Orientation of manifolds in generalized cohomology theories
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Contents |
1 Preliminaries
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 definitions of generalized (co)homology and their relation to spectra see [Rudyak2008].
The sign denotes an isomorphism of groups or homeomorphism of spaces.
I reserve the term ``classical orientation`` for orientation in ordinary (co)homology, see e.g. [Kreck2013].
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
![\displaystyle \pi_0(E) = \widetilde E_0(S^0)\cong \widetilde E_n(S^n)= E_n(S^n,*).](/images/math/8/1/7/817f1dbc688dd5117e0ebd1d262f500e.png)
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 a disk neighborhood of a point
. If an element
is such that
, then
is an
-orientation of
.
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
![\displaystyle \pm u+ \Ker (\varepsilon_* : E_n(M, \partial M)\to E_n(S^n,*))\subset E_n(M, \partial M),](/images/math/e/c/0/ec03ff2e55772f649b33b5a09863ecac.png)
where is any
-orientation of
.
4 Relation to normal and tangent bundles
Classical orientability of a smooth manifold is equivalent to the existence of 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
.
Note that the results of this sections hold for topological manifolds as well, if we are careful with the concept of Thom spaces for normal (micro)bundles for topological manifolds, see [Rudyak2008, Section V.2].
5 Products
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
![\displaystyle M\times N \stackrel{\varepsilon_M \times \varepsilon_N}\longrightarrow S^m\times S^n \to S^m\wedge S^n =S^{m+n}.](/images/math/9/a/d/9addbaf65acec10a8a73c055ce20734a.png)
It is easy to see that this composition is (homotopic to) .
Now, let and
be
-orientations of
and
, respectively. Consider the commutative diagram
![\displaystyle \xymatrix{ E_m(M)\otimes E_n(N) \ar[d] \ar[r]^{\mu} & E_{m+n}(M\times N) \ar[r]^{\varepsilon_*} \ar[d]^{\varepsilon_M)_*\otimes (\varepsilon_N)_*V} & E_{m+n}(S^{m+n}) \ar[d]^{=} \\ E_m(S^m)\otimes E_n(S^n) \ar[r]^{\mu'} & E_{m+n}(S^m\times S^n) \ar[r] & E_{m+n}(S^{m+n}) }](/images/math/7/4/1/7413b649b557efda3ff5102b97f18b5b.png)
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
![\displaystyle \frown [M]_E: F^i(M)\to F_{n-i}(M)](/images/math/f/3/f/f3f2d32a94ffecda3bd08de12cf6ee39.png)
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:
![\displaystyle P=P_{[M]_E}: F^i(M) \stackrel{\varphi}\longrightarrow F^i(T\nu) \cong \widetilde F_{n-i}(M^+) =F_{n-i}(M).](/images/math/7/a/d/7adbc656efad121c0281c808e48d2c81.png)
Here 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.
7 Transfer
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)
![\displaystyle f^!: F^i(M)\to F^{n-m+i}(N), \qquad f_!: F_i(N)\to F_{m-n+i}(M)](/images/math/7/d/e/7dece7b29c408dd88d3eb602cdb4beea.png)
to be the compositions
![\displaystyle f^!: F^i(M) \cong F_{m-i}(M) \stackrel{f_*}{\longrightarrow} F_{m-i}(N) \cong F^{n-m+i}(M),\ f^!=P_{[N]}^{-1}f_*P_{[M]} \\ f_!: F_i(N) \cong F^{n-i}(N) \stackrel{f^*}{\longrightarrow} F^{n-i}(M) \cong F_{m-n+i}(N),\ f_!=P_{[M]}f^*P_{[N]}^{-1}.](/images/math/0/0/1/0013865417becd77a18ab321df545c3c.png)
The reader can find many good properties of transfers in Dold [Dold1972], Dyer [Dyer1969], Rudyak [Rudyak2008].
If is a map of closed
-oriented manifolds then
![\displaystyle f_*f_!(x) =(\deg f)x](/images/math/a/3/5/a353a6f0865feacecdeb5b45f8694cf1.png)
for every . In particular, if
then
is epic. Similarly,
is a monomorphism if
. Theorem 7.2 below generalizes this fact.
Theorem 7.2.
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
.Moreover, in this case
is monic and
is epic for every
-module spectrum
.
8 Examples
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--Mac Lane spectrum and
.
(b) An ordinary (co)homology. Represented by the Eilenberg--MacLane spectrum . By IV.5.8(ii), orientability as defined in IV.5.6 is just
-orientability. In particular, a smooth manifold is
-orientable iff its structure group of its normald 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 piecewies linearly (PL) 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
![\displaystyle \dots \to H^*(X) \stackrel{2}{\longrightarrow} H^*(X)\stackrel{\textup{mod}~2}{\longrightarrow} H^*(X;\Bbb Z/2) \stackrel{\delta}{\longrightarrow} H^*(X) \to \dots\,.](/images/math/5/5/a/55a7fa3b5261cc416610359ac6fc3fca.png)
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).
Note that -orientability implies
-orientability implies
-orientability that implies
-orientability that implies
-orientability.
(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 that implies
-orientability that 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.
9 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
10 References
- [Atiyah&Bott&Shapiro1964] M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), suppl. 1, 3–38. MR0167985 (29 #5250) Zbl 0146.19001
- [Dold1972] A. Dold, Lectures on algebraic topology, Springer-Verlag, Berlin-Heidelberg-New York 1972. MR0415602 (54 #3685) Zbl 0234.55001
- [Dyer1969] E. Dyer, Cohomology theories, Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0268883 (42 #3780) Zbl 0182.57002
- [Kreck2013] M. Kreck, Orientation of manifolds, Bull. Man. Atl. (2013).
- [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002
- [Rudyak2008] Y. B. Rudyak, On Thom spectra, orientability, and cobordism, Springer-Verlag, 1998, Corrected reprint 2008. MR1627486 (99f:55001) Zbl 0906.55001
![[M]\in H_n(M)](/images/math/7/5/3/753a7f8d85351048ad0e89da15ce68f2.png)
For definitions of generalized (co)homology and their relation to spectra see [Rudyak2008].
The sign denotes an isomorphism of groups or homeomorphism of spaces.
I reserve the term ``classical orientation`` for orientation in ordinary (co)homology, see e.g. [Kreck2013].
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
![\displaystyle \pi_0(E) = \widetilde E_0(S^0)\cong \widetilde E_n(S^n)= E_n(S^n,*).](/images/math/8/1/7/817f1dbc688dd5117e0ebd1d262f500e.png)
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 a disk neighborhood of a point
. If an element
is such that
, then
is an
-orientation of
.
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
![\displaystyle \pm u+ \Ker (\varepsilon_* : E_n(M, \partial M)\to E_n(S^n,*))\subset E_n(M, \partial M),](/images/math/e/c/0/ec03ff2e55772f649b33b5a09863ecac.png)
where is any
-orientation of
.
4 Relation to normal and tangent bundles
Classical orientability of a smooth manifold is equivalent to the existence of 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
.
Note that the results of this sections hold for topological manifolds as well, if we are careful with the concept of Thom spaces for normal (micro)bundles for topological manifolds, see [Rudyak2008, Section V.2].
5 Products
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
![\displaystyle M\times N \stackrel{\varepsilon_M \times \varepsilon_N}\longrightarrow S^m\times S^n \to S^m\wedge S^n =S^{m+n}.](/images/math/9/a/d/9addbaf65acec10a8a73c055ce20734a.png)
It is easy to see that this composition is (homotopic to) .
Now, let and
be
-orientations of
and
, respectively. Consider the commutative diagram
![\displaystyle \xymatrix{ E_m(M)\otimes E_n(N) \ar[d] \ar[r]^{\mu} & E_{m+n}(M\times N) \ar[r]^{\varepsilon_*} \ar[d]^{\varepsilon_M)_*\otimes (\varepsilon_N)_*V} & E_{m+n}(S^{m+n}) \ar[d]^{=} \\ E_m(S^m)\otimes E_n(S^n) \ar[r]^{\mu'} & E_{m+n}(S^m\times S^n) \ar[r] & E_{m+n}(S^{m+n}) }](/images/math/7/4/1/7413b649b557efda3ff5102b97f18b5b.png)
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
![\displaystyle \frown [M]_E: F^i(M)\to F_{n-i}(M)](/images/math/f/3/f/f3f2d32a94ffecda3bd08de12cf6ee39.png)
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:
![\displaystyle P=P_{[M]_E}: F^i(M) \stackrel{\varphi}\longrightarrow F^i(T\nu) \cong \widetilde F_{n-i}(M^+) =F_{n-i}(M).](/images/math/7/a/d/7adbc656efad121c0281c808e48d2c81.png)
Here 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.
7 Transfer
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)
![\displaystyle f^!: F^i(M)\to F^{n-m+i}(N), \qquad f_!: F_i(N)\to F_{m-n+i}(M)](/images/math/7/d/e/7dece7b29c408dd88d3eb602cdb4beea.png)
to be the compositions
![\displaystyle f^!: F^i(M) \cong F_{m-i}(M) \stackrel{f_*}{\longrightarrow} F_{m-i}(N) \cong F^{n-m+i}(M),\ f^!=P_{[N]}^{-1}f_*P_{[M]} \\ f_!: F_i(N) \cong F^{n-i}(N) \stackrel{f^*}{\longrightarrow} F^{n-i}(M) \cong F_{m-n+i}(N),\ f_!=P_{[M]}f^*P_{[N]}^{-1}.](/images/math/0/0/1/0013865417becd77a18ab321df545c3c.png)
The reader can find many good properties of transfers in Dold [Dold1972], Dyer [Dyer1969], Rudyak [Rudyak2008].
If is a map of closed
-oriented manifolds then
![\displaystyle f_*f_!(x) =(\deg f)x](/images/math/a/3/5/a353a6f0865feacecdeb5b45f8694cf1.png)
for every . In particular, if
then
is epic. Similarly,
is a monomorphism if
. Theorem 7.2 below generalizes this fact.
Theorem 7.2.
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
.Moreover, in this case
is monic and
is epic for every
-module spectrum
.
8 Examples
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--Mac Lane spectrum and
.
(b) An ordinary (co)homology. Represented by the Eilenberg--MacLane spectrum . By IV.5.8(ii), orientability as defined in IV.5.6 is just
-orientability. In particular, a smooth manifold is
-orientable iff its structure group of its normald 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 piecewies linearly (PL) 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
![\displaystyle \dots \to H^*(X) \stackrel{2}{\longrightarrow} H^*(X)\stackrel{\textup{mod}~2}{\longrightarrow} H^*(X;\Bbb Z/2) \stackrel{\delta}{\longrightarrow} H^*(X) \to \dots\,.](/images/math/5/5/a/55a7fa3b5261cc416610359ac6fc3fca.png)
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).
Note that -orientability implies
-orientability implies
-orientability that implies
-orientability that implies
-orientability.
(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 that implies
-orientability that 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.
9 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on orientability.
- The Wikipedia page on orientations of vector spaces.
10 References
- [Atiyah&Bott&Shapiro1964] M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), suppl. 1, 3–38. MR0167985 (29 #5250) Zbl 0146.19001
- [Dold1972] A. Dold, Lectures on algebraic topology, Springer-Verlag, Berlin-Heidelberg-New York 1972. MR0415602 (54 #3685) Zbl 0234.55001
- [Dyer1969] E. Dyer, Cohomology theories, Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0268883 (42 #3780) Zbl 0182.57002
- [Kreck2013] M. Kreck, Orientation of manifolds, Bull. Man. Atl. (2013).
- [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002
- [Rudyak2008] Y. B. Rudyak, On Thom spectra, orientability, and cobordism, Springer-Verlag, 1998, Corrected reprint 2008. MR1627486 (99f:55001) Zbl 0906.55001