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허곡신거 [902596] · MS 2019 (수정됨) · 쪽지

2025-02-01 15:04:44
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1

Whitehead Torsion

게시글 주소: https://image.orbi.kr/00071714315

Motivation: "그들의 대화" 에서 최근에 나오는 핵심 용어들 중 하나가 Whitehead torsion이라는 것인데, 이러한 것을 고려하는 이유에 대해서 먼저 설명하기로. 모든 것의 기원은 소위 "cobordism theory"에 기반을 함: Let $M$ and $N$ be smooth closed manifolds of dimension $n$. An \textit{$h$-cobordism} from $M$ to $N$ is a compact smooth manifold $B$ of dimension $(n+1)$ with boundary $\partial B \cong M\coprod N$ having the property that the inclusion maps from $M$ and $N$ to $B$ are homotopy equivalences. If $n\geq 5$ and the manifold $M$ is simply connected, then the Smale's $h$-cobordism theorem says that $B$ is diffeomorphic to a product $M\times [0,1]$ (and, in particular, $M$ is diffeomorphic to $N$).

다시 말해서, cobordism은 두 다양체 M,N을 자연스럽게 interpolate하는 것을 말함. 여기서 $h$는 homotopy를 말하고, 그 이유는 up to homotopy로 interpolate을 했기 때문. 5차원 이상에서는 이것이 어떤 면에서 ``trivial'' 하다는 것을 말함. Smale이 이 정리를 이용해서 5차원 이상에서의 Poincare Conjecture를 풀었음 (예에에전에 한번 이거 관련 글 썼던 것 같음).

이러한 좋은 이유에 의해서 cobordism theory를 not simply connected인 경우에는 어떻게 사용할 수 있을까 사람들이 고심을 하고, 그렇게 나온 것이 s-cobordism theory임. 이것을 좀 더 자세히 설명하기 위해서는 몇몇 정의들이 필요함:

Definition. Let $X$ be a finite simplicial complex. Suppose that there is a simplex $\sigma\subset X$ containing a face $\sigma_0\subset\sigma$ such that $\sigma$ is not contained in any larger simplex of $X$, and $\sigma_0$ is not contained in any larger simplex other than $\sigma$. Let $Y\subset X$ be the subcomplex obtained by removing the interiors of $\sigma$ and $\sigma_0$. Then the inclusion $\iota:Y\hookrightarrow X$ is a homotopy equivalence. In this situation, we will say that $\iota$ is an \textit{elementary expansion}. Note that $Y$ is a retract of $X$; a retraction $X$ onto $Y$ will be called the \textit{elementary collapse}.

Definition. Let $f:Y\to X$ be a map between finite simplicial complexes. We will say that $f$ is a \textit{simple homotopy equivalence} if it is homotopic to a finite composition of elementary expansions and elementary collapses.

모든 compact smooth manifold는 PL 이기 때문에 finite simplicial complex structure를 갖게 됨. 따라서, smooth manifold의 경우에는 simple homotopy equivalence라는 것을 이야기할 수 있음.

s-cobordism theorem. Let $B$ be an $h$-cobordism theorem between smooth manifolds $M$ and $N$ of dimension $\geq 5$. Then $B$ is diffeomorphic to a product $M\times[0,1]$ if and only if the inclusion map $M\hookrightarrow B$ is a simple homotopy equivalence.

이제 이 s-cobordism theorem을 적용하기 위해서는 언제 homotopy equivalence of smooth manifolds $f:X\to Y$가 simple homotopy equivalence인지 알아내는 것. 이걸 Whitehead가 해결했는데, 각각의 homotopy equivalence $f:X\to Y$에 대해서, 어떤 algebraic invariant $\tau(f)$ called the \textit{Whitehead torsion} of $f$ 라고 하고, 이 torsion은 \textit{Whitehead group} of $X$라고 불리는 특정 abelian group $\mathrm{Wh}(X)$에 존재함. 이 torsion이 정확히 simple homotopy equivalence의 obtruction임. 다시 말해서, $\tau(f)$ vanishes if and only if $f$ is a simple homotopy equivalence.

이제 이 Whitehead torsion이 구체적으로 무엇인지 알아보기로. 먼저 앞에서 정의한 simple homotopy equivalence의 정의를 조금 더 구체적으로 적어봄.

Construction 1. Let $D^n$ denote the closed unit ball of dimension $n$ and let $S^{n-1} = \partial D^n$ denote its boundary. We will regard $S^{n-1}$ as decomposed into hemispheres $S^{n-1}_-$ and $S^{n-1}_+$ which meet along the ``equator'' $S^{n-2} = S^{n-1}_-\cap S^{n-1}_+$.

Let $Y$ be a CW complex equipped with a map $f:(S^{n-1}_-,S^{n-2})\to (Y^{n-1},Y^{n-2})$. Then the pushout $Y\coprod_{S^{n-1}_-}D^n$ has the structure of a CW complex which is obtained from $Y$ by adding two more cells: an $(n-1)$-cell given by the image of the interior of $S^{n-1}_+$ (attached via the map $f|_{S^{n-2}}:S^{n-2}\to Y^{n-2}$) and an $n$-cell given by the image of the interior of $D^n$ attached via the map

$$S^{n-1} = S^{n-1}_-\coprod_{S^{n-2}}S^{n-1}_+\to Y^{n-1}\coprod_{S^{n-2}}S^{n-1}_+.$$

In this case, we will refer to the CW complex $Y\coprod_{S^{n-1}_-}D^n$ as an \textit{elementary expension} of $Y$, and to the inclusion map $Y\hookrightarrow Y\coprod_{S^{n-1}_-}D^n$ as an \textit{elementary expansion}.

The hemisphere $S^{n-1}_-\subset D^n$ is a (deformation) retract of $D^n$. Composition with any retraction induces a (celluler) $c:Y\coprod_{S^{n-1}_-}D^n\to Y$, which we will refer to as an \textit{elementary collapse}. Note that the homotopy class of $c$ does not depend on the choice of retraction $D^n\to S^{n-1}_-$.

Definition 2. Let $f:X\to Y$ be a map of CW complexes. We will say that $f$ is a \textit{simple homotopy equivalence} if it is homotopic to a finite composition

$$X = X_0\xrightarrow{f_1}X_1\xrightarrow{f_2}X_2\to\cdots\xrightarrow{f_n}X_n = Y,$$

where each $f_i$ is either an elementary expansion or an elementary collapse.

We say that two finite CW complexes are \textit{simple homotopy equivalent} if there exists a simple homotopy equivalence between them.

Example. Let $X$ and $Y$ be finite CW complexes and let $f:X\to Y$ be a continuous map. We let $M(f) = (X\times[0,1])\coprod_{X\times\{1\}}Y$ denote the mapping cylinder of $f$. If $f$ is a celluler map, then we can regard $M(f)$ as a finite CW complex (taking the cells of $M(f)$ to be the cells of $Y$ together with cells of the form $e\times\{0\}$ and $e\times(0,1)$, where $e$ is a cell of $X$). The inclusion $Y\hookrightarrow M(f)$ is always a simple homotopy equivalence: in fact, it can be obtained by a finite sequence of elementary expansions which simultaneously add pairs of cells $e\times\{0\}$ and $e\times(0,1)$ (where we add cells in order of increasing dimension).

Note that the map $f$ is homotopic to a composition

$$X\simeq X\times\{0\}\xrightarrow{\iota}M(f)\xrightarrow{r}Y,$$

where $r$ is the canonical retraction from $M(f)$ onto $Y$ (which can be obtained by composing a finite sequence elementary collapses). It follows that $f$ is a simple homotopy equivalence if and only if $\iota$ is a simple homotopy equivalence. Consequently, when we are studying the question of whether or not some map $f$ is a simple homotopy equivalence, there is no real loss of generality in assuming that $f$ is the inclusion of a subcomplex.

Rmk. Celluler approximation theorem says that any continuous map between CW complexes can be homotoped to be a celluler map. In particular, the above example holds for general continuous map $f$.

Simple homotopy equivalence는 homotopy equivalence인 것은 눈으로 쉽게 확인할 수 있다. s-cobordism theorem을 적용하기 위해서, 우리는 그 역이 필요하다.

Question. Let $f:X\to Y$ be a homotopy equivalence between finite CW complexes. Is $f$ a simple homotopy equivalence? If not, how can we tell?

앞서 말했듯이, 이 질문에 대한 대답은 정확히 Whitehead torsion. 이걸 만들기 위해서 먼저 몇몇 정의들이 필요함. 지금까지는 상당히 자명한 것들만 나왔는데 지금부터는 약간 익숙치 않은 것들이 등장하기 시작함.

Definition.

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Bonzo · 1001001 · 02/01 15:05 · MS 2020

세줄요약좀

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허곡신거 · 902596 · 02/01 15:08 · MS 2019

1. 멋진 대화를 하고 있는 사람들 대화에 끼고 싶다
2. 대화에 끼려면 그 사람들이 무슨 말을 하는지 이해해야 한다
3. 따라서 그들의 대화 중에 나오는 용어들을 먼저 알아볼까 고민중이다

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