Classifying Poincaré complexes via fundamental triples

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Let $\textup{CW}_0$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\textup{CW}_0$ be the category of reduced CW-complexes, that is CW-complexes with $0$$0$-skeleton a point which is the base-point, and base-point preserving cellular maps. Given an object, $X$$X$, in $\textup{CW}_0$$\textup{CW}_0$, let $\widehat X$$\widehat X$ be the universal cover of $X$$X$, and let $C(\widehat X)$$C(\widehat X)$ be its cellular chain complex viewed as a complex of left modules over the integer group ring $\mathbb Z[\pi_1 X]$$\mathbb Z[\pi_1 X]$. To obtain a functor we assume that each object $X$$X$ in $\textup{CW}_0$$\textup{CW}_0$ is endowed with a base point in the universal covering $\widehat X$$\widehat X$ over the base point of $X$$X$. Then a map $f: X \rightarrow Y$$f: X \rightarrow Y$ in $\textup{CW}_0$$\textup{CW}_0$ induces a unique base point preserving covering map which, in turn, induces a map $f_{\ast}: C(\widehat X) \rightarrow C(\widehat Y)$$f_{\ast}: C(\widehat X) \rightarrow C(\widehat Y)$ ensuring that $C(\widehat X)$$C(\widehat X)$ is functorial in $X$$X$.

Given a homomorphism $\omega: \pi_1 X \rightarrow {\mathbb{Z}} / 2 {\mathbb{Z}} = \{0,1\}$$\omega: \pi_1 X \rightarrow {\mathbb{Z}} / 2 {\mathbb{Z}} = \{0,1\}$, we define the anti-isomorphism of group rings, $\overline{\phantom{x}}: {\mathbb Z[\pi_1 X]} \rightarrow {\mathbb Z[\pi_1 X]}$$\overline{\phantom{x}}: {\mathbb Z[\pi_1 X]} \rightarrow {\mathbb Z[\pi_1 X]}$, by $\overline g = (-1)^{\omega(g)} g^{-1}$$\overline g = (-1)^{\omega(g)} g^{-1}$ for $g \in \pi$$g \in \pi$ and extending linearly to all of $\Zz[\pi]$$\Zz[\pi]$. Then, for a left ${\mathbb Z[\pi_1 X]}$${\mathbb Z[\pi_1 X]}$-module, $M$$M$, the right module $M^{\omega}$$M^{\omega}$ has the same underlying abelian group and action given by $m.\lambda = \overline{\lambda}.m$$m.\lambda = \overline{\lambda}.m$ for $m \in M$$m \in M$ and $\lambda \in \mathbb Z[\pi_1 X]$$\lambda \in \mathbb Z[\pi_1 X]$. For $n, k \in \mathbb Z$$n, k \in \mathbb Z$, we put

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$$\displaystyle \begin{array}{rcl} {\rm H}_n(X;M^{\omega}) & = & {\rm H}_n(M^{\omega} \otimes_{\mathbb Z[\pi_1 X]} C(\widehat X)) \quad \rm{and} \\ {\rm H}^k(X;M) & = & {\rm H}_{-k}(\mbox{Hom}_{\mathbb Z[\pi_1 X]}(C(\widehat X),M)). \end{array}$$

As first defined by Wall [Wall1967a], a Poincaré duality complex of formal dimension $n$$n$ (

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$\rm PD^n$-complex), $X = (X, \omega_X, [X])$$X = (X, \omega_X, [X])$, consists of an object $X$$X$ in

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${\rm\bf CW}_0$ with finitely presented fundamental group $\pi_1 X$$\pi_1 X$, an orientation character, $\omega_X$$\omega_X$, viewed as a group homomorphism $\omega_X: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z$$\omega_X: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z$ and a fundamental class

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$[X] \in {\rm H}_n(X; \mathbb Z^{\omega})$, such that

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$$\displaystyle \cap [X]: {\rm H}^{r}(X; M) \rightarrow {\rm H}_{n-r}(X; M^{\omega}); \quad \alpha \mapsto \alpha \cap [X]$$

is an isomorphism of abelian groups for every $r \in \mathbb Z$$r \in \mathbb Z$ and every left $\mathbb Z [\pi_1 X]$$\mathbb Z [\pi_1 X]$-module $M$$M$. An oriented morphism of

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$\rm PD^n$-complexes $f: (X, \omega_X, [X]) \rightarrow (Y, \omega_Y, [Y])$$f: (X, \omega_X, [X]) \rightarrow (Y, \omega_Y, [Y])$ is a morphism $f: X \rightarrow Y$$f: X \rightarrow Y$ in

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${\rm'''CW'''}_0$, such that $\omega_X = \omega_Y\pi_1(f)$$\omega_X = \omega_Y\pi_1(f)$ and $f_{\ast}[X] = [Y]$$f_{\ast}[X] = [Y]$. The category

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${\rm\bf PD}^n_+$ is the category consisting of

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${\rm PD}^n$-complexes and oriented or degree $1$$1$ morphisms of

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${\rm PD}^n$—complexes.
Let $k$$k$-types be the full subcategory of the homotopy category

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${\rm\bf CW}_0 /\simeq$ consisting of

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$\rm CW$-complexes $X$$X$ in

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${\rm\bf CW}_0$ with $\pi_i(X) = 0$$\pi_i(X) = 0$ for $i > k$$i > k$ and let

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$$\displaystyle P_k: {\rm\bf CW}_0 \rightarrow k{\rm\bf-types}$$

be the $k$$k$-th Postnikov functor. For $n \geq 3$$n \geq 3$, a fundamental triple $T = (X, \omega, t)$$T = (X, \omega, t)$ of formal dimension $n$$n$ consists of an $(n-2)$$(n-2)$-type $X$$X$, a homomorphism $\omega: \pi_1X \rightarrow \mathbb Z / 2 \mathbb Z$$\omega: \pi_1X \rightarrow \mathbb Z / 2 \mathbb Z$ and an element

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$t \in {\rm H}_n(X; \mathbb Z^{\omega})$. A morphism $(X, \omega_X, t_X) \rightarrow (Y, \omega_Y, t_Y)$$(X, \omega_X, t_X) \rightarrow (Y, \omega_Y, t_Y)$ between fundamental triples is a homotopy class $\{f\}: X \rightarrow Y$$\{f\}: X \rightarrow Y$ of maps of the $(n-2)$$(n-2)$-types, such that $\omega_X = \omega_Y \pi_1(f)$$\omega_X = \omega_Y \pi_1(f)$ and $f_{\ast}(t_X) = t_Y$$f_{\ast}(t_X) = t_Y$. We obtain the category

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${\rm\bf Trp}^n$ of fundamental triples $T$$T$ of formal dimension $n$$n$.
Every degree $1$$1$ morphism $Y \rightarrow X$$Y \rightarrow X$ in

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${\rm\bf PD}^n_+$ induces a surjection $\pi_1Y \rightarrow \pi_1X$$\pi_1Y \rightarrow \pi_1X$ on fundamental groups, see for example [Browder1972a]. The category

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${\rm\bf Trp}^n_+ \subset {\rm\bf Trp}^n$ is the subcategory consisting of all morphisms inducing surjections on fundamental groups, and we obtain the functor

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$$\displaystyle \tau_+: {\rm\bf PD}^n_+ /\simeq \ \longrightarrow {\rm\bf Trp}^n_+ \quad X \longmapsto (P_{n-2}X, \omega_X, p_{n-2 \ast}[X]).$$

Theorem 1.1. The functor $\tau_+$$\tau_+$ reflects isomorphisms and is full for $n \geq 3$$n \geq 3$, that is, $\tau$$\tau$ is surjective onto sets of morphisms and $\tau(f)$$\tau(f)$ is an isomorphism if and only if $f$$f$ is an isomorphism..

Theorem 1.1 is Theorem 3.1 in [Baues&Bleile2008]. It follows directly from Poincaré duality and Whitehead's Theorem that the functor $\tau_+$$\tau_+$ reflects isomorphisms. To show that $\tau_+$$\tau_+$ is full requires work. Given

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${\rm PD}^n$-complexes $Y$$Y$ and $X$$X$, $n \geq 3$$n \geq 3$, and a morphism $f: \tau_+ Y \rightarrow \tau_+ X$$f: \tau_+ Y \rightarrow \tau_+ X$ in

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${\rm\bf Trp}^n_+$, we first construct a chain map $\xi: \widehat C(Y) \rightarrow \widehat C(X)$$\xi: \widehat C(Y) \rightarrow \widehat C(X)$ preserving fundamental classes, that is, $\xi_{\ast}[Y] = [X]$$\xi_{\ast}[Y] = [X]$. Then we use the category

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${\rm\bf H}^c_{k+1}$ of homotopy systems of order $(k+1)$$(k+1)$ introduced in [Baues1991] to realize $\xi$$\xi$ by a map $\overline f: Y \rightarrow X$$\overline f: Y \rightarrow X$ with $\tau_+(\overline f) = f$$\tau_+(\overline f) = f$.

Theorem 1.2. Take $n \geq 3$$n \geq 3$. Two

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${\rm PD}^n$-complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic.

Theorem 1.2 is Theorem 3.2 in [Baues&Bleile2008] and extends results for dimension $3$$3$ by Thomas [Thomas1969], Swarup [Swarup1974], and Hendriks [Hendriks1977], to arbitrary dimension. It also establishes Turaev's conjecture [Turaev1989] on

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${\rm PD}^n$-complexes whose $(n-2)$$(n-2)$-type is an Eilenberg-Mac Lane space $K(\pi_1X,1)$$K(\pi_1X,1)$. Theorem 1.1 also yields a criterion for the existence of a map of degree one between

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${\rm PD}^n$-complexes, recovering Swarup's result for maps between $3$$3$-manifolds and Hendriks' result for maps between

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${\rm PD}^3$-complexes.

Special cases of Theorem 1.1 and 1.2 were proved by Hambleton and Kreck [Hambleton&Kreck1988] for $n=4$$n=4$. Teichner extended their approach to the non-oriented case in his thesis [Teichner1992]. Cavicchioli and Spaggiari [Cavicchioli&Spaggiari2001] studied the homotopy type of finite oriented Poincar\'e complexes in even dimensions.

By early work of Milnor [Milnor1958] and Whitehead [Whitehead1949], the homotopy type of a simply-connected

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${\rm PD}^4$-complex, $X$$X$, is completely determined by its quadratic form. The $2$$2$-type of such an $X$$X$, with $\pi_2(X) = A$$\pi_2(X) = A$, is an Eilenberg-Mac Lane space

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${\rm K}(A, 2)$ with

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${\rm H}_4({\rm K}(A, 2)) = \Gamma(A)$. The image of the fundamental class, $[X]$$[X]$, under the secondary boundary homomorphism in Whitehead's Certain Exact Sequence is the quadratic form of $X$$X$. Hence, in this case, the functor, $\tau$$\tau$, coincides with the functor

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${\rm H}_{\ast}$ of Theorem 2.1.8 in [Baues2003].

Corollary 1.3. For $n \geq 3$$n \geq 3$,

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${\rm PD}^n$-complexes $X$$X$ and $Y$$Y$ and a map $f: P_{n-2}X \rightarrow P_{n-2}Y$$f: P_{n-2}X \rightarrow P_{n-2}Y$, there is a degree $1$$1$ map, $\overline f$$\overline f$, rendering

$\displaystyle \xymatrix{ X \ar[r]^-{p_{n-2}} \ar@{..>}[d]_{\overline f} & P_{n-2}X \ar[d]^f \\ Y \ar[r]^-{p_{n-2}} & P_{n-2}Y}$

homotopy commutative, if and only if $f$$f$ induces a surjection on fundamental groups, is compatible with the orientations $\omega_X$$\omega_X$ and $\omega_Y$$\omega_Y$, that is, $\omega_X = \omega_Y \pi_1(f)$$\omega_X = \omega_Y \pi_1(f)$, and

$\displaystyle f_{\ast}p_{n-2 \ast}[X] = p_{n-2 \ast}[Y].$ $$\displaystyle f_{\ast}p_{n-2 \ast}[X] = p_{n-2 \ast}[Y].$$

Corollary 1.3 is Corollary 3.3 in [Baues&Bleile2008].

Corollary 1.4. Given a

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${\rm PD}^3$-complex $X$$X$, let

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${\rm Aut}_+(X)$ be the group of oriented homotopy equivalences of $X$$X$ in

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${\rm\bf PD}^3_+ / \simeq$ and

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${\rm Aut}(\tau(X))$ the group of automorphisms of the triple $\tau(X)$$\tau(X)$ in

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${\rm\bf Trp}^3_+$. Then the latter is a subgroup of

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${\rm Aut}(\pi_1X)$ and there is a surjection of groups

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$$\displaystyle \tau_+: {\rm Aut}_+(X) \rightarrow {\rm Aut}(\tau(X)).$$

Corollary 1.4 is included in Corollary 4.4 in [Baues&Bleile2008].

Remark 1.5. For $n \geq 3$$n \geq 3$, let $[\frac{n}{2}]$$[\frac{n}{2}]$ be the integer part of $\frac{n}{2}$$\frac{n}{2}$. Associating with a

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${\rm PD}^n$-complex, $X$$X$, the pre-fundamental triple $(P_{[\frac{n}{2}]}X, \omega_X, p_{[\frac{n}{2}]\ast}[X])$$(P_{[\frac{n}{2}]}X, \omega_X, p_{[\frac{n}{2}]\ast}[X])$, an orientation preserving map between

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${\rm PD}^n$-complexes is a homotopy equivalence if and only if the induced map between pre-fundamental triples is an isomorphism. However, pre-fundamental triples do not determine the homotopy type of a

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${\rm PD}^n$-complex, as is demonstrated by the fake products $X = (S^n \vee S^n) \cup_{\alpha} e^{2n}$$X = (S^n \vee S^n) \cup_{\alpha} e^{2n}$, where $\alpha$$\alpha$ is the sum of the Whitehead product, $[\iota_1 , \iota_2]$$[\iota_1 , \iota_2]$, of the inclusions of the factors in the wedge product, and an element $\iota_1 \beta$$\iota_1 \beta$ with a non-trivial element $\beta \in \pi_{2n-1}(S^n)$$\beta \in \pi_{2n-1}(S^n)$ having trivial Hopf invariant. Pre-fundamental triples coincide with the fundamental triple for $n = 3$$n = 3$ and $n = 4$$n = 4$. It remains an open problem to enrich the structure of a pre-fundamental triple to obtain an analogue of Theorem 1.2.

1 References

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2 External links

• The Encyclopedia of Mathematics article about Poincaré complexes
• The Wikipedia page about Poincaré complexes
• Poincaré complex in n-Lab