B-Bordism
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | + | 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|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 | + | |
$$ | $$ | ||
\xymatrix{ | \xymatrix{ | ||
& B \ar[d]^{\gamma} \\ | & B \ar[d]^{\gamma} \\ | ||
− | + | 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 | ||
+ | $$ \Omega_n^B := \{ (M, \bar \nu) \}/\equiv.$$ | ||
+ | |||
+ | Alternative notations are $\Omega_n(B)$ and also $\Omega_n^G$ when $(B \to BO) = (BG \to BO)$ for $G \to O$ a stable representation of a topological group $G$. Details of the definition and some important theorems for computing $\Omega_n^B$ follow. | ||
</wikitex> | </wikitex> | ||
− | == The Pontrjagin Thom isomorphism == | + | === Examples === |
− | <wikitex> | + | <wikitex>; |
− | $$ \Omega_n^B \cong \pi_n^S(MB)$$ | + | We list some fundamental examples with common notation and also indicate the fibration B. |
+ | * [[Unoriented bordism|Unoriented bordism]]: $\mathcal{N}_*$; $B = (BO = BO)$. | ||
+ | * [[Oriented bordism|Oriented bordism]]: $\Omega_*$, $\Omega_*^{SO}$; $B = (BSO \to BO)$. | ||
+ | * [[Spin bordism|Spin bordism]]: $\Omega_*^{Spin}$; $B = (BSpin \to BO)$. | ||
+ | * [[Spin^c bordism|Spin<sup>$c$</sup> bordism]]: $\Omega_*^{Spin^{c}}$; $B = (BSpin^{c} \to BO)$. | ||
+ | * [[String bordism|String bodism]] : $\Omega_*^{String}, \Omega_*^{BO\langle 8 \rangle}$; $B = (BO\langle 8 \rangle \to BO)$. | ||
+ | * [[Complex bordism|Complex bordism]] : $\Omega_*^U$; $B = (BU \to BO)$. | ||
+ | * [[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. | ||
+ | </wikitex> | ||
+ | |||
+ | == B-structures and bordisms == | ||
+ | <wikitex>; | ||
+ | In this section we give a compressed accont of parts of {{cite|Stong1968|Chapter II}}. Let $G_{r, m}$ denote the [[Wikipedia:Grassman_manifold|Grassmann manifold]] of unoriented $r$-planes in $\Rr^m$ and let $BO(r) = \text{lim}_{m \to \infty} G_{r, m}$ be the infinite Grassmannian and fix a fibration $\gamma_r : B_r \to BO(r)$. | ||
+ | |||
+ | {{beginthm|Definition}} | ||
+ | Let $\xi: E \to X$ be a rank r vector bundle classified by $\xi : X \to BO(r)$. A $B_r$-structure on $\xi$ is a vertical homotopy class of maps $\bar \xi : X \to B_r$ such that $\gamma_r \circ \bar \xi = \xi$. | ||
+ | {{endthm}} | ||
+ | |||
+ | Note that if $\xi_0$ and $\xi_1$ are isomorphic vector bundles over $X$ then the sets of $B_r$-structures on each are in bijective equivalence. However $B_r$-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 $B_r$ structures. Happily this is the case for the normal bundle of an embedding as we now explain. Let $M$ be a compact manifold and let $i_0 : M \to \Rr^{n+r}$ be an embedding. Equipping $\Rr^{n+r}$ with the standard metric, the [[Wikipedia:Normal_bundle|normal bundle]] of $i_0$ is a rank r vector bundle over $M$ classified by its normal Gauss map $\nu(i_0) : M \to G_{r, n+r} \subset BO(r)$. If $i_1$ is another such embedding and $r >> n$, then $i_1$ is [[Wikipedia:Regular_homotopy|regularly homotopic]] to $i_0$ and all regular homotopies are regularly homotopic relative to their endpoints (see {{cite|Hirsch1959}}). A regular homotopy $H$ defines an isomorphism $\alpha_H :\nu(i_0) \cong \nu(i_1)$ and a regular homotopy of regular homotopies gives a homotopy between these isomorphisms. Taking care one proves the following | ||
+ | |||
+ | {{beginthm|Lemma|{{cite|Stong1968|p 15}}}} | ||
+ | For r sufficiently large, (depending only on n) there is a 1-1 correspondence between the set of $B_r$ structures of the normal bundles of any two embeddings $i_0, i_1 : M \to \Rr^{n+r}$. | ||
+ | {{endthm}} | ||
+ | |||
+ | This lemma is one motivation for the useful but subtle notion of a fibred stable vector bundle. | ||
+ | |||
+ | {{beginthm|Definition}} | ||
+ | A fibred stable vector bundle $B = (B_r, \gamma_r, g_r)$ consists of the following data: a sequence of fibrations $\gamma_r : B_r \to BO(r)$ together with a sequence of maps $g_r : B_r \to B_{r+1}$ fitting into the following commutative diagram | ||
+ | $$ | ||
+ | \xymatrix{ | ||
+ | B_r \ar[r]^{g_r} \ar[d]^{\gamma_r} & B_{r+1} \ar[d]^{\gamma_{r+1}} \\ | ||
+ | BO(r) \ar[r]^{j_r} & BO(r+1) | ||
+ | } | ||
+ | $$ | ||
+ | where $j_r$ is the standard inclusion. We let $B = \text{lim}_{r \to \infty}(B_r)$. | ||
+ | {{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 $B_r$. | ||
+ | {{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 | ||
+ | |||
+ | {{beginthm|Definition|{{cite|Stong1968|p 15}}}} | ||
+ | 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}} | ||
+ | |||
+ | If $W$ is a compact manifold with boundary $\partial W$ then by choosing the inward-pointing normal vector along $\partial W$, a $B$-structure on $W$ restricts to a $B$-structure on $\partial W$. In particular, if $(M, \bar \nu_M)$ is a closed $B$ manifold then $W = M \times [0, 1]$ has a canonical $B$-structure $\bar \nu_{M \times [0, 1]}$ which restricts to $(M, \bar \nu_M)$ on $M \times \{ 0 \}$. The restriction of this $B$-structure to $M \times \{ 1 \}$ is denoted $-\bar \nu$: by construction $(M \sqcup M, \bar \nu \sqcup - \bar \nu)$ is the boundary of $(M \times [0, 1], \bar \nu_{M \times [0, 1]})$. | ||
+ | |||
+ | {{beginthm|Definition}} | ||
+ | Closed $B$-manifolds $(M_0, \bar \nu_0)$ and $(M_1, \bar \nu_1)$ are $B$-bordant if there is a compact $B$-manifold $(W, \bar \nu)$ such that $\partial(W, \bar \nu) = (M_0 \sqcup M_1, \bar \nu_0 \sqcup -\bar \nu_1)$. We write $[M, \bar \nu]$ for the bordism class of $(M, \bar \nu)$. | ||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Proposition|{{cite|Stong1968|p 17}}}} | ||
+ | The set of $B$-bordism classes of closed n-manifolds with $B$-structure, | ||
+ | $$ \Omega_n^B := \{ [M, \bar \nu ] \},$$ | ||
+ | forms an abelian group under the operation of disjoint union with inverse $-[M,\bar \nu] = [M, -\bar \nu]$. | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
+ | == Singular bordism == | ||
+ | <wikitex>; | ||
+ | $B$-bordism gives rise to a generalised homology theory. If $X$ is a space then the $n$-cycles of this homology theory are pairs | ||
+ | $$ ((M, \bar \nu),~ f: M \to X) $$ | ||
+ | where $(M, \bar \nu)$ is a closed $n$-dimensional $B$-manifold and $f$ is any continuous map. Two cycles $((M_0, \bar \nu_0), f_0)$ and $((M_1, \bar \nu_1), f_1)$ are homologous if there is a pair | ||
+ | $$ ((W, \bar \nu),~ g : W \to X)$$ | ||
+ | where $(W, \bar \nu)$ is a $B$-bordism from $(M_0, \bar \nu_0)$ to $(M_1, \bar \nu_1)$ and $g : W \to X$ is a continuous map extending $f_0 \sqcup f_1$. Writing $[(M, \bar \nu), f]$ for the equivalence class of $((M, \bar \nu) ,f)$ we obtain an abelian group | ||
+ | $$ \Omega_n^B(X) : = \{ [(M, \bar \nu), f] \} $$ | ||
+ | with group operation disjoint union and inverse $-[(M, \bar \nu), f] = [(M, - \bar \nu), f]$. | ||
+ | {{beginthm|Proposition}} | ||
+ | The mapping $X \to \Omega_n^B(X)$ defines a generalised homology theory with coefficients $\Omega_n^B(\text{pt}) = \Omega_n^B$. | ||
+ | {{endthm}} | ||
+ | Given a stable vector bundle $B = (B_r, \gamma_r, g_r)$ we can form the stable vector bundle $B \times X := (B_r \times X, \gamma_r \times X, g_r \times \id_X)$. The following simple lemma is clear but often useful. | ||
+ | {{beginthm|Lemma}} | ||
+ | For any space $X$ there is an isomorphism $\Omega_n^B(X) \cong \Omega_n^{B \times X}$. | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
+ | == The orientation homomorphism == | ||
+ | <wikitex>; | ||
+ | We fix a local orientation at the base-point of $BO$. It then follows that every closed $B$-manifold $(M, \bar \nu)$ is given a local orientation. This amounts to a choice of fundamental class of $M$ which is a generator | ||
+ | $$[M] \in H_n(M; \underline{\Zz})$$ | ||
+ | where $\underline{\Zz}$ denotes the local coefficient system defined by the [[Wikipedia:Orientation_character|orientation character]] of $M$. | ||
+ | |||
+ | Given a closed $B$-manifold $(M, \bar \nu)$ we can use $\bar \nu$ to push the fundamental class of $[M]$ to $\bar \nu_*[M] \in H_n(B; \underline{\Zz})$. Now the local coefficient system is defined by the orientation character of the stable bundle $B$. It is easy to check that $\bar \nu_*[M]$ depends only on the $B$-bordism class of $(M, \bar \nu)$ and is additive with respect to the operations $+/-$ on $\Omega_n^B$. | ||
+ | {{beginthm|Definition}} | ||
+ | Let $B$ be a fibred stable vector bundle. The orientation homomorphism is defined as follows: | ||
+ | $$ \rho : \Omega_n^B \to H_n(B; \underline{\Zz}), ~~~[M, \bar \nu] \mapsto \bar \nu_*[M].$$ | ||
+ | {{endthm}} | ||
+ | For the singular bordism groups $\Omega_n^B(X)$ we have no bundle over $X$ so in general there is only a $\Zz/2$-valued orientation homomorphism. However, if the first [[Wikipedia:Stiefel-Whitney_class|Stiefel-Whitney class]] of $B$ vanishes, $w_1(B) = 0$, then all $B$-manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to $\Zz$. | ||
+ | {{beginthm|Definition}} | ||
+ | Let $B$ be a fibred stable vector bundle. The orientation homomorphism in singular bordism is defined as follows: | ||
+ | $$ \rho : \Omega_n^B(X) \to H_n(X; \Zz/2), ~~~ [(M, \bar \nu), f] \mapsto f_*[M].$$ | ||
+ | If $w_1(B) = 0$ then for all closed $B$-manifolds $[M] \in H_n(M; \Zz)$ and we can replace the $\Zz/2$-coefficients with $\Zz$-coefficients above. | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
+ | == The Pontrjagin-Thom isomorphism == | ||
+ | <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: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 | ||
+ | $$ c(M, \bar \nu) : S^{n+r} \to T(E_r)$$ | ||
+ | where we identify $S^{n+r}$ with the [[Wikipedia:One-point_compactification|one-point compatificiation]] of $\Rr^{n+r}$, we map via $\bar \nu_r$ on a tubular neighbourhood of $i(M) \subset \Rr^{n+r}$ and we map all other points to the base-point of $T(E_r)$. As r increases these maps are compatibly related by suspension and the structure maps of the spectrum $MB$. Hence we obtain a homotopy class | ||
+ | $$ [c(M, \bar \nu)] =: P((M, \bar \nu)) \in \text{lim}_{r \to \infty}(\pi_{n+r}(T(E_r)) = \pi_n(MB).$$ | ||
+ | The celebrated theorem of Pontrjagin and Thom states in part that $P((M, \bar \nu))$ depends only on the bordism class of $(M, \bar \nu)$. | ||
+ | |||
+ | {{beginthm|Theorem}} \label{thm:PT-iso} | ||
+ | There is an isomorphism of abelian groups | ||
+ | $$ P : \Omega_n^B \cong \pi_n^S(MB), ~~~[M, \bar \nu] \longmapsto P([M, \bar \nu]).$$ | ||
+ | {{endthm}} | ||
+ | |||
+ | For the proof see {{cite|Bröcker&tom Dieck1970|Satz 3.1 and Satz 4.9}}. | ||
+ | |||
+ | For example, if $B = PBO$ is the path fibration over $BO$, then $MB$ is homotopic to the sphere spectrum $S$ and $\pi_n(S) = \pi_n^S$ is the [[Wikipedia:Stable_homotopy_groups_of_spheres|$n$-th stable homotopy group]]. On the other hand, in this case $\Omega_n^B = \Omega_n^{fr}$ is the framed bordism group and as a special case of Theorem \ref{thm:PT-iso} we have | ||
+ | {{beginthm|Theorem}} | ||
+ | There is an isomorphism $P : \Omega_n^{fr} \cong \pi_n^S$. | ||
+ | {{endthm}} | ||
+ | The Pontrjagin-Thom isomorphism generalises to singular bordism. | ||
+ | {{beginthm|Theorem}} | ||
+ | For any space $X$ there is an isomorphism of abelian groups | ||
+ | $$ P : \Omega_n^B(X) \cong \pi_n^S(MB \wedge X_+)$$ | ||
+ | where $MB \wedge X_+$ denotes the smash produce of the specturm $MB$ and the space $X$ with a disjoint basepoint added. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
== Spectral sequences == | == Spectral sequences == | ||
− | <wikitex> | + | <wikitex>; |
− | $$ H_p(B; \pi_q^S) \Longrightarrow \Omega_{p+q}^B$$ | + | For any generalised homology theory $h_*$ there is a spectral sequence, called the [[Wikipedia:Atiyah-Hirzebruch_spectral_sequence|Atiyah-Hirzebruch spectral sequence]] (AHSS) which can be used to compute $h_*(X)$. The $E_2$ term of the AHSS is $H_p(X; h_q(\text{pt}))$ and one writes |
+ | $$ \bigoplus_{p+q = n} H_p(X; h_q(\text{pt})) \Longrightarrow h_{n}(X).$$ | ||
+ | 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: $H_*(MB;A)\cong H_*(B;A_\omega)$. | ||
+ | |||
+ | {{beginthm|Theorem}} <label>SS1</label> | ||
+ | Let $B$ be a fibred stable vector bundle. There is a spectral sequence | ||
+ | $$ \bigoplus_{p+q = n} H_p(B;\underline{\pi_q^S}) \Longrightarrow \Omega_{n}^B.$$ | ||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Theorem}}<label>SS2</label> | ||
+ | Let $B$ be a fibred stable vector bundle and $X$ a space. There is a spectral sequence | ||
+ | $$ \bigoplus_{p+q = n} H_p(X; \Omega_q^B) \Longrightarrow \Omega_n^B(X).$$ | ||
+ | {{endthm}} | ||
+ | |||
+ | Next recall [[Wikipedia:Stable_homotopy_groups_of_spheres#Finiteness_and_torsion|Serre's theorem]] {{cite|Serre1951}} that $\pi_i^S \otimes \Qq$ vanishes unless $i=0$ in which case $\pi_0^S \otimes \Qq \cong \Qq$. From the above spectral sequences of Theorems \ref{SS1} and \ref{SS2} we deduce the following | ||
+ | |||
+ | {{beginthm|Theorem|Cf. {{cite|Kreck&Lück2005|Thm 2.1}}}} | ||
+ | If $w_1(B) = 0$ then the orientation homomorphism induces an isomorphism | ||
+ | $$ \rho \otimes \id_{\Qq} : \Omega_n^B \otimes \Qq \cong H_n(B; \Qq).$$ | ||
+ | Moreover for any space $X$, $\Omega_n^B(X) \otimes \Qq \cong \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq))$ and if $B$ is connected, the rationalised orientation homomorphism $\rho \otimes \id_{\Qq} : \Omega_n^B(X) \otimes \Qq \to H_n(X; \Qq)$ may be identified with the projection | ||
+ | $$ \bigoplus_{p+q = n} H_p(X; H_q(B; \Qq)) \to H_n(B; H_0(B; \Qq)) = H_n(B; \Qq).$$ | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
+ | == Piecewise linear and topological bordism == | ||
+ | <wikitex>; | ||
+ | Let $BPL$ and $BTOP$ 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 $TOP(n) = \text{Homeo}(\Rr^n, *)$ and $TOP = \text{lim}_{n \to\infty} TOP(n)$, the piecewise linear case requires more care. | ||
+ | If $CAT = PL$ or $TOP$, and $\gamma : B \to BCAT$ is a fibration, and $M$ is a compact $CAT$ manifold then just as above, we can define an $B$-structure on $M$ to be an equivalence class of lifts of of the classifying map of the stable normal bundle of $M$: | ||
+ | $$ | ||
+ | \xymatrix{ | ||
+ | & B \ar[d]^{\gamma} \\ | ||
+ | M \ar[r]^{\nu_M} \ar[ur]^{\bar \nu} & BCAT.} | ||
+ | $$ | ||
+ | Note that $CAT$ manifolds have stable normal $CAT$ bundles classified by $\nu_M \to BCAT$. | ||
+ | |||
+ | Just as before we obtain bordism groups $\Omega_n^B$ of closed n-dimensional $CAT$-manifolds with $B$ structure | ||
+ | $$ \Omega_n^B : = \{ [M, \bar \nu ]\}.$$ | ||
+ | The fibration $B$ again defines a Thom spectrum $MB$ 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 $PL$-category, it is was a major breakthrough to achieve this for topological manifolds: achieved in {{cite|Kirby&Siebenmann1977}} for dimensions other than 4 and then in {{cite|Freedman&Quinn1990}} in dimension 4. Thus one has | ||
+ | |||
+ | {{beginthm|Theorem}} | ||
+ | There is an isomorphism $\Omega_n^B \cong \pi_n^S(MB)$. | ||
+ | {{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}}. | ||
+ | |||
+ | However, working rationally, the natural maps $O \to PL$ and $O \to TOP$ induce isomorphisms $$\pi_i(O) \otimes \Qq \cong \pi_i(PL) \otimes \Qq ~~ \text{and} ~~ \pi_i(O) \otimes \Qq \cong \pi_i(TOP) \otimes \Qq ~~\forall i.$$ | ||
+ | As a consequence one has | ||
+ | |||
+ | {{beginthm|Theorem}} | ||
+ | There are isomorphisms | ||
+ | <!--$$\Omega_i^{TOP} \otimes \Qq = \Omega_i^{PL} \otimes \Qq = 0 ~~ \forall i, $$--> | ||
+ | $$ \Omega_i^{SO} \otimes \Qq \cong \Omega_i^{SPL} \otimes \Qq \cong \Omega_i^{STOP} \otimes \Qq ~~ \forall i.$$ | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
== 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]] |
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 |
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 and is homotopy equivalent to a CW complex of finite type. Abusing notation, one writes for the fibration . Speaking somewhat imprecisely (precise details are below) a -manifold is a compact manifold together with a lift to of a classifying map for the stable normal bundle of :
The -dimensional -bordism group is defined to be the set of closed -manifolds modulo the relation of bordism via compact -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.
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.
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 bundle gives 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]. Let be a fibred stable vector bundle. A -structure on is an equivalence class of -structure on where two such structures are equivalent if they become equivalent for r sufficiently large. A -manifold is a pair where is a compact manifold and is a -structure on .
If is a compact manifold with boundary then by choosing the inward-pointing normal vector along , a -structure on restricts to a -structure on . In particular, if is a closed manifold then has a canonical -structure which restricts to on . The restriction of this -structure to is denoted : by construction is the boundary of .
Definition 2.6. Closed -manifolds and are -bordant if there is a compact -manifold such that . We write for the bordism class of .
Proposition 2.7 [Stong1968, p 17]. The set of -bordism classes of closed n-manifolds with -structure,
forms an abelian group under the operation of disjoint union with inverse .
3 Singular bordism
-bordism gives rise to a generalised homology theory. If is a space then the -cycles of this homology theory are pairs
where is a closed -dimensional -manifold and is any continuous map. Two cycles and are homologous if there is a pair
where is a -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 .
4 The orientation homomorphism
We fix a local orientation at the base-point of . It then follows that every closed -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 closed -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 . It is easy to check that depends only on the -bordism class of and is additive with respect to the operations on .
Definition 4.1. Let be a fibred stable vector bundle. The orientation homomorphism is defined as follows:
For the singular bordism groups we have no bundle over so in general there is only a -valued orientation homomorphism. However, if the first Stiefel-Whitney class of vanishes, , then all -manifolds are oriented in the usual sense and the orientation homomorphism can be lifted to .
Definition 4.2. Let be a fibred stable vector bundle. The orientation homomorphism in singular bordism is defined as follows:
If then for all closed -manifolds and we can replace the -coefficients with -coefficients above.
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 a -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.
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. Let be a fibred stable vector bundle. There is a spectral sequence
Theorem 6.2. Let be 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
Moreover for any space , and if is connected, the rationalised orientation homomorphism may be identified with the projection
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 an -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 with structure
The fibration again 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
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