Whitehead Torsion

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. Let $R$ be a ring (not necessarily commutative). For each integer $n\geq 0$, we let $\mathrm{GL}_n(R)$ denote the group of automorphisms of $R^n$ as a right $R$-module. Every automorphism $\alpha$ of $R^n$ extends to an automorphism $\alpha\oplus 1_R$ of $R^{n+1}$; this construction yields inclusions

$$\mathrm{GL}_1(R)\hookrightarrow\mathrm{GL}_2(R)\hookrightarrow\mathrm{GL}_3(R)\hookrightarrow\cdots.$$

We let $\mathrm{GL}_\infty(R)$ denote the direct limit of this sequence, and we define $K_1(R)$ to be the abelianization of $\mathrm{GL}_{\infty}(R)$.

Remark. 자 이 construction이 뭔가 natural 하면서도, 한편으로는 좀 어색한데, 만약 $R$이 commutative ring이라면, determinant function

$$\det:\mathrm{GL}_n(R)\to R^\times$$

이 group homomorphism을 줌. 자명히 det은 위의 direct system과 compatible 하기 때문에, direct limit으로 pass가 가능하고, $\det:K_1(R)\to R^\times$ 라는 group homomorphism을 induce함. 만약 $R$이 field 이거나 $\Bbb Z$ 라면, 이 det은 isomorphism이라는 것을 알 수 있음. (하지만 우리의 경우에 $R$은 group ring $\Bbb ZG$라 이 경우는 아님.)

Let $R$ be a ring. A \textit{based chain complex over $R$} is a bounded chain complex of $R$-modules

$$\cdots F_n\xrightarrow{d}F_{n-1}\xrightarrow{d}F_{n-2}\to\cdots,$$

together with a choice of unordered basis for each $F_m$ (so that each $F_m$ is a free $R$-module). In this case, we let $\chi(F_\ast)$ denote the sum $\sum(-1)^mr_m$, where $r_m$ denotes the cardinality of the (chosen) basis of $F_m$. We will refer to $\chi(F_\ast)$ as the \textit{Euler characteristic} of $(F_\ast,d)$.

Remark. If $R$ is a nonzero commutative ring, then the Euler characteristic $\chi(F_\ast)$ is independent of the choice of basis of the modules $F_\ast$. For a general noncommutative ring $R$, this need not be the case.

Let $(F_\ast,d)$ be a based chain complex over $R$ which is \textit{acyclic}, i.e., the homology of $(F_\ast,d)$ vanishes. Since each $F_m$ is a free $R$-module, it then follows that the identity map $1:F_\ast\to F_\ast$ is chain homotopic to zero, i.e., there exists a map $h:F_\ast\to F_{\ast+1}$ satisfying $dh+hd = 1$. We let $F_{\text{even}}:=\bigoplus_n F_{2n}$ and $F_{\text{odd}}:=\bigoplus_{n}F_{2n+1}$.

Lemma. In the above setting, the map $d+h:F_{\text{even}}\to F_{\text{odd}}$ is an isomorphism. $\square$

The specification of a basis for each $F_m$ determines isomorphisms

$$F_{\text{even}}\simeq R^a,\quad F_{\text{odd}}\simeq R^b$$

for some integers $a,b\geq 0$, which are well-defined up to the action of permutation matrices.

Definition. Let $\tilde{K}_1(R)$ denote the quotient of $K_1(R)$ by the subgroup $\langle \pm 1\rangle$. If $(F_\ast,d)$ is an acyclic based complex with $\chi(F_\ast) = 0$, we define the \textit{torsion} of $(F_\ast,d)$ to be the image of $d+h\in\mathrm{GL}_a(R)$ under the map $\mathrm{GL}_a(R)\to\mathrm{GL}_\infty(R)\to\tilde{K}_1(R)$. It can be shown that this definition does not depend on the ordering of the basis elements of $F_\ast$. We will denote the torsion of $(F_\ast,d)$ by $\tau(F_\ast)$.

Lemma. In the above definition, the torsion $\tau(F_\ast)$ is well-defined, i.e., does not depend on the choice of nulhomotopy $h$. $\square$

Definition. Let $f:X_\ast\to Y_\ast$ be a map of chain complexes over a ring $R$. The \textit{mapping cone of $f$} is defined to be the chain complex

$$C(f)_\ast = X_{\ast -1}\oplus Y_\ast$$

with differential $d(x,y) = (-dx,f(x)+dy)$. Note that if $X_\ast$ and $Y_\ast$ are based complexes, then we can regard $C(f)_\ast$ as a based complex (where we fix some convention for how our bases should be ordered).

Suppose that we have $\chi(X_\ast,d) = \chi(Y_\ast,d)$ and that $f$ is a \textit{quasi-isomorphism}, i.e. it induces an isomorphism on homology. Then $\chi(C(f)_\ast,d) = 0$ and $C(f)_\ast$ is acyclic. We define the \textit{torsion of $f$} to be the element $\tau(f) = \tau(C(f)_\ast,d)\in K_1(R)$.

이제 Whitehead torsion을 정의할 것인데, 정의의 동기를 주는 example을 먼저 보기로 함.

Example. Let $(F_\ast,d)$ be an acyclic based complex with $\chi(F_\ast) = 0$, and let $f$ be the identity map from $F_\ast$ to itself. Then the mapping cone $C(f)_\ast$ has an explicit nulhomotopy given by $(x,y)\mapsto(y,0)$. Identify $C(f)_{\text{even}}$ and $C(f)_{\text{odd}}$ with $F_\ast$, so that $d+h$ is given by

$$(x,y)\mapsto (y-dx,x+dy).$$

This map is given by a permutation matrix modulo the filtration by degree, so we have $\tau(f) = 1\in\tilde{K}_1(R)$.

Suppose $X,Y$ are finite CW complexes and that we are given a homotopy equivalence $f:X\to Y$. For simplicity, we assume that $X,Y$ are connected. We fix a base point $x\in X$ and set $G =\pi_1(X,x)\simeq \pi_1(Y,f(x))$. Let $\tilde{Y}$ be a universal cover of $Y$ and let $\tilde{X} = X_{\times_Y}\tilde{Y}$ be the corresponding universal cover of $X$, so that $G$ acts on $\tilde{X}$ and $\tilde{Y}$ by deck transformations. Let us further assume that $f$ is a celluler map. Then $f$ induces a map of cellular chain complexes

$$\lambda:C_\ast(\tilde{X};\Bbb Z)\to C_\ast(\tilde{Y};\Bbb Z).$$

Note that we can regard $C_\ast(\tilde{X},\Bbb Z)$ and $C_\ast(\tilde{Y};\Bbb Z)$ as chain complexes of free $\Bbb Z[G]$-modules, with basis elements in bijection with the cells of $X$ and $Y$ respectively. Since $f$ is a homotopy equivalence, the map $\lambda$ is a quasi-isomorphism. We may therefore consider the torsion $\tau(\lambda)\in\tilde{K}_1(\Bbb Z[G])$. However, it is not well-defined: in order to extract an element of $C_\ast(\tilde{X};\Bbb Z)$ from a cell $e\subset X$, we need to choose a cell of $\tilde{X}$ lying over $e$ (which is ambiguous up to the action of $G$) and an orientation of the cell $e$ (which is ambiguous up to a sign). This motivates the following:

Definition. Let $G$ be a group. The \textit{Whitehead group} $\mathrm{Wh}(G)$ of $G$ is the quotient of $K_1(\Bbb Z[G])$ by elements of the form $[\pm g]$, where $g\in G$.

If $f:X\to Y$ is a celluler homotopy equivalence of connected finite CW complexes, we define the \textit{Whitehead torsion} $\tau(f)\in\mathrm{Wh}(G)$ to be the image in $\mathrm{Wh}(G)$ of the torsion of the induced map

$$\lambda:C_\ast(\tilde{X};\Bbb Z)\to C_\ast(\tilde{Y};\Bbb Z).$$