepoint $b\in M$$b\in M$ and $w:I\rightarrow M$$w:I\rightarrow M$ is a path from $b$$b$ to $g(1)$$g(1)$. Two pairs $(f,w), (f',w')$$(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$$H:S^k\times I\rightarrow M$ between $f$$f$ and $f'$$f'$ such that $w\ast H(1,-)$$w\ast H(1,-)$ and $w'$$w'$ are homotopic relative endpoints. The sum of $[(f_0,w_0)]$$[(f_0,w_0)]$ and $[(f_1,w_1)]$$[(f_1,w_1)]$ is defined by forming the (class of the) connected sum immersion $f_1\# f_2$$f_1\# f_2$ along with the (class of the) path $w_1\ast w_0^{-1}$$w_1\ast w_0^{-1}$. The action of $\pi_1(M,b)$$\pi_1(M,b)$ is given by mapping $[(f,w)]$$[(f,w)]$ to $[(f,w\ast\omega)]$$[(f,w\ast\omega)]$ where $\omega$$\omega$ is a loop at $b$$b$ representing a $g\in\pi_1(M)$$g\in\pi_1(M)$.

The equivariant intersection of $(g,w), (g',w')$$(g,w), (g',w')$ is described as follows. Choose representavives with $g_0$$g_0$ and $g_1$$g_1$ transverse. For every double point $(x_0,x_1)$$(x_0,x_1)$ with $g_0(x_0)=g_1(x_1)=d$$g_0(x_0)=g_1(x_1)=d$ determine the sign $\epsilon(d)$$\epsilon(d)$ in the usual way, i.e. by comparing orientations of $T_{x_0}S^k\oplus T_{x_1}S^k$$T_{x_0}S^k\oplus T_{x_1}S^k$ and $T_dM$$T_dM$. The element $g(d)$$g(d)$ is given by the class of the loop

$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$ $$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$$

where $u_i$$u_i$ is a path in $S^k$$S^k$ from $1$$1$ to $x_i$$x_i$.

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

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

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

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

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