Manifolds with singularities

Revision as of 17:03, 7 June 2010

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.

2 Definitions

2.1 Cone-like singularities

A manifold with singularities of Baas-Sullivan type is a topological space $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; 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. </wikitex> == Definitions == ===Cone-like singularities=== <wikitex>; 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 {{cite|Baas1973}}) is a space of the form $$\overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P(1)$$ $$\partial A = A(1) \times P_1 $$ Here, $A$ is a manifold with boundary $A(1)$. </wikitex> ===<wikitex>$\Sigma$-manifolds</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 $P^I := P_{i_1} \times ...\times P_{i_q}$. {{beginthm|Definition}}A manifold $M$ is a $\Sigma_k$-Manifold if # 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 $$ \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M $$. # for each $I \subset \{0,...,k\}$ there is a manifold $\beta_I M$ and a diffeomorphism $$ \phi_I: \partial_I M \rightarrow \beta_I M \times P^I$$ , such that if $J \subset I$ and $\iota: \partial_I M \rightarrow \partial_J M$ is the inclusion, then the composition $$ \phi_J \circ \iota \circ \phi_I^{-1}: \beta_I M \times P^I \rightarrow \beta_J M \times P^J$$ restricts to the identity on the factor $P^J$ in $P^I$. The diffeomorphisms $\phi_I$ are called product structures. {{endthm|Definition}} On a $\Sigma_k$-manifold $M$, 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 $$ \overline{A} = A / \sim $$ 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 $A$. </wikitex> == Some examples == ===Intersecting spheres=== <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 $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$. </wikitex> ===Inverse images of critical points=== <wikitex>; Let $f:\Rr^n \rightarrow \Rr$ be a Morse-function with <script type="math/tex">\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

$M$ is a $\Sigma_k$-Manifold if

1. 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

$$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$$

.

1. for each $I \subset \{0,...,k\}$ there is 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.

Tex syntax error

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 $A$.

3 Some examples

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 $\overline{S} := S^n \cup_{j(S^{n-1}} S^n$. 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 $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$.

3.2 Inverse images of critical points

Tex syntax error

Tex syntax error

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

• [Baas1973] N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302 (1974). MR0346824 (49 #11547b) Zbl 0281.57027
• [Botvinnik1992] B. I. Botvinnik, Manifolds with singularities and the Adams-Novikov spectral sequence, Cambridge University Press, Cambridge, 1992. MR1192127 (93h:55002) Zbl 0764.55001
• [Botvinnik2001] B. Botvinnik, Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683–718 (electronic). MR1857524 (2002j:57045) Zbl 1002.57055
• [Sullivan1967] D. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600. MR0212811 (35 #3676) Zbl 0153.54002
• [Sullivan1996] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, 1 (1996), 69–103. MR1434103 (98c:57027) Zbl 0871.57021

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

$$\displaystyle \partial A = A(1) \times P_1$$

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

$M$ is a $\Sigma_k$-Manifold if

1. 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

$$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$$

.

1. for each $I \subset \{0,...,k\}$ there is 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.

Tex syntax error

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 $A$.

3 Some examples

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 $\overline{S} := S^n \cup_{j(S^{n-1}} S^n$. 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 $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$.

3.2 Inverse images of critical points

Tex syntax error

Tex syntax error

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

• [Baas1973] N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302 (1974). MR0346824 (49 #11547b) Zbl 0281.57027
• [Botvinnik1992] B. I. Botvinnik, Manifolds with singularities and the Adams-Novikov spectral sequence, Cambridge University Press, Cambridge, 1992. MR1192127 (93h:55002) Zbl 0764.55001
• [Botvinnik2001] B. Botvinnik, Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683–718 (electronic). MR1857524 (2002j:57045) Zbl 1002.57055
• [Sullivan1967] D. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600. MR0212811 (35 #3676) Zbl 0153.54002
• [Sullivan1996] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, 1 (1996), 69–103. MR1434103 (98c:57027) Zbl 0871.57021

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

$$\displaystyle \partial A = A(1) \times P_1$$

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

$M$ is a $\Sigma_k$-Manifold if

1. 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

$$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$$

.

1. for each $I \subset \{0,...,k\}$ there is 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.

Tex syntax error

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 $A$.

3 Some examples

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 $\overline{S} := S^n \cup_{j(S^{n-1}} S^n$. 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 $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$.

3.2 Inverse images of critical points

Tex syntax error

Tex syntax error

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

• [Baas1973] N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302 (1974). MR0346824 (49 #11547b) Zbl 0281.57027
• [Botvinnik1992] B. I. Botvinnik, Manifolds with singularities and the Adams-Novikov spectral sequence, Cambridge University Press, Cambridge, 1992. MR1192127 (93h:55002) Zbl 0764.55001
• [Botvinnik2001] B. Botvinnik, Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683–718 (electronic). MR1857524 (2002j:57045) Zbl 1002.57055
• [Sullivan1967] D. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600. MR0212811 (35 #3676) Zbl 0153.54002
• [Sullivan1996] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, 1 (1996), 69–103. MR1434103 (98c:57027) Zbl 0871.57021

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

$$\displaystyle \partial A = A(1) \times P_1$$

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

$M$ is a $\Sigma_k$-Manifold if

1. 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

$$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$$

.

1. for each $I \subset \{0,...,k\}$ there is 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.

Tex syntax error

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 $A$.

3 Some examples

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 $\overline{S} := S^n \cup_{j(S^{n-1}} S^n$. 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 $\overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4$.

3.2 Inverse images of critical points

Tex syntax error

Tex syntax error

4 Invariants

...

5 Classification/Characterization

...

6 Further discussion

...

7 References

• [Baas1973] N. A. Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302 (1974). MR0346824 (49 #11547b) Zbl 0281.57027
• [Botvinnik1992] B. I. Botvinnik, Manifolds with singularities and the Adams-Novikov spectral sequence, Cambridge University Press, Cambridge, 1992. MR1192127 (93h:55002) Zbl 0764.55001
• [Botvinnik2001] B. Botvinnik, Manifolds with singularities accepting a metric of positive scalar curvature, Geom. Topol. 5 (2001), 683–718 (electronic). MR1857524 (2002j:57045) Zbl 1002.57055
• [Sullivan1967] D. Sullivan, On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600. MR0212811 (35 #3676) Zbl 0153.54002
• [Sullivan1996] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, 1 (1996), 69–103. MR1434103 (98c:57027) Zbl 0871.57021