Classifying Poincaré complexes via fundamental triples

From Manifold Atlas
Jump to: navigation, search

The user responsible for this page is Bea Bleile. No other user may edit this page at present.

This page is being refereed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments.

Let \textup{CW}_0 be the category of reduced CW-complexes, that is CW-complexes with 0-skeleton a point which is the base-point, and base-point preserving cellular maps. Given an object, X, in \textup{CW}_0, let \widehat X be the universal cover of X, and let 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]. To obtain a functor we assume that each object X in \textup{CW}_0 is endowed with a base point in the universal covering \widehat X over the base point of X. Then a map f: X \rightarrow Y in \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) ensuring that C(\widehat X) is functorial in X.

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

Tex syntax error
As first defined by Wall [Wall1967a], a Poincaré duality complex of formal dimension n (
Tex syntax error
-complex), X = (X, \omega_X, [X]), consists of an object X in
Tex syntax error
with finitely presented fundamental group \pi_1 X, an orientation character, \omega_X, viewed as a group homomorphism \omega_X: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z and a fundamental class
Tex syntax error
, such that
Tex syntax error
is an isomorphism of abelian groups for every r \in \mathbb Z and every left \mathbb Z [\pi_1 X]-module M. An oriented morphism of
Tex syntax error
-complexes f: (X, \omega_X, [X]) \rightarrow (Y, \omega_Y, [Y]) is a morphism f: X \rightarrow Y in
Tex syntax error
, such that \omega_X = \omega_Y\pi_1(f) and f_{\ast}[X] = [Y]. The category
Tex syntax error
is the category consisting of
Tex syntax error
-complexes and oriented or degree 1 morphisms of
Tex syntax error
—complexes.
Let k-types be the full subcategory of the homotopy category
Tex syntax error
consisting of
Tex syntax error
-complexes X in
Tex syntax error
with \pi_i(X) = 0 for i > k and let
Tex syntax error
be the k-th Postnikov functor. For n \geq 3, a fundamental triple T = (X, \omega, t) of formal dimension n consists of an (n-2)-type X, a homomorphism \omega: \pi_1X \rightarrow \mathbb Z / 2 \mathbb Z and an element
Tex syntax error
. A morphism (X, \omega_X, t_X) \rightarrow (Y, \omega_Y, t_Y) between fundamental triples is a homotopy class \{f\}: X \rightarrow Y of maps of the (n-2)-types, such that \omega_X = \omega_Y \pi_1(f) and f_{\ast}(t_X) = t_Y. We obtain the category
Tex syntax error
of fundamental triples T of formal dimension n.
Every degree 1 morphism Y \rightarrow X in
Tex syntax error
induces a surjection \pi_1Y \rightarrow \pi_1X on fundamental groups, see for example [Browder1972a]. The category
Tex syntax error
is the subcategory consisting of all morphisms inducing surjections on fundamental groups, and we obtain the functor
Tex syntax error

Theorem 1.1. The functor \tau_+ reflects isomorphisms and is full for n \geq 3, that is, \tau is surjective onto sets of morphisms and \tau(f) is an isomorphism if and only if 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_+ reflects isomorphisms. To show that \tau_+ is full requires work. Given
Tex syntax error
-complexes Y and X, n \geq 3, and a morphism f: \tau_+ Y \rightarrow \tau_+ X in
Tex syntax error
, we first construct a chain map \xi: \widehat C(Y) \rightarrow \widehat C(X) preserving fundamental classes, that is, \xi_{\ast}[Y] = [X]. Then we use the category
Tex syntax error
of homotopy systems of order (k+1) introduced in [Baues1991] to realize \xi by a map \overline f: Y \rightarrow X with \tau_+(\overline f) = f.
Theorem 1.2. Take n \geq 3. Two
Tex syntax error
-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 by Thomas [Thomas1969], Swarup [Swarup1974], and Hendriks [Hendriks1977], to arbitrary dimension. It also establishes Turaev's conjecture [Turaev1989] on
Tex syntax error
-complexes whose (n-2)-type is an Eilenberg-Mac Lane space K(\pi_1X,1). Theorem 1.1 also yields a criterion for the existence of a map of degree one between
Tex syntax error
-complexes, recovering Swarup's result for maps between 3-manifolds and Hendriks' result for maps between
Tex syntax error
-complexes.

Special cases of Theorem 1.1 and 1.2 were proved by Hambleton and Kreck [Hambleton&Kreck1988] for 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
Tex syntax error
-complex, X, is completely determined by its quadratic form. The 2-type of such an X, with \pi_2(X) = A, is an Eilenberg-Mac Lane space
Tex syntax error
with
Tex syntax error
. The image of the fundamental class, [X], under the secondary boundary homomorphism in Whitehead's Certain Exact Sequence is the quadratic form of X. Hence, in this case, the functor, \tau, coincides with the functor
Tex syntax error
of Theorem 2.1.8 in [Baues2003].
Corollary 1.3. For n \geq 3,
Tex syntax error
-complexes X and Y and a map f: P_{n-2}X \rightarrow P_{n-2}Y, there is a degree 1 map, \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 induces a surjection on fundamental groups, is compatible with the orientations \omega_X and \omega_Y, that is, \omega_X = \omega_Y \pi_1(f), and

\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
Tex syntax error
-complex X, let
Tex syntax error
be the group of oriented homotopy equivalences of X in
Tex syntax error
and
Tex syntax error
the group of automorphisms of the triple \tau(X) in
Tex syntax error
. Then the latter is a subgroup of
Tex syntax error
and there is a surjection of groups
Tex syntax error

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

Remark 1.5. For n \geq 3, let [\frac{n}{2}] be the integer part of \frac{n}{2}. Associating with a
Tex syntax error
-complex, X, the pre-fundamental triple (P_{[\frac{n}{2}]}X, \omega_X, p_{[\frac{n}{2}]\ast}[X]), an orientation preserving map between
Tex syntax error
-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
Tex syntax error
-complex, as is demonstrated by the fake products X = (S^n \vee S^n) \cup_{\alpha} e^{2n}, where \alpha is the sum of the Whitehead product, [\iota_1 , \iota_2], of the inclusions of the factors in the wedge product, and an element \iota_1 \beta with a non-trivial element \beta \in \pi_{2n-1}(S^n) having trivial Hopf invariant. Pre-fundamental triples coincide with the fundamental triple for n = 3 and 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

2 External links

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox