Talk:Normal bordism - definitions (Ex)
In both parts let be a connected finite Poincare complex of dimension
and let
.
Part 1
The following definition of the set of normal maps is similar to [Lück2001, Definition 3.46].
We define
![\displaystyle \mathcal{N}_n(X,k):= \left\{ [\xi,M,i,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, i:\,M\to\mathbb{R}^{n+k}\textrm{ embedding},\, (f,\overline{f}):\,\nu(M,i)\to\xi\textrm{ bundle map},\, f:\,M\to X\textrm{ of degree }1 \end{array} \right\}](/images/math/9/5/e/95eac2d8c047fc766e68218c6e2cd0fc.png)
where we identify iff
1) There exists a compact manifold of dimension
whose boundary can be written as
.
2) There exists an embedding :
such that for
we have
and
meets
transversally.
3) There exists a vector bundle :
of rank
and for
there exist vector bundle isomorphisms
:
.
4) There exists a bundle map :
such that for
we have
and such that
:
has degree one as a map between Poincare pairs.
5) For there exist diffeomorphisms
:
such that
a) :
is a diffeomorphism
b)
c) the induced bundle map :
satisfies
.
Part 2
The following definition of the set of tangential normal maps differs from [Lück2001, Definition 3.50].
We define
![\displaystyle \mathcal{N}^T_n(X,k):= \left\{ [\xi,M,a,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, a\in\mathbb{N}_0,\, (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, f:\,M\to X\textrm{ of degree }1 \end{array} \right\}](/images/math/1/c/d/1cd45eeb72d7fe8f65eded6c4e63162e.png)
where we identify iff
1) There exists a compact manifold of dimension
whose boundary can be written as
.
2) There exists a vector bundle :
and there exist
and
a bundle map
:
such that for
we have
and such that
:
has degree one as a map between Poincare pairs.
3) For there exist diffeomorphisms
:
such that
.
4) For there exist bundle isomorphisms
:
such that
![\displaystyle \xymatrix{TM_j\oplus\underline{\mathbb{R}}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}}} \ar[d]_{TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}}} & & \xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}} \ar[d]^{V_j}\\ TW|_{\partial_jW}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{F}|_{\partial_jW}} & & \eta|_{X\times\{j\}} }](/images/math/7/4/e/74e20705c840b6ca855d5ac331779b90.png)
commutes.
Here :
is the differential of
and
:
is given by an inward normal field of
.