Fake lens spaces
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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
- We cite mainly from
[Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which
are a special sort of fake lens spaces. LetTex syntax error, , where
Tex syntax errorare such that . When
Tex syntax errordefine a representation of on
Tex syntax errorby . Any free representation of on a
-dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the
groupTex syntax erroron for some as above.\footnote{In the notation of [Wall(1999), chapter
14E] we have .} \end{defn}
The lens space is a -dimensional
manifold withTex syntax error. Its
universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension .
Moreover, we haveTex syntax errorwhen ,
Tex syntax errorwhen is odd and
when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in
terms of Reidemeister torsion which is a unit inTex syntax error. This ring is defined as
Tex syntax errorwith
Tex syntax errorwhere
Tex syntax errorbe the group ring of and
is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There
is also an augmentation mapTex syntax error
[Wall(1999), page 214]. The homotopy classification is stated
in terms of a certain unit inTex syntax error. These invariants also suffice
for the homotopy and simple homotopy classification of finite
CW-complexes withTex syntax errorand 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. \begin{defn} A {\it polarization} of a CW-complex as above is a pair where is a choice of a generator of and is a
choice of a homotopy equivalenceTex syntax error.
\end{defn} \noindent Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 . \begin{prop}Wall(1999) Let be a finite CW-complex as above polarized by . Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] 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
Tex syntax errorwhich maps to a unit . Furthermore, the
complex is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such are in
one-to-one correspondence with the units inTex syntax error. The correspondence is given by
Tex syntax error.
\item The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by . \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.4]. Since the
unitsTex syntax errorare exhausted by the lens spaces
we obtain the following corollary. \begin{cor}
For any fake lens space there existsTex syntax error
and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}
5 Homeomorphism classification
- ...
6 Further discussion
- ...
7 References
This page has not been refereed. The information given here might be incomplete or provisional. |
2 Construction and examples
- ...
3 Invariants
- , for
- , , for , for all other values of .
- , , ...
4 Homotopy Classification
- We cite mainly from
[Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which
are a special sort of fake lens spaces. LetTex syntax error, , where
Tex syntax errorare such that . When
Tex syntax errordefine a representation of on
Tex syntax errorby . Any free representation of on a
-dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the
groupTex syntax erroron for some as above.\footnote{In the notation of [Wall(1999), chapter
14E] we have .} \end{defn}
The lens space is a -dimensional
manifold withTex syntax error. Its
universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension .
Moreover, we haveTex syntax errorwhen ,
Tex syntax errorwhen is odd and
when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in
terms of Reidemeister torsion which is a unit inTex syntax error. This ring is defined as
Tex syntax errorwith
Tex syntax errorwhere
Tex syntax errorbe the group ring of and
is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There
is also an augmentation mapTex syntax error
[Wall(1999), page 214]. The homotopy classification is stated
in terms of a certain unit inTex syntax error. These invariants also suffice
for the homotopy and simple homotopy classification of finite
CW-complexes withTex syntax errorand 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. \begin{defn} A {\it polarization} of a CW-complex as above is a pair where is a choice of a generator of and is a
choice of a homotopy equivalenceTex syntax error.
\end{defn} \noindent Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 . \begin{prop}Wall(1999) Let be a finite CW-complex as above polarized by . Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] 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
Tex syntax errorwhich maps to a unit . Furthermore, the
complex is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such are in
one-to-one correspondence with the units inTex syntax error. The correspondence is given by
Tex syntax error.
\item The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by . \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.4]. Since the
unitsTex syntax errorare exhausted by the lens spaces
we obtain the following corollary. \begin{cor}
For any fake lens space there existsTex syntax error
and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}
5 Homeomorphism classification
- ...
6 Further discussion
- ...
7 References
This page has not been refereed. The information given here might be incomplete or provisional. |
2 Construction and examples
- ...
3 Invariants
- , for
- , , for , for all other values of .
- , , ...
4 Homotopy Classification
- We cite mainly from
[Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which
are a special sort of fake lens spaces. LetTex syntax error, , where
Tex syntax errorare such that . When
Tex syntax errordefine a representation of on
Tex syntax errorby . Any free representation of on a
-dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the
groupTex syntax erroron for some as above.\footnote{In the notation of [Wall(1999), chapter
14E] we have .} \end{defn}
The lens space is a -dimensional
manifold withTex syntax error. Its
universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension .
Moreover, we haveTex syntax errorwhen ,
Tex syntax errorwhen is odd and
when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in
terms of Reidemeister torsion which is a unit inTex syntax error. This ring is defined as
Tex syntax errorwith
Tex syntax errorwhere
Tex syntax errorbe the group ring of and
is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There
is also an augmentation mapTex syntax error
[Wall(1999), page 214]. The homotopy classification is stated
in terms of a certain unit inTex syntax error. These invariants also suffice
for the homotopy and simple homotopy classification of finite
CW-complexes withTex syntax errorand 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. \begin{defn} A {\it polarization} of a CW-complex as above is a pair where is a choice of a generator of and is a
choice of a homotopy equivalenceTex syntax error.
\end{defn} \noindent Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 . \begin{prop}Wall(1999) Let be a finite CW-complex as above polarized by . Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] 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
Tex syntax errorwhich maps to a unit . Furthermore, the
complex is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such are in
one-to-one correspondence with the units inTex syntax error. The correspondence is given by
Tex syntax error.
\item The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by . \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.4]. Since the
unitsTex syntax errorare exhausted by the lens spaces
we obtain the following corollary. \begin{cor}
For any fake lens space there existsTex syntax error
and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}
5 Homeomorphism classification
- ...
6 Further discussion
- ...
7 References
This page has not been refereed. The information given here might be incomplete or provisional. |
2 Construction and examples
- ...
3 Invariants
- , for
- , , for , for all other values of .
- , , ...
4 Homotopy Classification
- We cite mainly from
[Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which
are a special sort of fake lens spaces. LetTex syntax error, , where
Tex syntax errorare such that . When
Tex syntax errordefine a representation of on
Tex syntax errorby . Any free representation of on a
-dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the
groupTex syntax erroron for some as above.\footnote{In the notation of [Wall(1999), chapter
14E] we have .} \end{defn}
The lens space is a -dimensional
manifold withTex syntax error. Its
universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension .
Moreover, we haveTex syntax errorwhen ,
Tex syntax errorwhen is odd and
when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in
terms of Reidemeister torsion which is a unit inTex syntax error. This ring is defined as
Tex syntax errorwith
Tex syntax errorwhere
Tex syntax errorbe the group ring of and
is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There
is also an augmentation mapTex syntax error
[Wall(1999), page 214]. The homotopy classification is stated
in terms of a certain unit inTex syntax error. These invariants also suffice
for the homotopy and simple homotopy classification of finite
CW-complexes withTex syntax errorand 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. \begin{defn} A {\it polarization} of a CW-complex as above is a pair where is a choice of a generator of and is a
choice of a homotopy equivalenceTex syntax error.
\end{defn} \noindent Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 . \begin{prop}Wall(1999) Let be a finite CW-complex as above polarized by . Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] 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
Tex syntax errorwhich maps to a unit . Furthermore, the
complex is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such are in
one-to-one correspondence with the units inTex syntax error. The correspondence is given by
Tex syntax error.
\item The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by . \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.4]. Since the
unitsTex syntax errorare exhausted by the lens spaces
we obtain the following corollary. \begin{cor}
For any fake lens space there existsTex syntax error
and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}
5 Homeomorphism classification
- ...
6 Further discussion
- ...
7 References
This page has not been refereed. The information given here might be incomplete or provisional. |