# Manifolds with singularities

## 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}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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$$P_1$ be a closed manifold. A manifold with a $P_1$$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$$A$ is a manifold with boundary $A(1)$$A(1)$.

More complex singularities occur if, instead of taking a cone over only one manifold $P_1$$P_1$, we allow a collection $\{P_1,...,P_k\}$$\{P_1,...,P_k\}$ of several closed manifolds. In this case, we define a a manifold with a $\{P_1,...,P_k\}$$\{P_1,...,P_k\}$-singularity to be a (second-countable and Hausdorff) topological space $\overline{A}$$\overline{A}$ locally homeomorphic to one of the spaces $\Rr^n , \Rr^{n_1} \times C P_1 , \Rr^{n_2} \times C P_1 \times C P_2 , ...$$\Rr^n , \Rr^{n_1} \times C P_1 , \Rr^{n_2} \times C P_1 \times C P_2 , ...$.

An alternative approach to manifolds with singularities would be to remove the singular set and to define an equivalence relation on the remaining manifold that 'remembers' the singularities. This view is taken in ([Botvinnik1992],[Botvinnik2001]). We describe it in the next section.

### 2.2 $\Sigma$$\Sigma$-manifolds

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

Definition 2.1.A manifold $M$$M$ is a $\Sigma_k$$\Sigma_k$-Manifold if there is given

1. a partition $\partial M = \partial_0 M \cup ... \cup \partial_k M$$\partial M = \partial_0 M \cup ... \cup \partial_k M$, such that $\partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M$$\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\}$$I = \{i_1,...,i_q\} \subset \{0,...,k\}$, and such that
$\displaystyle \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M$
.
1. for each $I \subset \{0,...,k\}$$I \subset \{0,...,k\}$ a manifold $\beta_I M$$\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$$J \subset I$ and $\iota: \partial_I M \rightarrow \partial_J M$$\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$$P^J$ in $P^I$$P^I$. The diffeomorphisms $\phi_I$$\phi_I$ are called product structures.

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

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

for a $\Sigma_k$$\Sigma_k$-manifold $A$$A$.

The spaces defined above as manifolds with a $(P_1,...,P_k)$$(P_1,...,P_k)$-singularity are contained in this new definition. Given manifolds $P_1,...,P_k$$P_1,...,P_k$, set $\Sigma_k = (P_0, P_1,...,P_k)$$\Sigma_k = (P_0, P_1,...,P_k)$. Removing a neighborhood of the cone-tips in a manifold with $(P_1,...,P_k)$$(P_1,...,P_k)$-singularity $\overline{A}$$\overline{A}$ gives a $\Sigma_k$$\Sigma_k$-manifold $M$$M$. Now the collapsing of the equivalence relation in $M$$M$ corresponds to the re-attachement of the cone-ends.

When dealing with manifolds with singularities it is convenient to work with the underlying $\Sigma$$\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$$i: S^{n-1} \rightarrow S^n$. Then define $\overline{S} := S^n \cup_{i(S^{n-1})} S^n$$\overline{S} := S^n \cup_{i(S^{n-1})} S^n$. Outside of the intersecting sphere $i(S^{n-1})$$i(S^{n-1})$ this is an $n$$n$-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of $S^{n-1}$$S^{n-1}$ and a cone over $\Zz_4$$\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$$\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$$f:\Rr^n \rightarrow \Rr$ be a Morse-function with $0$$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$$f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n$ near $0$$0$. Setting $M := f^{-1}(\{0\})$$M := f^{-1}(\{0\})$, we see that the cone $C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \}$$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$$0$ in $M$$M$. It follows that $M$$M$ is of the form $N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2})$$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$$\Sigma$-singularity $\overline{M}$$\overline{M}$ with underlying $\Sigma$$\Sigma$-manifold $M$$M$ carries the structure in question if all manifolds involved in $\textbf{Definition 2.1}$$\textbf{Definition 2.1}$ carry this structure and the product diffeomorphisms preserves it.

For example, $\overline{M}$$\overline{M}$ is orientable if $M$$M$ as well as all manifolds in $\Sigma$$\Sigma$ and the manifolds $\beta_I M$$\beta_I M$ are orientable and the product diffeomorphisms are orientation-preserving. As another example, $\overline{M}$$\overline{M}$ becomes a Riemannian manifold with singularities if we put a Riemannian metric on $M$$M$ as well as on the manifolds in $\Sigma$$\Sigma$ and on the manifolds $\beta_I M$$\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}$$\overline{E}$ on a manifold with singularities $\overline{M}$$\overline{M}$ as a bundle $E$$E$ on the underlying manifold $M$$M$ subject to the following additional condition: there are bundles $E_I$$E_I$ over the manifolds $\beta_I M \times P^I$$\beta_I M \times P^I$ and bundle equivalences $E|_{\partial_I M} \rightarrow E_I$$E|_{\partial_I M} \rightarrow E_I$ covering the product diffeomorphisms.

## 4 Bordism theory for manifolds with singularities

A $\Sigma_k$$\Sigma_k$-manifold $M$$M$ induces the structure of a $\Sigma_k$$\Sigma_k$-manifold on $\partial_0 M$$\partial_0 M$. We call $\partial_0 M$$\partial_0 M$ the boundary of $M$$M$. Given a manifold with a $\Sigma_k$$\Sigma_k$-singularity $\overline{M} := M / \sim$$\overline{M} := M / \sim$, we define $\partial \overline{M} = \partial_0 M / \sim$$\partial \overline{M} = \partial_0 M / \sim$ to be its boundary. A theory of bordism with $\Sigma_k$$\Sigma_k$-singularities can now be developed just as for ordinary manifolds.

For illustration we pick up the case of a $P_1$$P_1$-singularity considered above. $\overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P_1$$\overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P_1$ is bordant to zero if there exists $\overline{B} = B \cup_{B(1) \times P_1} B(1) \times C P_1$$\overline{B} = B \cup_{B(1) \times P_1} B(1) \times C P_1$, such that

$\displaystyle \partial B = A \cup_{A(1) \times P_1} B(1) \times P_1$
$\displaystyle \partial B(1) = A(1)$
.

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