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 $M$ $I_k(M)$ and the intersection/self-intersection form on it. Below are two definitions of it. Both are used in {{citeD|Wall1999}}. | + | 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 $1\in S^k$ 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 $g\in\pi_1(M)$. | 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 $1\in S^k$ 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 $$w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$$where $u_i$ is a path in $S^k$ from $1$ to $x_i$. | 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 $1$ 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. | 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 | + | 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. | '''1)''' Show that the above definitions of $\mathbb{Z}[\pi_1(M)]$-modules are equivalent. |
Revision as of 14:41, 19 March 2012
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 .
The equivariant intersection of is described as follows. Choose representavives with and transverse. For every double point with determine the sign in the usual way, i.e. by comparing orientations of and . The element is given by the class of the loopDefinition 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 selfintersection form) coincide up to possible conjugation by a fixed element .
\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 , 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 .
The equivariant intersection of is described as follows. Choose representavives with and transverse. For every double point with determine the sign in the usual way, i.e. by comparing orientations of and . The element is given by the class of the loopDefinition 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 selfintersection form) coincide up to possible conjugation by a fixed element .
$ to $x_i$. 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 exist 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 selfintersection form) coincide up to possible conjugation by a fixed element $\alpha\in\pi_1(M)$. [[Category:Exercises]]k-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 .
The equivariant intersection of is described as follows. Choose representavives with and transverse. For every double point with determine the sign in the usual way, i.e. by comparing orientations of and . The element is given by the class of the loopDefinition 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 selfintersection form) coincide up to possible conjugation by a fixed element .