B-Bordism
(→The Pontrjagin-Thom isomorphism) |
m (→Examples) |
||
(11 intermediate revisions by one user not shown) | |||
Line 4: | Line 4: | ||
On this page we recall the definition of the bordism groups of closed smooth manifolds, with extra topological structure: orientation, spin-structure, weakly almost complex structure etc. The situation for [[Wikipedia:Piecewise_linear_manifold|piecewise linear]] and [[Wikipedia:Topological_manifold|topological manifolds]] is similar and we discuss it briefly [[B-Bordism#Piecewise linear and topological bordism|below]]. | On this page we recall the definition of the bordism groups of closed smooth manifolds, with extra topological structure: orientation, spin-structure, weakly almost complex structure etc. The situation for [[Wikipedia:Piecewise_linear_manifold|piecewise linear]] and [[Wikipedia:Topological_manifold|topological manifolds]] is similar and we discuss it briefly [[B-Bordism#Piecewise linear and topological bordism|below]]. | ||
− | The formulation of the general set-up for B-Bordism dates back to {{cite|Lashof1963}}. There are detailed treatments in {{cite|Stong1968|Chapter II}} and {{cite|Bröcker&tom Dieck1970}} as well as summaries in {{cite| | + | The formulation of the general set-up for B-Bordism dates back to {{cite|Lashof1963}}. There are detailed treatments in {{cite|Stong1968|Chapter II}} and {{cite|Bröcker&tom Dieck1970}} as well as summaries in {{cite|Teichner1992|Part 1: 1}}, {{cite|Kreck1999|Section 1}}, {{cite|Kreck&Lück2005|18.10}}. See also the [[Wikipedia:Bordism|Wikipedia bordism page]]. |
We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$ where $BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$ for the fibration $\gamma$. Speaking somewhat imprecisely (precise details are below) a $B$-manifold is a compact manifold $M$ together with a lift to $B$ of a classifying map for the stable normal bundle of $M$: | We specify extra topological structure universally by means of a fibration $\gamma : B \to BO$ where $BO$ denotes the classifying space of the stable orthogonal group and $B$ is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes $B$ for the fibration $\gamma$. Speaking somewhat imprecisely (precise details are below) a $B$-manifold is a compact manifold $M$ together with a lift to $B$ of a classifying map for the stable normal bundle of $M$: | ||
Line 10: | Line 10: | ||
\xymatrix{ | \xymatrix{ | ||
& B \ar[d]^{\gamma} \\ | & B \ar[d]^{\gamma} \\ | ||
− | W \ar[r] | + | W \ar[r]^{\nu_W} \ar[ur]^{\bar \nu} & BO.} |
$$ | $$ | ||
The $n$-dimensional $B$-bordism group is defined to be the set of closed $B$-manifolds modulo the relation of bordism via compact $B$-manifolds. Addition is given by disjoint union and in fact for each $n \geq 0$ there is a group | The $n$-dimensional $B$-bordism group is defined to be the set of closed $B$-manifolds modulo the relation of bordism via compact $B$-manifolds. Addition is given by disjoint union and in fact for each $n \geq 0$ there is a group | ||
Line 24: | Line 24: | ||
* [[Oriented bordism|Oriented bordism]]: $\Omega_*$, $\Omega_*^{SO}$; $B = (BSO \to BO)$. | * [[Oriented bordism|Oriented bordism]]: $\Omega_*$, $\Omega_*^{SO}$; $B = (BSO \to BO)$. | ||
* [[Spin bordism|Spin bordism]]: $\Omega_*^{Spin}$; $B = (BSpin \to BO)$. | * [[Spin bordism|Spin bordism]]: $\Omega_*^{Spin}$; $B = (BSpin \to BO)$. | ||
− | * [[Spin^ | + | * [[Spin^c bordism|Spin<sup>$c$</sup> bordism]]: $\Omega_*^{Spin^{c}}$; $B = (BSpin^{c} \to BO)$. |
* [[String bordism|String bodism]] : $\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$; $B = (BO\langle 8 \rangle \to BO)$. | * [[String bordism|String bodism]] : $\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$; $B = (BO\langle 8 \rangle \to BO)$. | ||
* [[Complex bordism|Complex bordism]] : $\Omega_*^U$; $B = (BU \to BO)$. | * [[Complex bordism|Complex bordism]] : $\Omega_*^U$; $B = (BU \to BO)$. | ||
* [[Special unitary bordism|Special unitary bordism]] : $\Omega_*^{SU}$; $B = (BSU \to BO)$. | * [[Special unitary bordism|Special unitary bordism]] : $\Omega_*^{SU}$; $B = (BSU \to BO)$. | ||
− | * [[Framed bordism|Framed bordism]] : $\Omega_*^{fr}$; $B = PBO \to BO$, the path space fibration. | + | * [[Framed bordism|Framed bordism]] : $\Omega_*^{fr}$; $B = (PBO \to BO)$, the path space fibration. |
</wikitex> | </wikitex> | ||
Line 58: | Line 58: | ||
{{endthm}} | {{endthm}} | ||
− | {{ | + | {{beginrem|Remark}} |
A fibred stable vector bundle $B$ gives rise to a stable vector bundle as defined in {{cite|Kreck&Lück2005|18.10}}. One defines $E_r \to B_r$ to be the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$ is the universal r-plane bundle over $BO(r)$. The diagram above gives rise to bundle maps $\bar g_r : E_r \oplus \underline{\Rr} \to E_{r+1}$ covering the maps $g_r$; where $\underline{\Rr}$ denotes the trivial rank 1 bundle over $B_r$. | A fibred stable vector bundle $B$ gives rise to a stable vector bundle as defined in {{cite|Kreck&Lück2005|18.10}}. One defines $E_r \to B_r$ to be the pullback bundle $\gamma_r^*(EO(r))$ where $EO(r)$ is the universal r-plane bundle over $BO(r)$. The diagram above gives rise to bundle maps $\bar g_r : E_r \oplus \underline{\Rr} \to E_{r+1}$ covering the maps $g_r$; where $\underline{\Rr}$ denotes the trivial rank 1 bundle over $B_r$. | ||
− | {{ | + | {{endrem}} |
Now a $B_r$-structure on the normal bundle of an embedding $i: M \to \Rr^{n+r}$ defines a unique $B_{r+1}$-structure on the composition of $i$ with the standard inclusion $\Rr^{n+r} \to \Rr^{n+r+1}$. Hence we can make the following | Now a $B_r$-structure on the normal bundle of an embedding $i: M \to \Rr^{n+r}$ defines a unique $B_{r+1}$-structure on the composition of $i$ with the standard inclusion $\Rr^{n+r} \to \Rr^{n+r+1}$. Hence we can make the following | ||
{{beginthm|Definition|{{cite|Stong1968|p 15}}}} | {{beginthm|Definition|{{cite|Stong1968|p 15}}}} | ||
− | Let $B$ be a fibred stable | + | Let $B$ be a fibred stable vector bundle. A $B$-structure on $M$ is an equivalence class of $B_r$-structure on $M$ where two such structures are equivalent if they become equivalent for r sufficiently large. A $B$-manifold is a pair $(M, \bar \nu)$ where $M$ is a compact manifold and $\bar \nu$ is a $B$-structure on $M$. |
{{endthm}} | {{endthm}} | ||
Line 120: | Line 120: | ||
== The Pontrjagin-Thom isomorphism == | == The Pontrjagin-Thom isomorphism == | ||
<wikitex>; | <wikitex>; | ||
− | If $E$ is a vector bundle, let $T(E)$ denote its [[Wikipedia:Thom_space|Thom space]]. Recall that that a fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$ defines a stable vector bundle $(E_r, \gamma_r, \bar g_r)$ where $E_r = \gamma_r^*(EO(r))$. This stable vector bundle defines a Thom [[Wikipedia: | + | If $E$ is a vector bundle, let $T(E)$ denote its [[Wikipedia:Thom_space|Thom space]]. Recall that that a fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$ defines a stable vector bundle $(E_r, \gamma_r, \bar g_r)$ where $E_r = \gamma_r^*(EO(r))$. This stable vector bundle defines a Thom [[Wikipedia:Spectrum_(homotopy_theory)|spectrum]] which we denote $MB$. The $r$-th space of $MB$ is $T(E_r)$. |
By definition a $B$-manifold, $(M, \bar \nu)$, is an equivalence class of $B_r$-structures on $\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$ gives rise to the collapse map | By definition a $B$-manifold, $(M, \bar \nu)$, is an equivalence class of $B_r$-structures on $\nu(i)$, the normal bundle of an embedding $i : M \to \Rr^{n+r}$. Hence $(M, \bar \nu)$ gives rise to the collapse map | ||
Line 181: | Line 181: | ||
\xymatrix{ | \xymatrix{ | ||
& B \ar[d]^{\gamma} \\ | & B \ar[d]^{\gamma} \\ | ||
− | M \ar[r] | + | M \ar[r]^{\nu_M} \ar[ur]^{\bar \nu} & BCAT.} |
$$ | $$ | ||
Note that $CAT$ manifolds have stable normal $CAT$ bundles classified by $\nu_M \to BCAT$. | Note that $CAT$ manifolds have stable normal $CAT$ bundles classified by $\nu_M \to BCAT$. | ||
Line 193: | Line 193: | ||
{{endthm}} | {{endthm}} | ||
− | The basic bordism groups for $PL$ and $TOP$ manifolds, $B = (BCAT = BCAT)$ and $B = (BSCAT \to BCAT)$, are denoted by $\Omega_*^{PL}$, $\Omega_*^{SPL}$, $\Omega_*^{TOP}$ and $\Omega_*{STOP}$. Their computation is significantly more difficult than the corresponding bordism groups of smooth manifolds: there is no analogue of Bott periodicity for $\pi_i(PL)$ and $\pi_i(TOP)$ and so the spectra $MPL$ and $MTOP$ are far more complicated. For now we simply refer the reader to {{cite|Madsen&Milgram1979|Chapters 5 & 14}} and {{cite|Brumfiel&Madsen&Milgram1973}}. | + | The basic bordism groups for $PL$ and $TOP$ manifolds, $B = (BCAT = BCAT)$ and $B = (BSCAT \to BCAT)$, are denoted by $\Omega_*^{PL}$, $\Omega_*^{SPL}$, $\Omega_*^{TOP}$ and $\Omega_*^{STOP}$. Their computation is significantly more difficult than the corresponding bordism groups of smooth manifolds: there is no analogue of Bott periodicity for $\pi_i(PL)$ and $\pi_i(TOP)$ and so the spectra $MPL$ and $MTOP$ are far more complicated. For now we simply refer the reader to {{cite|Madsen&Milgram1979|Chapters 5 & 14}} and {{cite|Brumfiel&Madsen&Milgram1973}}. |
However, working rationally, the natural maps $O \to PL$ and $O \to TOP$ induce isomorphisms $$\pi_i(O) \otimes \Qq \cong \pi_i(PL) \otimes \Qq ~~ \text{and} ~~ \pi_i(O) \otimes \Qq \cong \pi_i(TOP) \otimes \Qq ~~\forall i.$$ | However, working rationally, the natural maps $O \to PL$ and $O \to TOP$ induce isomorphisms $$\pi_i(O) \otimes \Qq \cong \pi_i(PL) \otimes \Qq ~~ \text{and} ~~ \pi_i(O) \otimes \Qq \cong \pi_i(TOP) \otimes \Qq ~~\forall i.$$ | ||
Line 200: | Line 200: | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
There are isomorphisms | There are isomorphisms | ||
− | $$\Omega_i^{TOP} \otimes \Qq = \Omega_i^{PL} \otimes \Qq = 0 ~~ \forall i, $$ | + | <!--$$\Omega_i^{TOP} \otimes \Qq = \Omega_i^{PL} \otimes \Qq = 0 ~~ \forall i, $$--> |
$$ \Omega_i^{SO} \otimes \Qq \cong \Omega_i^{SPL} \otimes \Qq \cong \Omega_i^{STOP} \otimes \Qq ~~ \forall i.$$ | $$ \Omega_i^{SO} \otimes \Qq \cong \Omega_i^{SPL} \otimes \Qq \cong \Omega_i^{STOP} \otimes \Qq ~~ \forall i.$$ | ||
{{endthm}} | {{endthm}} | ||
Line 207: | Line 207: | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
+ | |||
+ | == External links == | ||
+ | * The Encyclopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Bordism bordism]. | ||
+ | * The Wikipedia page on [[Wikipedia:Cobordism|cobordism]]. | ||
+ | |||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:Bordism]] | [[Category:Bordism]] |
Latest revision as of 22:14, 11 September 2019
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Introduction
On this page we recall the definition of the bordism groups of closed smooth manifolds, with extra topological structure: orientation, spin-structure, weakly almost complex structure etc. The situation for piecewise linear and topological manifolds is similar and we discuss it briefly below.
The formulation of the general set-up for B-Bordism dates back to [Lashof1963]. There are detailed treatments in [Stong1968, Chapter II] and [Bröcker&tom Dieck1970] as well as summaries in [Teichner1992, Part 1: 1], [Kreck1999, Section 1], [Kreck&Lück2005, 18.10]. See also the Wikipedia bordism page.
We specify extra topological structure universally by means of a fibration where denotes the classifying space of the stable orthogonal group andTex syntax erroris homotopy equivalent to a CW complex of finite type. Abusing notation, one writes
Tex syntax errorfor the fibration . Speaking somewhat imprecisely (precise details are below) a
Tex syntax error-manifold is a compact manifold together with a lift to
Tex syntax errorof a classifying map for the stable normal bundle of :
Tex syntax error-bordism group is defined to be the set of closed
Tex syntax error-manifolds modulo the relation of bordism via compact
Tex syntax error-manifolds. Addition is given by disjoint union and in fact for each there is a group
Alternative notations are and also when for a stable representation of a topological group . Details of the definition and some important theorems for computing follow.
[edit] 1.1 Examples
We list some fundamental examples with common notation and also indicate the fibration B.
- Unoriented bordism: ; .
- Oriented bordism: , ; .
- Spin bordism: ; .
- Spin bordism: ; .
- String bodism : ; .
- Complex bordism : ; .
- Special unitary bordism : ; .
- Framed bordism : ; , the path space fibration.
[edit] 2 B-structures and bordisms
In this section we give a compressed accont of parts of [Stong1968, Chapter II]. Let denote the Grassmann manifold of unoriented -planes in and let be the infinite Grassmannian and fix a fibration .
Definition 2.1. Let be a rank r vector bundle classified by . A -structure on is a vertical homotopy class of maps such that .
Note that if and are isomorphic vector bundles over then the sets of -structures on each are in bijective equivalence. However -structures are defined on specific bundles, not isomorphism classes of bundles: a specific isomorphism, up to appropriate equivalence, is required to give a bijection between the sets of structures. Happily this is the case for the normal bundle of an embedding as we now explain. Let be a compact manifold and let be an embedding. Equipping with the standard metric, the normal bundle of is a rank r vector bundle over classified by its normal Gauss map . If is another such embedding and , then is regularly homotopic to and all regular homotopies are regularly homotopic relative to their endpoints (see [Hirsch1959]). A regular homotopy defines an isomorphism and a regular homotopy of regular homotopies gives a homotopy between these isomorphisms. Taking care one proves the following
Lemma 2.2 [Stong1968, p 15]. For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the set of structures of the normal bundles of any two embeddings .
This lemma is one motivation for the useful but subtle notion of a fibred stable vector bundle.
Definition 2.3. A fibred stable vector bundle consists of the following data: a sequence of fibrations together with a sequence of maps fitting into the following commutative diagram
where is the standard inclusion. We let .
Remark 2.4.
A fibred stable vector bundleTex syntax errorgives rise to a stable vector bundle as defined in [Kreck&Lück2005, 18.10]. One defines to be the pullback bundle where is the universal r-plane bundle over . The diagram above gives rise to bundle maps covering the maps ; where denotes the trivial rank 1 bundle over .
Now a -structure on the normal bundle of an embedding defines a unique -structure on the composition of with the standard inclusion . Hence we can make the following
Definition 2.5 [Stong1968, p 15].
LetTex syntax errorbe a fibred stable vector bundle. A
Tex syntax error-structure on is an equivalence class of -structure on where two such structures are equivalent if they become equivalent for r sufficiently large. A
Tex syntax error-manifold is a pair where is a compact manifold and is a
Tex syntax error-structure on .
Tex syntax error-structure on restricts to a
Tex syntax error-structure on . In particular, if is a closed
Tex syntax errormanifold then has a canonical
Tex syntax error-structure which restricts to on . The restriction of this
Tex syntax error-structure to is denoted : by construction is the boundary of .
Definition 2.6.
ClosedTex syntax error-manifolds and are
Tex syntax error-bordant if there is a compact
Tex syntax error-manifold such that . We write for the bordism class of .
Proposition 2.7 [Stong1968, p 17].
The set ofTex syntax error-bordism classes of closed n-manifolds with
Tex syntax error-structure,
forms an abelian group under the operation of disjoint union with inverse .
[edit] 3 Singular bordism
Tex syntax error-bordism gives rise to a generalised homology theory. If is a space then the -cycles of this homology theory are pairs
Tex syntax error-manifold and is any continuous map. Two cycles and are homologous if there is a pair
Tex syntax error-bordism from to and is a continuous map extending . Writing for the equivalence class of we obtain an abelian group
with group operation disjoint union and inverse .
Proposition 3.1. The mapping defines a generalised homology theory with coefficients .
Given a stable vector bundle we can form the stable vector bundle . The following simple lemma is clear but often useful.
Lemma 3.2. For any space there is an isomorphism .
[edit] 4 The orientation homomorphism
Tex syntax error-manifold is given a local orientation. This amounts to a choice of fundamental class of which is a generator
where denotes the local coefficient system defined by the orientation character of .
Given a closedTex syntax error-manifold we can use to push the fundamental class of to . Now the local coefficient system is defined by the orientation character of the stable bundle
Tex syntax error. It is easy to check that depends only on the
Tex syntax error-bordism class of and is additive with respect to the operations on .
Definition 4.1.
LetTex syntax errorbe a fibred stable vector bundle. The orientation homomorphism is defined as follows:
Tex syntax errorvanishes, , then all
Tex syntax error-manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to .
Definition 4.2.
LetTex syntax errorbe a fibred stable vector bundle. The orientation homomorphism in singular bordism is defined as follows:
Tex syntax error-manifolds and we can replace the -coefficients with -coefficients above.
[edit] 5 The Pontrjagin-Thom isomorphism
If is a vector bundle, let denote its Thom space. Recall that that a fibred stable vector bundle defines a stable vector bundle where . This stable vector bundle defines a Thom spectrum which we denote . The -th space of is .
By definition aTex syntax error-manifold, , is an equivalence class of -structures on , the normal bundle of an embedding . Hence gives rise to the collapse map
where we identify with the one-point compatificiation of , we map via on a tubular neighbourhood of and we map all other points to the base-point of . As r increases these maps are compatibly related by suspension and the structure maps of the spectrum . Hence we obtain a homotopy class
The celebrated theorem of Pontrjagin and Thom states in part that depends only on the bordism class of .
Theorem 5.1. There is an isomorphism of abelian groups
For the proof see [Bröcker&tom Dieck1970, Satz 3.1 and Satz 4.9].
For example, if is the path fibration over , then is homotopic to the sphere spectrum and is the -th stable homotopy group. On the other hand, in this case is the framed bordism group and as a special case of Theorem 5.1 we have
Theorem 5.2. There is an isomorphism .
The Pontrjagin-Thom isomorphism generalises to singular bordism.
Theorem 5.3. For any space there is an isomorphism of abelian groups
where denotes the smash produce of the specturm and the space with a disjoint basepoint added.
[edit] 6 Spectral sequences
For any generalised homology theory there is a spectral sequence, called the Atiyah-Hirzebruch spectral sequence (AHSS) which can be used to compute . The term of the AHSS is and one writes
The Pontrjagin-Thom isomorphisms above therefore give rise to the following theorems. For the first we recall that stable homotopy defines a generalised homology theory, and we use the Thom isomorphism with local coefficients: .
Theorem 6.1.
LetTex syntax errorbe a fibred stable vector bundle. There is a spectral sequence
Theorem 6.2.
LetTex syntax errorbe a fibred stable vector bundle and a space. There is a spectral sequence
Next recall Serre's theorem [Serre1951] that vanishes unless in which case . From the above spectral sequences of Theorems 6.1 and 6.2 we deduce the following
Theorem 6.3 Cf. [Kreck&Lück2005, Thm 2.1]. If then the orientation homomorphism induces an isomorphism
Tex syntax erroris connected, the rationalised orientation homomorphism may be identified with the projection
[edit] 7 Piecewise linear and topological bordism
Let and denote respectively the classifying spaces for stable piecewise linear homeomorphisms of Euclidean space and origin-preserving homeomorphisms of Euclidean space. Note that while there are honest groups and , the piecewise linear case requires more care.
If or , and is a fibration, and is a compact manifold then just as above, we can define anTex syntax error-structure on to be an equivalence class of lifts of of the classifying map of the stable normal bundle of :
Note that manifolds have stable normal bundles classified by .
Just as before we obtain bordism groups of closed n-dimensional -manifolds withTex syntax errorstructure
Tex syntax erroragain defines a Thom spectrum and one asks if there is a Pontrjagin-Thom isomorphism. The proof of the Pontrjagin-Thom theorem relies on transversality for manifolds and while this is comparatively easy in the -category, it is was a major breakthrough to achieve this for topological manifolds: achieved in [Kirby&Siebenmann1977] for dimensions other than 4 and then in [Freedman&Quinn1990] in dimension 4. Thus one has
Theorem 7.1. There is an isomorphism .
The basic bordism groups for and manifolds, and , are denoted by , , and . Their computation is significantly more difficult than the corresponding bordism groups of smooth manifolds: there is no analogue of Bott periodicity for and and so the spectra and are far more complicated. For now we simply refer the reader to [Madsen&Milgram1979, Chapters 5 & 14] and [Brumfiel&Madsen&Milgram1973].
However, working rationally, the natural maps and induce isomorphismsAs a consequence one has
Theorem 7.2. There are isomorphisms
[edit] 8 References
- [Brumfiel&Madsen&Milgram1973] G. Brumfiel, I. Madsen and R. J. Milgram, characteristic classes and cobordism, Ann. of Math. (2) 97 (1973), 82–159. MR0310881 (46 #9979) Zbl 0248.57006
- [Bröcker&tom Dieck1970] T. Bröcker and T. tom Dieck, Kobordismentheorie, Springer-Verlag, Berlin, 1970. MR0275446 (43 #1202) Zbl 0211.55501
- [Freedman&Quinn1990] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton University Press, Princeton, NJ, 1990. MR1201584 (94b:57021) Zbl 0705.57001
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [Kreck&Lück2005] M. Kreck and W. Lück, The Novikov conjecture, Birkhäuser Verlag, Basel, 2005. MR2117411 (2005i:19003) Zbl 1058.19001
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Lashof1963] R. Lashof, Poincaré duality and cobordism, Trans. Amer. Math. Soc. 109 (1963), 257–277. MR0156357 (27 #6281) Zbl 0137.42803
- [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
- [Serre1951] J. Serre, Homologie singulière des espaces fibrès. Applications, Ann. of Math. (2) 54 (1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
- [Teichner1992] P. Teichner, Topological 4-manifolds with finite fundamental group PhD Thesis, University of Mainz, Germany, Shaker Verlag 1992, ISBN 3-86111-182-9.