Fake lens spaces
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 15:18, 25 April 2013 and the changes since publication. |
Contents |
1 Introduction
A fake lens space is the orbit space of a free action of a finite cyclic group
on a sphere
. It is a generalization of the notion of a lens space which is
the orbit space of a free action which comes from a unitary representation.
2 Construction and examples
- ...
3 Invariants
-
,
for
-
,
,
for
,
for all other values of
.
-
,
, ...
4 Homotopy Classification
All the results are taken from chapter 14E of [Wall1999].
Notation
Recall the arithmetic (Rim) square:
![\displaystyle \xymatrix{ \Zz G \ar[r]^{\eta} \ar[d]_{\varepsilon} & R_G \ar[d]^{\varepsilon'} \\ \Zz \ar[r]_{\eta'} & \Zz_N }](/images/math/3/4/d/34da3249bf4eb2eac0505f1c05c1a32f.png)
where with
be the group ring of
and
is the ideal generated by the norm element
of
. The maps
,
are the augmentation maps.
Recall that the Reidemeister torsion is a unit in where
.
The homotopy classification is stated in the a priori broader context of finite CW-complexes with
and with the universal cover homotopy equivalent to
of which fake lens spaces are obviously a special case. It is convenient to make the
following definition.
Definition 4.1.
A polarization of a CW-complex with
and with the universal cover homotopy equivalent to
is a pair
where
is a choice of a generator of
and
is a choice of a homotopy equivalence
.
Recall the classical lens space . By
is denoted its
-skeleton with respect to the standard cell decomposition. If
is odd this is a lens space, if
is even this is a CW-complex obtained by attaching an
-cell to the lens space of dimension
.
Proposition 4.2.
Let be a finite CW-complex with
and universal cover
polarized by
. Then there exists a simple homotopy equivalence
![\displaystyle h \colon L \rightarrow L^{2d-2}(N,1,\ldots,1) \cup_\phi e^{2d-1}](/images/math/8/5/9/859dfd1a13049d27d30567c8ad21da1e.png)
preserving the polarization. It is unique up to homotopy and the
action of . The chain complex differential on the right hand side
is given by
for some
which maps to a unit
. Then
is a simple Poincare complex with Reidemeister torsion
.
- The polarized homotopy types of such
are in one-to-one correspondence with the units in
. The correspondence is given by
. The invariant
can be identified with the first non-trivial
-invariant of
(in the sense of homotopy theory)
.
- The polarized simple homotopy types of such
are in one-to-one correspondence with the units in
. The correspondence is given by
.
See Theorem 14E.3 in [Wall1999].
The existence of a fake lens space in the homotopy type of such
is addressed in [Theorem 14E.4] of [Wall1999].
Since the units are exhausted by the lens spaces
we obtain the following corollary.
Corollary 4.3.
For any fake lens space there exists
and a homotopy equivalence
![\displaystyle h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1).](/images/math/f/7/4/f7452a4efe890ba1ac37885e48d654c3.png)
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
![\displaystyle \xymatrix{ 0 \ar[r] & {\widetilde L}^s_{2d} (G) \ar[r]^(0.4){\partial} \ar[d]_{\cong}^{G-sign} & {\mathcal S}^s (L^{2d-1}(\alpha)) \ar[r]^{\eta} \ar[d]^{\widetilde \rho}& \widetilde {\mathcal N} (L^{2d-1}(\alpha)) \ar[r] \ar[d]^{[\widetilde \rho]}& 0 \\ 0 \ar[r] & 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G}/ 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & 0 }](/images/math/c/e/0/ce0adafd80f8f574c18c1ece6266bff7.png)
where is the homomorphism induced by
(see ?).
The map is injective if
with
odd (compare [Wall1999, Corollary on page 222]?).
The following theorem is taken from [Wall1999, Theorem 14E.7].
Theorem 5.1.
Let and
be oriented fake lens spaces with fundamental group
cyclic of odd order
. Then there is an orientation preserving homeomorphism
inducing the identity on
if and only if
and
.
Given and
, there exists a corresponding fake lens space
if and only if the following four statements hold:
-
and
are both real (
even) or imaginary (
odd).
-
generates ...
- ...
- ...
The following theorem is proved in [Macko&Wegner2010, Theorem 1.2]).
Theorem 5.2.
Let be a fake lens space with
where
with
,
odd and
. Then we have
![\displaystyle {\mathcal S}^s (L^{2d-1}(\alpha)) \cong \bar \Sigma_N (d) \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,1\}}} \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,2i\}}}](/images/math/f/6/3/f6344e3373c335dbe886fe14bf63a150.png)
where is a free abelian group. If
is odd then its rank is
. If
is even then its rank is
if
and
if
. In the torsion summand we have
.
6 Further discussion
- ...
7 References
- [Macko&Wegner2010] Template:Macko&Wegner2010
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
This page has not been refereed. The information given here might be incomplete or provisional. |
![S^{2d-1}](/images/math/8/f/7/8f719df09bfc2311b6a672809262d654.png)
2 Construction and examples
- ...
3 Invariants
-
,
for
-
,
,
for
,
for all other values of
.
-
,
, ...
4 Homotopy Classification
All the results are taken from chapter 14E of [Wall1999].
Notation
Recall the arithmetic (Rim) square:
![\displaystyle \xymatrix{ \Zz G \ar[r]^{\eta} \ar[d]_{\varepsilon} & R_G \ar[d]^{\varepsilon'} \\ \Zz \ar[r]_{\eta'} & \Zz_N }](/images/math/3/4/d/34da3249bf4eb2eac0505f1c05c1a32f.png)
where with
be the group ring of
and
is the ideal generated by the norm element
of
. The maps
,
are the augmentation maps.
Recall that the Reidemeister torsion is a unit in where
.
The homotopy classification is stated in the a priori broader context of finite CW-complexes with
and with the universal cover homotopy equivalent to
of which fake lens spaces are obviously a special case. It is convenient to make the
following definition.
Definition 4.1.
A polarization of a CW-complex with
and with the universal cover homotopy equivalent to
is a pair
where
is a choice of a generator of
and
is a choice of a homotopy equivalence
.
Recall the classical lens space . By
is denoted its
-skeleton with respect to the standard cell decomposition. If
is odd this is a lens space, if
is even this is a CW-complex obtained by attaching an
-cell to the lens space of dimension
.
Proposition 4.2.
Let be a finite CW-complex with
and universal cover
polarized by
. Then there exists a simple homotopy equivalence
![\displaystyle h \colon L \rightarrow L^{2d-2}(N,1,\ldots,1) \cup_\phi e^{2d-1}](/images/math/8/5/9/859dfd1a13049d27d30567c8ad21da1e.png)
preserving the polarization. It is unique up to homotopy and the
action of . The chain complex differential on the right hand side
is given by
for some
which maps to a unit
. Then
is a simple Poincare complex with Reidemeister torsion
.
- The polarized homotopy types of such
are in one-to-one correspondence with the units in
. The correspondence is given by
. The invariant
can be identified with the first non-trivial
-invariant of
(in the sense of homotopy theory)
.
- The polarized simple homotopy types of such
are in one-to-one correspondence with the units in
. The correspondence is given by
.
See Theorem 14E.3 in [Wall1999].
The existence of a fake lens space in the homotopy type of such
is addressed in [Theorem 14E.4] of [Wall1999].
Since the units are exhausted by the lens spaces
we obtain the following corollary.
Corollary 4.3.
For any fake lens space there exists
and a homotopy equivalence
![\displaystyle h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1).](/images/math/f/7/4/f7452a4efe890ba1ac37885e48d654c3.png)
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
![\displaystyle \xymatrix{ 0 \ar[r] & {\widetilde L}^s_{2d} (G) \ar[r]^(0.4){\partial} \ar[d]_{\cong}^{G-sign} & {\mathcal S}^s (L^{2d-1}(\alpha)) \ar[r]^{\eta} \ar[d]^{\widetilde \rho}& \widetilde {\mathcal N} (L^{2d-1}(\alpha)) \ar[r] \ar[d]^{[\widetilde \rho]}& 0 \\ 0 \ar[r] & 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G}/ 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & 0 }](/images/math/c/e/0/ce0adafd80f8f574c18c1ece6266bff7.png)
where is the homomorphism induced by
(see ?).
The map is injective if
with
odd (compare [Wall1999, Corollary on page 222]?).
The following theorem is taken from [Wall1999, Theorem 14E.7].
Theorem 5.1.
Let and
be oriented fake lens spaces with fundamental group
cyclic of odd order
. Then there is an orientation preserving homeomorphism
inducing the identity on
if and only if
and
.
Given and
, there exists a corresponding fake lens space
if and only if the following four statements hold:
-
and
are both real (
even) or imaginary (
odd).
-
generates ...
- ...
- ...
The following theorem is proved in [Macko&Wegner2010, Theorem 1.2]).
Theorem 5.2.
Let be a fake lens space with
where
with
,
odd and
. Then we have
![\displaystyle {\mathcal S}^s (L^{2d-1}(\alpha)) \cong \bar \Sigma_N (d) \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,1\}}} \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,2i\}}}](/images/math/f/6/3/f6344e3373c335dbe886fe14bf63a150.png)
where is a free abelian group. If
is odd then its rank is
. If
is even then its rank is
if
and
if
. In the torsion summand we have
.
6 Further discussion
- ...
7 References
- [Macko&Wegner2010] Template:Macko&Wegner2010
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
This page has not been refereed. The information given here might be incomplete or provisional. |