Π-trivial map
m (Change to emphasise that a choice of lift is required for a Pi-trivial map to represent homology in the cover and that this lift should be part of the data in being a Pi-trivial map) |
(→Examples) |
||
(12 intermediate revisions by 2 users not shown) | |||
Line 2: | Line 2: | ||
== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | This page is based on \cite{Ranicki2002}. A map $f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$ be an [[Oriented | + | This page is based on \cite{Ranicki2002}. A map $f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|oriented cover]] with covering map $p:\widetilde{M} \to M$. If $f$ factors through $\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$. Note that a '''choice''' of lift $\widetilde{f}$ is required in order to represent a homology class. |
− | By covering space theory (c.f. \cite{Hatcher2002|Proposition 1.33}) a map $f:N\to M$ can be lifted to $\widetilde{M}$ if and only if $f_*(\pi_1(N)) \subset p_*(\pi_1(\widetilde{M}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N)\to \pi_1(M) \to \pi$ is trivial for $q:\pi_1(M)\to \pi_1(M)/\pi_1(\widetilde{M} | + | Let $b_1$ be a basepoint of $N$. By covering space theory (c.f. \cite{Hatcher2002|Proposition 1.33}) a map $f:N\to M$ can be lifted to $\widetilde{M}$ if and only if $f_*(\pi_1(N,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi$ is trivial for $q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})$ the quotient map. |
+ | |||
+ | The group $\pi$ is well-defined for any choice of lift $\widetilde{f(b_1)}$ since $p:\widetilde{M}\to M$ is a regular covering and changing the basepoint in $\widetilde{M}$ to a different lift corresponds to conjugating $\pi_1(\widetilde{M},\widetilde{f(b_1)})$ by some $g\in \pi_1(M,f(b_1))$. | ||
</wikitex> | </wikitex> | ||
== Definition == | == Definition == | ||
<wikitex>; | <wikitex>; | ||
− | Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an [[Oriented | + | Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|oriented cover]]. A '''$\pi$-trivial map''' $f:N^n\to M^m$ is a map from an oriented manifold $N$ with basepoint $b_1$ such that the composite |
$$\xymatrix{ | $$\xymatrix{ | ||
− | \pi_1(N) \ar[r]^-{f_*} & \pi_1(M) \ar[r] & \pi | + | \pi_1(N,b_1) \ar[r]^-{f_*} & \pi_1(M,f(b_1)) \ar[r] & \pi |
}$$ | }$$ | ||
− | is trivial. | + | is trivial, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$. |
</wikitex> | </wikitex> | ||
== Properties == | == Properties == | ||
<wikitex>; | <wikitex>; | ||
A map $f:N\to M$ that factors through $\widetilde{M}$ must map all of $N$ to the same sheet of $\widetilde{M}$, hence the pullback satisfies $$f^*\widetilde{M} \cong N\times \pi.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$. | A map $f:N\to M$ that factors through $\widetilde{M}$ must map all of $N$ to the same sheet of $\widetilde{M}$, hence the pullback satisfies $$f^*\widetilde{M} \cong N\times \pi.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$. | ||
+ | </wikitex> | ||
+ | |||
+ | == Lifts and paths - two alternative perspectives == | ||
+ | <wikitex>; | ||
+ | Rather than taking a lift as part of the data for a $\pi$-trivial map we could instead take an equivalence class of paths in $M$ as is explained in this section. Since $\pi$ is the group of deck transformations of $\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$ with the group structure determined by the action of $\pi$ once a choice of lift $\widetilde{f}_{\!\id}$ has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a $\pi$-trivial map can be thought of as a choice of isomorphism | ||
+ | $$ \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g \; \; \mathrm{s.t.}\; \widetilde{f}(b_1) = g\widetilde{f}_{\!\id}(b_1).\end{array}$$ | ||
+ | Let $b$ be a basepoint of $M$. The set of homotopy classes of paths from $b$ to $f(b_1)$ is non-canonically isomophic to $\pi_1(M,b)$. An isomorphism is defined by a choice of path $[w_{id}]$ to represent the identity element: $$\begin{array}{rcl} \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}& \to & \pi_1(M,b)\\ [w] &\mapsto & [w_{id}^{-1}*w],\end{array}$$where $*$ denotes concatenation of paths and $w_{id}^{-1}$ is the path $w_{id}$ in reverse. Let $\widetilde{b}$ be a basepoint of $\widetilde{M}$ that is a lift of $b$. | ||
+ | |||
+ | Define an equivalence relation $\sim$ on this set by saying $$[w]\sim[w^\prime] \iff [w^{-1}*w^\prime] \in p_*(\pi_1(\widetilde{M},\widetilde{b})).$$ The above isomorphism given by choosing $[w_{id}]$ descends to give an isomorphism $$\{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \;\longrightarrow\; \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})\cong \pi,$$ where we use the same choice of path $[w_{id}]$ to identify $\pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b}) $ with $\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi$. | ||
+ | |||
+ | Thus a choice of lift $\widetilde{f}:N\to \widetilde{M}$ corresponds to a choice of homotopy class of paths from $b$ to $f(b_1)$ modulo $\pi_1(\widetilde{M})$. A choice of lift $\widetilde{b}$ defines a bijection of sets $$\{\widetilde{f}:N\to \widetilde{M}\} \longleftrightarrow \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$$ as follows. Given a choice of lift $\widetilde{f}$ choose any path $\widetilde{w}:I \to \widetilde{M}$ from $\widetilde{b}$ to $\widetilde{f}(b_1)$. Take the equivalence class of $p(\widetilde{w})$ which is a path in $M$ from $b$ to $f(b_1)$. Conversely given a choice of class $$[w]\in \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$$ choose any representative $w:I \to M$. This lifts uniquely to a path $\widetilde{w}$ starting at $\widetilde{b}$. Define a lift $\widetilde{f}$ by setting $\widetilde{f}(b_1):= \widetilde{w}(1)$. Note this map is well-defined since different choices of representative $w$ may differ by elements of $\pi_1(\widetilde{M},\widetilde{b})$ but their lifts will still end at the same point. | ||
+ | To sum up we have the following diagram of non-canonical isomorphisms and bijections | ||
+ | $$\xymatrix{ \{\widetilde{f}:N\to \widetilde{M}\} \ar@{<->}[rr]^-{\widetilde{b}} \ar[dr]_-{\widetilde{f}_{id}} && \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \ar[dl]^-{[w_{id}]} \\ & \pi & }.$$ Each map is obtained by making a choice and any two choices uniquely determine the third with the diagram commuting, so with two choices made the horizontal bijection is in fact an isomorphism of groups. | ||
+ | Since an oriented cover comes with a choice of lift $\widetilde{b}$ as '''part of the data''' a choice of identity lift corresponds to a choice of identity path, so it does not matter which we choose to include as part of the data for a $\pi$-trivial map. | ||
</wikitex> | </wikitex> | ||
== Examples == | == Examples == | ||
<wikitex>; | <wikitex>; | ||
− | ... | + | Let $f:S^n\looparrowright M^{2n}$ be an immersion and let $\widetilde{M}$ be the universal cover of $M$, let $s\in S^n$ and $b\in M$ be basepoints. For $n>1$, $\pi_1(S^n)=0$ so $f$ lifts to $\widetilde{M}$. An immersion $f:S^n\looparrowright M^{2n}$ is a $\pi_1(M)$-trivial immersion as soon as a lift $\widetilde{f}:S^n \looparrowright \widetilde{M}$ has been prescribed or, alternatively, once a homotopy class of paths $w:I\to M$ from $b$ to $f(s)$ has been prescribed. A pair $(f:S^n\looparrowright M^{2n}, w: I\to M | w(0)=b \; \text{and} \; w(1)=f(s))$ is often called a pointed immersion in the literature (See, for example, \cite{Lück2001|Section 4.1}). |
</wikitex> | </wikitex> | ||
− | == References == | + | ==References== |
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] |
Latest revision as of 17:26, 16 June 2014
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Introduction
This page is based on [Ranicki2002]. A map between manifolds represents a homology class . Let be an oriented cover with covering map . If factors through as then represents a homology class . Note that a choice of lift is required in order to represent a homology class.
Let be a basepoint of . By covering space theory (c.f. [Hatcher2002, Proposition 1.33]) a map can be lifted to if and only if , i.e. if and only if the composition is trivial for the quotient map.
The group is well-defined for any choice of lift since is a regular covering and changing the basepoint in to a different lift corresponds to conjugating by some .
[edit] 2 Definition
Let be an -dimensional manifold and let be an oriented cover. A -trivial map is a map from an oriented manifold with basepoint such that the composite
is trivial, together with a choice of lift .
[edit] 3 Properties
[edit] 4 Lifts and paths - two alternative perspectives
Rather than taking a lift as part of the data for a -trivial map we could instead take an equivalence class of paths in as is explained in this section. Since is the group of deck transformations of , the set of lifts is non-canonically isomorphic to with the group structure determined by the action of once a choice of lift has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a -trivial map can be thought of as a choice of isomorphism
Tex syntax error
To sum up we have the following diagram of non-canonical isomorphisms and bijections
Since an oriented cover comes with a choice of lift as part of the data a choice of identity lift corresponds to a choice of identity path, so it does not matter which we choose to include as part of the data for a -trivial map.
[edit] 5 Examples
Let be an immersion and let be the universal cover of , let and be basepoints. For , so lifts to . An immersion is a -trivial immersion as soon as a lift has been prescribed or, alternatively, once a homotopy class of paths from to has been prescribed. A pair is often called a pointed immersion in the literature (See, for example, [Lück2001, Section 4.1]).
[edit] 6 References
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001