Manifolds with singularities

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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
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Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ({{cite|Sullivan1996}},{{cite|Sullivan1967}}) and {{cite|Baas1973}}. Applications of the concept include representing cycles in homology theories with coefficients.
+
Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ({{cite|Sullivan1996}},{{cite|Sullivan1967}}) and {{cite|Baas1973}}. Applications of the concept include representing cycles in homology theories with coefficients. All manifolds in this article are understood to be smooth.
</wikitex>
</wikitex>
Line 32: Line 32:
<wikitex>
<wikitex>
Following ({{cite|Botvinnik2001}}, {{cite|Botvinnik1992}}), a more general definition can be given. Let $(P_1 , ..., P_k)$ be a (possibly empty) collection of closed manifolds and denote by $P_0$ the set containing only one point. Then define $\Sigma_k := (P_0, P_1, ... , P_k)$. For a subset $I = \{i_1,..., i_q\} \subset \{0,...,k\}$ define $P^I := P_{i_1} \times ...\times P_{i_q}$.
Following ({{cite|Botvinnik2001}}, {{cite|Botvinnik1992}}), a more general definition can be given. Let $(P_1 , ..., P_k)$ be a (possibly empty) collection of closed manifolds and denote by $P_0$ the set containing only one point. Then define $\Sigma_k := (P_0, P_1, ... , P_k)$. For a subset $I = \{i_1,..., i_q\} \subset \{0,...,k\}$ define $P^I := P_{i_1} \times ...\times P_{i_q}$.
−
{{beginthm|Definition}}A manifold $M$ is a $\Sigma_k$-Manifold if
+
{{beginthm|Definition}}A manifold $M$ is a $\Sigma_k$-Manifold if there is given
−
# there is a partition $\partial M = \partial_0 M \cup ... \cup \partial_k M$, such that $\partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M$ is a manifold for each $I = \{i_1,...,i_q\} \subset \{0,...,k\}$, and such that
+
# a partition $\partial M = \partial_0 M \cup ... \cup \partial_k M$, such that $\partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M$ is a manifold for each $I = \{i_1,...,i_q\} \subset \{0,...,k\}$, and such that
$$ \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M $$.
$$ \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M $$.
−
# for each $I \subset \{0,...,k\}$ there is a manifold $\beta_I M$ and a diffeomorphism
+
# for each $I \subset \{0,...,k\}$ a manifold $\beta_I M$ and a diffeomorphism
$$ \phi_I: \partial_I M \rightarrow \beta_I M \times P^I$$ ,
$$ \phi_I: \partial_I M \rightarrow \beta_I M \times P^I$$ ,
such that if $J \subset I$ and $\iota: \partial_I M \rightarrow \partial_J M$ is the inclusion, then the composition
such that if $J \subset I$ and $\iota: \partial_I M \rightarrow \partial_J M$ is the inclusion, then the composition
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The spaces defined above as manifolds with a $P_1$-singularity are contained in this more general definition. In fact, they give the manifolds with a $\Sigma_1$-singularity. For given a manifold $P_1$, set $\Sigma_1 = (P_0, P_1)$. Then the manifold $A$ with boundary $A(1) \times P_1$, which appears in the above definition, is a $\Sigma_1$-manifold. The attachement of the cone-end $A(1) \times C P_1$ now corresponds to the collapsing of the equivalence relation $\sim$ in $A$.
The spaces defined above as manifolds with a $P_1$-singularity are contained in this more general definition. In fact, they give the manifolds with a $\Sigma_1$-singularity. For given a manifold $P_1$, set $\Sigma_1 = (P_0, P_1)$. Then the manifold $A$ with boundary $A(1) \times P_1$, which appears in the above definition, is a $\Sigma_1$-manifold. The attachement of the cone-end $A(1) \times C P_1$ now corresponds to the collapsing of the equivalence relation $\sim$ in $A$.
+
+
When dealing with manifolds with singularities it is convenient to work with the underlying $\Sigma$-manifold and make sure that all operations one performs on them are compatible with the equivalence relation.
</wikitex>
</wikitex>
−
== Some examples ==
+
== Some examples and constructions ==
+
+
===Intersecting spheres===
===Intersecting spheres===
<wikitex>;
<wikitex>;
−
A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding $i: S^{n-1} \rightarrow S^n$. Then define $\overline{S} := S^n \cup_{j(S^{n-1}} S^n$. Outside of the intersecting sphere $j(S^{n-1})$ this is an $n$-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of $S^{n-1}$ and a cone over $\Zz_4$ . We can write $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$.
+
A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding $i: S^{n-1} \rightarrow S^n$. Then define $\overline{S} := S^n \cup_{i(S^{n-1})} S^n$. Outside of the intersecting sphere $i(S^{n-1})$ this is an $n$-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of $S^{n-1}$ and a cone over $\Zz_4$ . We can write $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$.
</wikitex>
</wikitex>
===Inverse images of critical points===
===Inverse images of critical points===
<wikitex>;
<wikitex>;
−
Let $f:\Rr^n \rightarrow \Rr$ be a Morse-function with $0$ as a single critical point. We can suppose that $f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n$ near $0$. Setting $M := f^{-1}(\{0\})$, we see that the cone $C S^{k-1} \times S^{n-k-1} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-1} \subset \Rr^n \}$ provides a neighborhood of $0$ in $M$. It follows that $M$ is of the form $
+
Let $f:\Rr^n \rightarrow \Rr$ be a Morse-function with $0$ as a single critical point. We can suppose that $f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n$ near $0$. Setting $M := f^{-1}(\{0\})$, we see that the cone $C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \}$ provides a neighborhood of $0$ in $M$. It follows that $M$ is of the form $N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2})$.
+
</wikitex>
+
===Structures on manifolds with singularities===
+
<wikitex>;
+
Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with $\Sigma$-singularity $\overline{M}$ with underlying $\Sigma$-manifold $M$ carries the structure in question if all manifolds involved in $\textbf{Definition 2.1}$ carry this structure and the product diffeomorphisms preserves it.
+
+
For example, $\overline{M}$ is orientable if $M$ as well as all manifolds in $\Sigma$ and the manifolds $\beta_I M$ are orientable and the product diffeomorphisms are orientation-preserving. As another example, $\overline{M}$ becomes a Riemannian manifold with singularities if we put a Riemannian metric on $M$ as well as on the manifolds in $\Sigma$ and on the manifolds $\beta_I M$ in such a way that the product diffeomorphism are isometries.
+
</wikitex>
+
+
===Bundles on manifolds with singularities===
+
<wikitex>;
+
As usual, we define a bundle $\overline{E}$ on a manifold with singularities $\overline{M}$ as a bundle $E$ on the underlying manifold $M$ subject to the following additional condition: there are bundles $E_I$ over the manifolds $\beta_I M \times P^I$ and bundle equivalences $E|_{\partial_I M} \rightarrow E_I$ covering the product diffeomorphisms.
+
</wikitex>
</wikitex>
+
===Bordism theory for manifolds with singularities===
== Invariants ==
== Invariants ==
<wikitex>;
<wikitex>;

Revision as of 13:56, 8 June 2010

Contents

1 Introduction

Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ([Sullivan1996],[Sullivan1967]) and [Baas1973]. Applications of the concept include representing cycles in homology theories with coefficients. All manifolds in this article are understood to be smooth.


2 Definitions

2.1 Cone-like singularities

A manifold with singularities of Baas-Sullivan type is a topological space \overline{A} that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let P_1 be a closed manifold. A manifold with a P_1-singularity (following [Baas1973]) is a space of the form

\displaystyle \overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)
\displaystyle \partial A = A(1) \times P_1

Here, A is a manifold with boundary A(1).


2.2 \Sigma-manifolds


Following ([Botvinnik2001], [Botvinnik1992]), a more general definition can be given. Let (P_1 , ..., P_k) be a (possibly empty) collection of closed manifolds and denote by P_0 the set containing only one point. Then define \Sigma_k := (P_0, P_1, ... , P_k). For a subset I = \{i_1,..., i_q\} \subset \{0,...,k\} define P^I := P_{i_1} \times ...\times P_{i_q}.

Definition 2.1.A manifold 
Tex syntax error
 is a \Sigma_k-Manifold if there is given
  1. a partition \partial M = \partial_0 M \cup ... \cup \partial_k M, such that \partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M is a manifold for each I = \{i_1,...,i_q\} \subset \{0,...,k\}, and such that
\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M
.
  1. for each I \subset \{0,...,k\} a manifold \beta_I M and a diffeomorphism
\displaystyle \phi_I: \partial_I M \rightarrow \beta_I M \times P^I
,

such that if J \subset I and \iota: \partial_I M \rightarrow \partial_J M is the inclusion, then the composition

\displaystyle \phi_J \circ \iota \circ \phi_I^{-1}: \beta_I M \times P^I \rightarrow \beta_J M \times P^J

restricts to the identity on the factor P^J in P^I. The diffeomorphisms \phi_I are called product structures.

On a \Sigma_k-manifold 
Tex syntax error
, there is a canonical equivalence relation \sim: two points x,y \in M are defined to be equivalent if there is an I \subset \{0,...,k\} such that x,y \in \partial_I M and pr \circ \phi_I(x) = pr \circ \phi_I(y), where pr: \beta_I M \times P^I \rightarrow \beta_I M is the projection.

Now we can give a general definition: a manifold with a \Sigma_k-singularity is a topological space \overline{A} of the form

\displaystyle \overline{A} = A / \sim

for a \Sigma_k-manifold A.

The spaces defined above as manifolds with a P_1-singularity are contained in this more general definition. In fact, they give the manifolds with a \Sigma_1-singularity. For given a manifold P_1, set \Sigma_1 = (P_0, P_1). Then the manifold A with boundary A(1) \times P_1, which appears in the above definition, is a \Sigma_1-manifold. The attachement of the cone-end A(1) \times C P_1 now corresponds to the collapsing of the equivalence relation \sim in A.

When dealing with manifolds with singularities it is convenient to work with the underlying \Sigma-manifold and make sure that all operations one performs on them are compatible with the equivalence relation.


3 Some examples and constructions

3.1 Intersecting spheres

A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding i: S^{n-1} \rightarrow S^n. Then define \overline{S} := S^n \cup_{i(S^{n-1})} S^n. Outside of the intersecting sphere i(S^{n-1}) this is an n-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of S^{n-1} and a cone over \Zz_4 . We can write \overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4.

3.2 Inverse images of critical points

Let f:\Rr^n \rightarrow \Rr be a Morse-function with 0 as a single critical point. We can suppose that f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n near 0. Setting M := f^{-1}(\{0\}), we see that the cone C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \} provides a neighborhood of 0 in 
Tex syntax error
. It follows that 
Tex syntax error
 is of the form N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2}).

3.3 Structures on manifolds with singularities

Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with \Sigma-singularity \overline{M} with underlying \Sigma-manifold 
Tex syntax error
 carries the structure in question if all manifolds involved in \textbf{Definition 2.1} carry this structure and the product diffeomorphisms preserves it. For example, \overline{M} is orientable if 
Tex syntax error
 as well as all manifolds in \Sigma and the manifolds \beta_I M are orientable and the product diffeomorphisms are orientation-preserving. As another example, \overline{M} becomes a Riemannian manifold with singularities if we put a Riemannian metric on 
Tex syntax error
 as well as on the manifolds in \Sigma and on the manifolds \beta_I M in such a way that the product diffeomorphism are isometries.

3.4 Bundles on manifolds with singularities

As usual, we define a bundle \overline{E} on a manifold with singularities \overline{M} as a bundle E on the underlying manifold 
Tex syntax error
 subject to the following additional condition: there are bundles E_I over the manifolds \beta_I M \times P^I and bundle equivalences E|_{\partial_I M} \rightarrow E_I covering the product diffeomorphisms.


3.5 Bordism theory for manifolds with singularities

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

This page has not been refereed. The information given here might be incomplete or provisional.

$ as a single critical point. We can suppose that $f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n$ near that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let P_1 be a closed manifold. A manifold with a P_1-singularity (following [Baas1973]) is a space of the form

\displaystyle \overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)
\displaystyle \partial A = A(1) \times P_1

Here, A is a manifold with boundary A(1).


2.2 \Sigma-manifolds


Following ([Botvinnik2001], [Botvinnik1992]), a more general definition can be given. Let (P_1 , ..., P_k) be a (possibly empty) collection of closed manifolds and denote by P_0 the set containing only one point. Then define \Sigma_k := (P_0, P_1, ... , P_k). For a subset I = \{i_1,..., i_q\} \subset \{0,...,k\} define P^I := P_{i_1} \times ...\times P_{i_q}.

Definition 2.1.A manifold 
Tex syntax error
 is a \Sigma_k-Manifold if there is given
  1. a partition \partial M = \partial_0 M \cup ... \cup \partial_k M, such that \partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M is a manifold for each I = \{i_1,...,i_q\} \subset \{0,...,k\}, and such that
\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M
.
  1. for each I \subset \{0,...,k\} a manifold \beta_I M and a diffeomorphism
\displaystyle \phi_I: \partial_I M \rightarrow \beta_I M \times P^I
,

such that if J \subset I and \iota: \partial_I M \rightarrow \partial_J M is the inclusion, then the composition

\displaystyle \phi_J \circ \iota \circ \phi_I^{-1}: \beta_I M \times P^I \rightarrow \beta_J M \times P^J

restricts to the identity on the factor P^J in P^I. The diffeomorphisms \phi_I are called product structures.

On a \Sigma_k-manifold 
Tex syntax error
, there is a canonical equivalence relation \sim: two points x,y \in M are defined to be equivalent if there is an I \subset \{0,...,k\} such that x,y \in \partial_I M and pr \circ \phi_I(x) = pr \circ \phi_I(y), where pr: \beta_I M \times P^I \rightarrow \beta_I M is the projection.

Now we can give a general definition: a manifold with a \Sigma_k-singularity is a topological space \overline{A} of the form

\displaystyle \overline{A} = A / \sim

for a \Sigma_k-manifold A.

The spaces defined above as manifolds with a P_1-singularity are contained in this more general definition. In fact, they give the manifolds with a \Sigma_1-singularity. For given a manifold P_1, set \Sigma_1 = (P_0, P_1). Then the manifold A with boundary A(1) \times P_1, which appears in the above definition, is a \Sigma_1-manifold. The attachement of the cone-end A(1) \times C P_1 now corresponds to the collapsing of the equivalence relation \sim in A.

When dealing with manifolds with singularities it is convenient to work with the underlying \Sigma-manifold and make sure that all operations one performs on them are compatible with the equivalence relation.


3 Some examples and constructions

3.1 Intersecting spheres

A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding i: S^{n-1} \rightarrow S^n. Then define \overline{S} := S^n \cup_{i(S^{n-1})} S^n. Outside of the intersecting sphere i(S^{n-1}) this is an n-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of S^{n-1} and a cone over \Zz_4 . We can write \overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4.

3.2 Inverse images of critical points

Let f:\Rr^n \rightarrow \Rr be a Morse-function with 0 as a single critical point. We can suppose that f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n near 0. Setting M := f^{-1}(\{0\}), we see that the cone C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \} provides a neighborhood of 0 in 
Tex syntax error
. It follows that 
Tex syntax error
 is of the form N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2}).

3.3 Structures on manifolds with singularities

Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with \Sigma-singularity \overline{M} with underlying \Sigma-manifold 
Tex syntax error
 carries the structure in question if all manifolds involved in \textbf{Definition 2.1} carry this structure and the product diffeomorphisms preserves it. For example, \overline{M} is orientable if 
Tex syntax error
 as well as all manifolds in \Sigma and the manifolds \beta_I M are orientable and the product diffeomorphisms are orientation-preserving. As another example, \overline{M} becomes a Riemannian manifold with singularities if we put a Riemannian metric on 
Tex syntax error
 as well as on the manifolds in \Sigma and on the manifolds \beta_I M in such a way that the product diffeomorphism are isometries.

3.4 Bundles on manifolds with singularities

As usual, we define a bundle \overline{E} on a manifold with singularities \overline{M} as a bundle E on the underlying manifold 
Tex syntax error
 subject to the following additional condition: there are bundles E_I over the manifolds \beta_I M \times P^I and bundle equivalences E|_{\partial_I M} \rightarrow E_I covering the product diffeomorphisms.


3.5 Bordism theory for manifolds with singularities

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

This page has not been refereed. The information given here might be incomplete or provisional.

$. Setting $M := f^{-1}(\{0\})$, we see that the cone $C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \}$ provides a neighborhood of that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let P_1 be a closed manifold. A manifold with a P_1-singularity (following [Baas1973]) is a space of the form

\displaystyle \overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)
\displaystyle \partial A = A(1) \times P_1

Here, A is a manifold with boundary A(1).


2.2 \Sigma-manifolds


Following ([Botvinnik2001], [Botvinnik1992]), a more general definition can be given. Let (P_1 , ..., P_k) be a (possibly empty) collection of closed manifolds and denote by P_0 the set containing only one point. Then define \Sigma_k := (P_0, P_1, ... , P_k). For a subset I = \{i_1,..., i_q\} \subset \{0,...,k\} define P^I := P_{i_1} \times ...\times P_{i_q}.

Definition 2.1.A manifold 
Tex syntax error
 is a \Sigma_k-Manifold if there is given
  1. a partition \partial M = \partial_0 M \cup ... \cup \partial_k M, such that \partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M is a manifold for each I = \{i_1,...,i_q\} \subset \{0,...,k\}, and such that
\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M
.
  1. for each I \subset \{0,...,k\} a manifold \beta_I M and a diffeomorphism
\displaystyle \phi_I: \partial_I M \rightarrow \beta_I M \times P^I
,

such that if J \subset I and \iota: \partial_I M \rightarrow \partial_J M is the inclusion, then the composition

\displaystyle \phi_J \circ \iota \circ \phi_I^{-1}: \beta_I M \times P^I \rightarrow \beta_J M \times P^J

restricts to the identity on the factor P^J in P^I. The diffeomorphisms \phi_I are called product structures.

On a \Sigma_k-manifold 
Tex syntax error
, there is a canonical equivalence relation \sim: two points x,y \in M are defined to be equivalent if there is an I \subset \{0,...,k\} such that x,y \in \partial_I M and pr \circ \phi_I(x) = pr \circ \phi_I(y), where pr: \beta_I M \times P^I \rightarrow \beta_I M is the projection.

Now we can give a general definition: a manifold with a \Sigma_k-singularity is a topological space \overline{A} of the form

\displaystyle \overline{A} = A / \sim

for a \Sigma_k-manifold A.

The spaces defined above as manifolds with a P_1-singularity are contained in this more general definition. In fact, they give the manifolds with a \Sigma_1-singularity. For given a manifold P_1, set \Sigma_1 = (P_0, P_1). Then the manifold A with boundary A(1) \times P_1, which appears in the above definition, is a \Sigma_1-manifold. The attachement of the cone-end A(1) \times C P_1 now corresponds to the collapsing of the equivalence relation \sim in A.

When dealing with manifolds with singularities it is convenient to work with the underlying \Sigma-manifold and make sure that all operations one performs on them are compatible with the equivalence relation.


3 Some examples and constructions

3.1 Intersecting spheres

A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding i: S^{n-1} \rightarrow S^n. Then define \overline{S} := S^n \cup_{i(S^{n-1})} S^n. Outside of the intersecting sphere i(S^{n-1}) this is an n-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of S^{n-1} and a cone over \Zz_4 . We can write \overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4.

3.2 Inverse images of critical points

Let f:\Rr^n \rightarrow \Rr be a Morse-function with 0 as a single critical point. We can suppose that f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n near 0. Setting M := f^{-1}(\{0\}), we see that the cone C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \} provides a neighborhood of 0 in 
Tex syntax error
. It follows that 
Tex syntax error
 is of the form N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2}).

3.3 Structures on manifolds with singularities

Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with \Sigma-singularity \overline{M} with underlying \Sigma-manifold 
Tex syntax error
 carries the structure in question if all manifolds involved in \textbf{Definition 2.1} carry this structure and the product diffeomorphisms preserves it. For example, \overline{M} is orientable if 
Tex syntax error
 as well as all manifolds in \Sigma and the manifolds \beta_I M are orientable and the product diffeomorphisms are orientation-preserving. As another example, \overline{M} becomes a Riemannian manifold with singularities if we put a Riemannian metric on 
Tex syntax error
 as well as on the manifolds in \Sigma and on the manifolds \beta_I M in such a way that the product diffeomorphism are isometries.

3.4 Bundles on manifolds with singularities

As usual, we define a bundle \overline{E} on a manifold with singularities \overline{M} as a bundle E on the underlying manifold 
Tex syntax error
 subject to the following additional condition: there are bundles E_I over the manifolds \beta_I M \times P^I and bundle equivalences E|_{\partial_I M} \rightarrow E_I covering the product diffeomorphisms.


3.5 Bordism theory for manifolds with singularities

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

This page has not been refereed. The information given here might be incomplete or provisional.

$ in $M$. It follows that $M$ is of the form $N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2})$. ===Structures on manifolds with singularities=== ; Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with $\Sigma$-singularity $\overline{M}$ with underlying $\Sigma$-manifold $M$ carries the structure in question if all manifolds involved in $\textbf{Definition 2.1}$ carry this structure and the product diffeomorphisms preserves it. For example, $\overline{M}$ is orientable if $M$ as well as all manifolds in $\Sigma$ and the manifolds $\beta_I M$ are orientable and the product diffeomorphisms are orientation-preserving. As another example, $\overline{M}$ becomes a Riemannian manifold with singularities if we put a Riemannian metric on $M$ as well as on the manifolds in $\Sigma$ and on the manifolds $\beta_I M$ in such a way that the product diffeomorphism are isometries. ===Bundles on manifolds with singularities=== ; As usual, we define a bundle $\overline{E}$ on a manifold with singularities $\overline{M}$ as a bundle $E$ on the underlying manifold $M$ subject to the following additional condition: there are bundles $E_I$ over the manifolds $\beta_I M \times P^I$ and bundle equivalences $E|_{\partial_I M} \rightarrow E_I$ covering the product diffeomorphisms. ===Bordism theory for manifolds with singularities=== == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]] {{Stub}}\overline{A} that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let P_1 be a closed manifold. A manifold with a P_1-singularity (following [Baas1973]) is a space of the form

\displaystyle \overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)
\displaystyle \partial A = A(1) \times P_1

Here, A is a manifold with boundary A(1).


2.2 \Sigma-manifolds


Following ([Botvinnik2001], [Botvinnik1992]), a more general definition can be given. Let (P_1 , ..., P_k) be a (possibly empty) collection of closed manifolds and denote by P_0 the set containing only one point. Then define \Sigma_k := (P_0, P_1, ... , P_k). For a subset I = \{i_1,..., i_q\} \subset \{0,...,k\} define P^I := P_{i_1} \times ...\times P_{i_q}.

Definition 2.1.A manifold 
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 is a \Sigma_k-Manifold if there is given
  1. a partition \partial M = \partial_0 M \cup ... \cup \partial_k M, such that \partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M is a manifold for each I = \{i_1,...,i_q\} \subset \{0,...,k\}, and such that
\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M
.
  1. for each I \subset \{0,...,k\} a manifold \beta_I M and a diffeomorphism
\displaystyle \phi_I: \partial_I M \rightarrow \beta_I M \times P^I
,

such that if J \subset I and \iota: \partial_I M \rightarrow \partial_J M is the inclusion, then the composition

\displaystyle \phi_J \circ \iota \circ \phi_I^{-1}: \beta_I M \times P^I \rightarrow \beta_J M \times P^J

restricts to the identity on the factor P^J in P^I. The diffeomorphisms \phi_I are called product structures.

On a \Sigma_k-manifold 
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, there is a canonical equivalence relation \sim: two points x,y \in M are defined to be equivalent if there is an I \subset \{0,...,k\} such that x,y \in \partial_I M and pr \circ \phi_I(x) = pr \circ \phi_I(y), where pr: \beta_I M \times P^I \rightarrow \beta_I M is the projection.

Now we can give a general definition: a manifold with a \Sigma_k-singularity is a topological space \overline{A} of the form

\displaystyle \overline{A} = A / \sim

for a \Sigma_k-manifold A.

The spaces defined above as manifolds with a P_1-singularity are contained in this more general definition. In fact, they give the manifolds with a \Sigma_1-singularity. For given a manifold P_1, set \Sigma_1 = (P_0, P_1). Then the manifold A with boundary A(1) \times P_1, which appears in the above definition, is a \Sigma_1-manifold. The attachement of the cone-end A(1) \times C P_1 now corresponds to the collapsing of the equivalence relation \sim in A.

When dealing with manifolds with singularities it is convenient to work with the underlying \Sigma-manifold and make sure that all operations one performs on them are compatible with the equivalence relation.


3 Some examples and constructions

3.1 Intersecting spheres

A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding i: S^{n-1} \rightarrow S^n. Then define \overline{S} := S^n \cup_{i(S^{n-1})} S^n. Outside of the intersecting sphere i(S^{n-1}) this is an n-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of S^{n-1} and a cone over \Zz_4 . We can write \overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4.

3.2 Inverse images of critical points

Let f:\Rr^n \rightarrow \Rr be a Morse-function with 0 as a single critical point. We can suppose that f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n near 0. Setting M := f^{-1}(\{0\}), we see that the cone C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \} provides a neighborhood of 0 in 
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. It follows that 
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 is of the form N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2}).

3.3 Structures on manifolds with singularities

Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with \Sigma-singularity \overline{M} with underlying \Sigma-manifold 
Tex syntax error
 carries the structure in question if all manifolds involved in \textbf{Definition 2.1} carry this structure and the product diffeomorphisms preserves it. For example, \overline{M} is orientable if 
Tex syntax error
 as well as all manifolds in \Sigma and the manifolds \beta_I M are orientable and the product diffeomorphisms are orientation-preserving. As another example, \overline{M} becomes a Riemannian manifold with singularities if we put a Riemannian metric on 
Tex syntax error
 as well as on the manifolds in \Sigma and on the manifolds \beta_I M in such a way that the product diffeomorphism are isometries.

3.4 Bundles on manifolds with singularities

As usual, we define a bundle \overline{E} on a manifold with singularities \overline{M} as a bundle E on the underlying manifold 
Tex syntax error
 subject to the following additional condition: there are bundles E_I over the manifolds \beta_I M \times P^I and bundle equivalences E|_{\partial_I M} \rightarrow E_I covering the product diffeomorphisms.


3.5 Bordism theory for manifolds with singularities

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

This page has not been refereed. The information given here might be incomplete or provisional.

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