Orientation of manifolds in generalized cohomology theories
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One of classical definitions of orientability of a closed connected manifold is the existence of the fundamental class . It is clear that this definition is very suitable to generalize it to generalize (co)homology theories, and this generalization turns out to be highly productive and fruitful.
For the definition of spectra, ring spectra, etc, see [Rudyak2008].
For definitions of generalized (co)homology and their relation to spectra see [Rudyak2008].
The sign denotes an isomorphism of groups or homeomorphism of spaces.
We denote the th Stiefel-Whitney class by .
2 Basic definition
Let be a topological -dimensional manifold, possibly with boundary. Consider a point and an open disk neighborhood of . Let be the map that collapses the complement of .
Let be a commutative ring spectrum, and let be the image of under the isomorphism
Definition 2.1. Let be a compact topological -dimensional manifold (not necessarily connected). An element is called an orientation of with respect to , or, briefly, an -orientation of , if for every and every disk neighborhood of .
Note that a non-connected is -orientable iff all its components are.
A manifold with a fixed -orientation is called -oriented, and a manifold which admits an -orientation is called -orientable. So, an -oriented manifold is in fact a pair .
It follows from the classical orientability that a classically oriented manifold is -orientable, see [Kreck2013]. Conversely, if a connected manifold is -orientable then (indeed, we know that either or , but the second case is impossible because must be surjective). Hence, a connected manifold is -orientable iff , i.e., iff is classically orientable. Thus, for arbitrary (not necessarily connected) is -orientable iff is classically orientable
Note that is a canonical -orientation of the sphere .
The following proposition holds because, for every two pairs and with connected, the maps and are homotopic.
Proposition 2.2. Let be a connected manifold, and let be an open disk neighborhood of a point . If an element is such that , then is an -orientation of .
For the proof, see [Rudyak2008, Proposition V.2.2].
3 Number of orientations
Let be a connected manifold. Let be and -orientation of with . Consider another -orientation with . Then , and so . Conversely, if and is an -orientation of then is an -orientation of because .
Furthermore, if is an -orientation of with then is an -orientation of with ,
Thus, if is a connected -oriented manifold, then there is a bijection between the set of all -orientations of and the set
where is any -orientation of .
4 Relation to normal and tangent bundles
Classical orientability of a smooth manifold is equivalent to the existence of a Thom class of the tangent (or normal) bundle of , see [Kreck2013, Theorem 7.1]. The similar claim holds for generalized (co)homology.
Given a vector -dimensional bundle over a compact space , consider the Thom space , the one-point compactification of the total space of . Then for every the inclusion of fiber to the total space of yields an inclusion , where is the one-point compactification of . Now, given a ring spectrum , note the canonical isomorphism and denote by the image of under this isomorphism.
Definition 4.1. A Thom-Dold class of with respect to (on a -orientation of ) is a class such that for all .
Theorem 4.2. A smooth manifold is -orientable if and only if the tangent or normal bundle of is -orientable. Moreover, -orientations of are in a bijective correspondence with -orientations of stable normal bundle of .
For the proof, see [Rudyak2008, Theorem V.2.4 and Corollary V.2.6].
Furthermore, Theorem 4.2 holds for topological manifolds as well, if we are careful with the concept of Thom spaces and their normal bundles for topological manifolds, see [Rudyak2008, Definitions IV.5.1 and IV.7.12]. To apply the theory to microbundles, use [Rudyak2008, Theorem IV.7.7].
Here we show that the product of two -oriented manifolds and admits a canonical -orientation. For sake of simplicity, assume and to be closed. Consider two collapsing maps and and form the map
It is easy to see that this composition is (homotopic to) .
Now, let and be -orientations of and , respectively. Consider the commutative diagram
where are given by the ring structure on . Because of the commutativity of the above diagram, we see that . Thus is an -orientation.
It is also worthy to note that if and are -orientable then is, cf. [Rudyak2008, V.1.10(ii)].
6 Poincaré Duality
Let be an -module spectrum. Given a closed -oriented manifold , consider the homomorphism
where is the cap product.
It turns out to be that is an isomorphism. This is called Poincaré duality and is frequently denoted by .
The Poincaré duality isomorphism admits the following alternative description:
Here is the Thom spectrum of the stable normal bundle of , and is the Thom-Dold isomorphism given by an -orientation (Thom-Dold class) of the (stable) normal bundle of , which, in turn, is given by the -orientation of according to Theorem 4.2.
For the proofs of the statements in this section, see [Rudyak2008, Theorem 2.9].
Definition 7.1. Let be a module spectrum over a ring spectrum . Let be a map of closed manifolds.
Suppose that both are -oriented, and let be the Poincaré duality isomorphisms, respectively. We define the transfers (other names: Umkehrs, Gysin homomorphisms)
to be the compositions
If is a map of closed -oriented manifolds then
for every . In particular, if then is epic. Similarly, is a monomorphism if . Theorem 7.2 below generalizes this fact.
Theorem 7.2 [Rudyak2008, Lemma V.2.12 and Theorem V.2.14]. Let be a ring spectrum. Let be a map of degree of closed -orientable manifolds. If is an -orientation of then is an -orientation of . In particular, the manifold is -orientable if is. Moreover, in this case is monic and is epic for every -module spectrum .
Here we list several examples.
(a) An ordinary (co)homology modulo 2. Represented by the Eilenberg-MacLane spectrum . Every manifold is -orientable; for connected the orientation is given be modulo 2 fundamental class. see [Dold1972]. Vice versa, if a ring spectrum is such that every manifold is -orientable, then is a graded Eilenberg-MacLane spectrum and .
(b) An ordinary (co)homology. Represented by the Eilenberg--MacLane spectrum . By Theorem 4.2 and [Rudyak2008, IV.5.8(ii)], classical orientability is just -orientability. In particular, a smooth manifold is -orientable iff the structure group of its normal and/or tangent bundle can be reduced to . Furthermore, -orientability of a manifold is equivalent to the equality .
(c) -theory. Atiyah-Bott-Shapiro [Atiyah&Bott&Shapiro1964] proved that a smooth manifold is -orientable if and only if it admits a -structure. This holds, in turn, iff . This condition is purely homotopic and can be formulated for every topological manifold (in fact, for Poincaré spaces) in view the equality where is the modulo 2 Thom class of the tangent bundle.
The equality is necessary for -orientability of topological manifolds, but it is not sufficient for -orientability even of piecewise linear manifolds, see [Rudyak2008, Ch. VI]. One the other hand, Sullivan proved that every topological manifold is -orientable, see Madsen-Milgram [Madsen&Milgram1979] for a good proof. Here is the -localized -theory.
Note that complex manifold are -oriented for all from (a,b,c) (but not (d, e) below).
(d) Complex -theory. The complexification induces a ring morphism . So, every -orientable manifold is -orientable.
Atiyah-Bott-Shapiro [Atiyah&Bott&Shapiro1964] proved that a smooth manifold is -orientable iff it admits a -structure. The last condition is equivalent to the purely homotopic conditions , where is the connecting homomorphism in the Bockstein exact sequence
This condition is necessary for -orientability of manifolds, but it is not sufficient for -orientability of piecewise linearly (and hence topological) manifolds, see [Rudyak2008, Ch. VI]. On the other hand, every classically oriented topological manifold is -orientable in view of Sullivan's result mentioned in example (c).
(e) Stable (co)homotopy groups, or frames (co)bordism theory. Represented by the spectrum . Because of Theorem 4.2, a manifold is orientable with respect to the sphere spectrum iff its tangent bundle has trivial stable fiber homotopy type, i.e., iff there exists such that is equivalent to where is a trivial -dimensional bundle. In particular, we have the following necessary (but not sufficient) condition: for all .
Note that -orientability implies -orientability implies -orientability implies -orientability implies -orientability. Furthermore, any -orientable manifold is -orientable for every ring spectrum , cf. [Rudyak2008, I.1.6]. So, (a) and (e) appear as two extremal cases.
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