Lecture 9: The Topology ofComplements

May 17, 2019

Gunnar Carlsson

Stanford University

The Topology of Complements

Suppose we have Y ⊆ X an embedding of topological spaces,and we have topological information about X and Y . What canbe said about the topology of X − Y ?

Robotics

Motion planning through obstacles

Sensor Nets

Covered region for a sensor net

Topology of Complements

I Why should this be possible?

I First results say that we can obtain homology in the casewhere the ambient space X is a manifold

I Alexander duality theorem

Topology of Complements

I Why should this be possible?

I First results say that we can obtain homology in the casewhere the ambient space X is a manifold

I Alexander duality theorem

Topology of Complements

I Why should this be possible?

I First results say that we can obtain homology in the casewhere the ambient space X is a manifold

I Alexander duality theorem

Manifolds

I A space X is an n-dimensional manifold if it is locally likeRn

I Means every point has a neighborhood homeomorphic toan open disc in Rn

Manifold

Not a manifold

Manifolds

I A space X is an n-dimensional manifold if it is locally likeRn

I Means every point has a neighborhood homeomorphic toan open disc in Rn

Manifold Not a manifold

Cohomology

I To state theorem, requires cohomology

I Apply Hom(−; k) to a chain complex, obtain a cochaincomplex with coboundary operator δ.

I Ker(δ)/im(δ) defined to be cohomology H∗(X ).

I H i (X ) ∼= Hi (X )∗

I Contravariant

Cohomology

I To state theorem, requires cohomology

I Apply Hom(−; k) to a chain complex, obtain a cochaincomplex with coboundary operator δ.

I Ker(δ)/im(δ) defined to be cohomology H∗(X ).

I H i (X ) ∼= Hi (X )∗

I Contravariant

Cohomology

I To state theorem, requires cohomology

I Apply Hom(−; k) to a chain complex, obtain a cochaincomplex with coboundary operator δ.

I Ker(δ)/im(δ) defined to be cohomology H∗(X ).

I H i (X ) ∼= Hi (X )∗

I Contravariant

Cohomology

I To state theorem, requires cohomology

I Apply Hom(−; k) to a chain complex, obtain a cochaincomplex with coboundary operator δ.

I Ker(δ)/im(δ) defined to be cohomology H∗(X ).

I H i (X ) ∼= Hi (X )∗

I Contravariant

Cohomology

I To state theorem, requires cohomology

I Apply Hom(−; k) to a chain complex, obtain a cochaincomplex with coboundary operator δ.

I Ker(δ)/im(δ) defined to be cohomology H∗(X ).

I H i (X ) ∼= Hi (X )∗

I Contravariant

Alexander DualityTheorems

Y ⊆ X , Y compact

X = Rn: Hi (Rn − Y ) ∼= Hn−i−1(Y )

X = Sn: Hi (Sn − Y ) ∼= Hn−i−1(Y )

X a general manifold, Hi (X ,X − Y ) ∼= Hn−i (Y )

Tells us we can recover the homology or cohomology ofcomplements. Can we recover the actual space from Y ?

Alexander DualityTheorems

Y ⊆ X , Y compact

X = Rn: Hi (Rn − Y ) ∼= Hn−i−1(Y )

X = Sn: Hi (Sn − Y ) ∼= Hn−i−1(Y )

X a general manifold, Hi (X ,X − Y ) ∼= Hn−i (Y )

Tells us we can recover the homology or cohomology ofcomplements. Can we recover the actual space from Y ?

Alexander DualityTheorems

Y ⊆ X , Y compact

X = Rn: Hi (Rn − Y ) ∼= Hn−i−1(Y )

X = Sn: Hi (Sn − Y ) ∼= Hn−i−1(Y )

X a general manifold, Hi (X ,X − Y ) ∼= Hn−i (Y )

Tells us we can recover the homology or cohomology ofcomplements. Can we recover the actual space from Y ?

Alexander DualityTheorems

Y ⊆ X , Y compact

X = Rn: Hi (Rn − Y ) ∼= Hn−i−1(Y )

X = Sn: Hi (Sn − Y ) ∼= Hn−i−1(Y )

X a general manifold, Hi (X ,X − Y ) ∼= Hn−i (Y )

Tells us we can recover the homology or cohomology ofcomplements. Can we recover the actual space from Y ?

Alexander DualityTheorems

Y ⊆ X , Y compact

X = Rn: Hi (Rn − Y ) ∼= Hn−i−1(Y )

X = Sn: Hi (Sn − Y ) ∼= Hn−i−1(Y )

X a general manifold, Hi (X ,X − Y ) ∼= Hn−i (Y )

Tells us we can recover the homology or cohomology ofcomplements. Can we recover the actual space from Y ?

Spanier-Whitehead duality

I If a compact Y is embedded in Sn, can consider thecomplement of Y in Sn+1 ⊇ Sn

I Sn+1 − Y ∼= Σ(Sn − Y )

I X and Y said to be stably homotopy equivalent ifΣiX ' ΣiY for some i

I A space determines a stable homotopy type

I Turns out Y determines the stable homotopy type ofcomplement of Y for an inclusion Y ↪→ SN

I Key fact is that for large N all embeddings of Y in SN areisotopic

Spanier-Whitehead duality

I If a compact Y is embedded in Sn, can consider thecomplement of Y in Sn+1 ⊇ Sn

I Sn+1 − Y ∼= Σ(Sn − Y )

I X and Y said to be stably homotopy equivalent ifΣiX ' ΣiY for some i

I A space determines a stable homotopy type

I Turns out Y determines the stable homotopy type ofcomplement of Y for an inclusion Y ↪→ SN

I Key fact is that for large N all embeddings of Y in SN areisotopic

Spanier-Whitehead duality

I If a compact Y is embedded in Sn, can consider thecomplement of Y in Sn+1 ⊇ Sn

I Sn+1 − Y ∼= Σ(Sn − Y )

I X and Y said to be stably homotopy equivalent ifΣiX ' ΣiY for some i

I A space determines a stable homotopy type

I Turns out Y determines the stable homotopy type ofcomplement of Y for an inclusion Y ↪→ SN

I Key fact is that for large N all embeddings of Y in SN areisotopic

Spanier-Whitehead duality

I If a compact Y is embedded in Sn, can consider thecomplement of Y in Sn+1 ⊇ Sn

I Sn+1 − Y ∼= Σ(Sn − Y )

I X and Y said to be stably homotopy equivalent ifΣiX ' ΣiY for some i

I A space determines a stable homotopy type

I Turns out Y determines the stable homotopy type ofcomplement of Y for an inclusion Y ↪→ SN

I Key fact is that for large N all embeddings of Y in SN areisotopic

Spanier-Whitehead duality

I If a compact Y is embedded in Sn, can consider thecomplement of Y in Sn+1 ⊇ Sn

I Sn+1 − Y ∼= Σ(Sn − Y )

I X and Y said to be stably homotopy equivalent ifΣiX ' ΣiY for some i

I A space determines a stable homotopy type

I Turns out Y determines the stable homotopy type ofcomplement of Y for an inclusion Y ↪→ SN

I Key fact is that for large N all embeddings of Y in SN areisotopic

Spanier-Whitehead duality

I If a compact Y is embedded in Sn, can consider thecomplement of Y in Sn+1 ⊇ Sn

I Sn+1 − Y ∼= Σ(Sn − Y )

I X and Y said to be stably homotopy equivalent ifΣiX ' ΣiY for some i

I A space determines a stable homotopy type

I Turns out Y determines the stable homotopy type ofcomplement of Y for an inclusion Y ↪→ SN

I Key fact is that for large N all embeddings of Y in SN areisotopic

Spanier-Whitehead Duality

For fixed dimensions, is there actually a dependence on theembedding?

Answer is yes.

Spanier-Whitehead Duality

For fixed dimensions, is there actually a dependence on theembedding?

Answer is yes.

Knot Theory

Embeddings of circle in R3

Knot Theory

Fundamental group of knot complement is a key invariant of aknot

Link Theory

Embeddings of disjoint union of circles in R3

Unstable Homotopy Types

I Homology or cohomology by itself cannot detect thedifference between homotopy types of complements by theAlexander duality theorem

I Fundamental group can in some cases

I Fundamental group is often complicated non-abeliangroup, not well suited for computation

I Can we impose additional structure on homology orcohomology which detects unstable phenomena?

Unstable Homotopy Types

I Homology or cohomology by itself cannot detect thedifference between homotopy types of complements by theAlexander duality theorem

I Fundamental group can in some cases

I Fundamental group is often complicated non-abeliangroup, not well suited for computation

I Can we impose additional structure on homology orcohomology which detects unstable phenomena?

Unstable Homotopy Types

I Homology or cohomology by itself cannot detect thedifference between homotopy types of complements by theAlexander duality theorem

I Fundamental group can in some cases

I Fundamental group is often complicated non-abeliangroup, not well suited for computation

I Can we impose additional structure on homology orcohomology which detects unstable phenomena?

Unstable Homotopy Types

I Homology or cohomology by itself cannot detect thedifference between homotopy types of complements by theAlexander duality theorem

I Fundamental group can in some cases

I Fundamental group is often complicated non-abeliangroup, not well suited for computation

I Can we impose additional structure on homology orcohomology which detects unstable phenomena?

Kunneth Formula

I Kunneth formula describes homology and cohomology ofproducts X × Y

I Hn(X × Y ) ∼= ⊕iHn−i (X )⊗ Hi (Y )

I Hn(X × Y ) ∼= ⊕iHn−1(X )⊗ H i (Y )

I More compact descriptions as graded vector spaces:

H∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

andH∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

I T a torus, H0(T) = k ,H1(T) = k2, and H2(T) = k .

Kunneth Formula

I Kunneth formula describes homology and cohomology ofproducts X × Y

I Hn(X × Y ) ∼= ⊕iHn−i (X )⊗ Hi (Y )

I Hn(X × Y ) ∼= ⊕iHn−1(X )⊗ H i (Y )

I More compact descriptions as graded vector spaces:

H∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

andH∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

I T a torus, H0(T) = k ,H1(T) = k2, and H2(T) = k .

Kunneth Formula

I Kunneth formula describes homology and cohomology ofproducts X × Y

I Hn(X × Y ) ∼= ⊕iHn−i (X )⊗ Hi (Y )

I Hn(X × Y ) ∼= ⊕iHn−1(X )⊗ H i (Y )

I More compact descriptions as graded vector spaces:

H∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

andH∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

I T a torus, H0(T) = k ,H1(T) = k2, and H2(T) = k .

Kunneth Formula

I Kunneth formula describes homology and cohomology ofproducts X × Y

I Hn(X × Y ) ∼= ⊕iHn−i (X )⊗ Hi (Y )

I Hn(X × Y ) ∼= ⊕iHn−1(X )⊗ H i (Y )

I More compact descriptions as graded vector spaces:

H∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

andH∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

I T a torus, H0(T) = k ,H1(T) = k2, and H2(T) = k .

Kunneth Formula

I Kunneth formula describes homology and cohomology ofproducts X × Y

I Hn(X × Y ) ∼= ⊕iHn−i (X )⊗ Hi (Y )

I Hn(X × Y ) ∼= ⊕iHn−1(X )⊗ H i (Y )

I More compact descriptions as graded vector spaces:

H∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

andH∗(X × Y ) ∼= H∗(X )⊗ H∗(Y )

I T a torus, H0(T) = k ,H1(T) = k2, and H2(T) = k .

Cup Products

I The cup product is a homomorphism

H∗(X )⊗ H∗(X ) −→ H∗(X )

I Given by the composite

H∗(X )⊗ H∗(X ) −→ H∗(X × X )H∗(∆)−→ H∗(X )

where ∆ denotes the diagonal map X → X × X , whichsends x to the pair (x , x)

I Image of x ⊗ x ′ is denoted by x ∪ x ′

I H∗(X ) becomes a graded k-algebra

Cup Products

I The cup product is a homomorphism

H∗(X )⊗ H∗(X ) −→ H∗(X )

I Given by the composite

H∗(X )⊗ H∗(X ) −→ H∗(X × X )H∗(∆)−→ H∗(X )

where ∆ denotes the diagonal map X → X × X , whichsends x to the pair (x , x)

I Image of x ⊗ x ′ is denoted by x ∪ x ′

I H∗(X ) becomes a graded k-algebra

Cup Products

I The cup product is a homomorphism

H∗(X )⊗ H∗(X ) −→ H∗(X )

I Given by the composite

H∗(X )⊗ H∗(X ) −→ H∗(X × X )H∗(∆)−→ H∗(X )

where ∆ denotes the diagonal map X → X × X , whichsends x to the pair (x , x)

I Image of x ⊗ x ′ is denoted by x ∪ x ′

I H∗(X ) becomes a graded k-algebra

Cup Products

I The cup product is a homomorphism

H∗(X )⊗ H∗(X ) −→ H∗(X )

I Given by the composite

H∗(X )⊗ H∗(X ) −→ H∗(X × X )H∗(∆)−→ H∗(X )

where ∆ denotes the diagonal map X → X × X , whichsends x to the pair (x , x)

I Image of x ⊗ x ′ is denoted by x ∪ x ′

I H∗(X ) becomes a graded k-algebra

Cup Products

I For X a suspension, and x and x ′ positive degree elementsin H∗(X ), x ∪ x ′ = 0. “Cup products of positive degreeelements vanish on suspensions”.

I Means that cup products can detect difference betweenunstable homotopy types that are the same as stablehomotopy types

I Torus gives an example

Cup Products

I For X a suspension, and x and x ′ positive degree elementsin H∗(X ), x ∪ x ′ = 0. “Cup products of positive degreeelements vanish on suspensions”.

I Means that cup products can detect difference betweenunstable homotopy types that are the same as stablehomotopy types

I Torus gives an example

Cup Products

I For X a suspension, and x and x ′ positive degree elementsin H∗(X ), x ∪ x ′ = 0. “Cup products of positive degreeelements vanish on suspensions”.

I Means that cup products can detect difference betweenunstable homotopy types that are the same as stablehomotopy types

I Torus gives an example

Cup Products on a Torus

I The graded ring H∗(T) is isomorphic to a Grassmannalgebra Λ(x , y)

I x ∪ x = 0, y ∪ y = 0, and x ∪ y = −y ∪ x

I x ∪ y is non-zero, so T is not a suspension

I Let B be the bouquet S1 ∨ S1 ∨ S2

Cup Products on a Torus

I The graded ring H∗(T) is isomorphic to a Grassmannalgebra Λ(x , y)

I x ∪ x = 0, y ∪ y = 0, and x ∪ y = −y ∪ x

I x ∪ y is non-zero, so T is not a suspension

I Let B be the bouquet S1 ∨ S1 ∨ S2

Cup Products on a Torus

I The graded ring H∗(T) is isomorphic to a Grassmannalgebra Λ(x , y)

I x ∪ x = 0, y ∪ y = 0, and x ∪ y = −y ∪ x

I x ∪ y is non-zero, so T is not a suspension

I Let B be the bouquet S1 ∨ S1 ∨ S2

Cup Products on a Torus

I The graded ring H∗(T) is isomorphic to a Grassmannalgebra Λ(x , y)

I x ∪ x = 0, y ∪ y = 0, and x ∪ y = −y ∪ x

I x ∪ y is non-zero, so T is not a suspension

I Let B be the bouquet S1 ∨ S1 ∨ S2

Cup Products on a Torus

I B is the suspension of S0 ∨ S0 ∨ S1

I Follows that cup products of positive elements vanish

I Homology and cohomology of B and T are identical asvector spaces

I Cup product shows them to be distinct as unstablehomotopy types. On the other hand, ΣT ' ΣB.

Cup Products on a Torus

I B is the suspension of S0 ∨ S0 ∨ S1

I Follows that cup products of positive elements vanish

I Homology and cohomology of B and T are identical asvector spaces

I Cup product shows them to be distinct as unstablehomotopy types. On the other hand, ΣT ' ΣB.

Cup Products on a Torus

I B is the suspension of S0 ∨ S0 ∨ S1

I Follows that cup products of positive elements vanish

I Homology and cohomology of B and T are identical asvector spaces

I Cup product shows them to be distinct as unstablehomotopy types. On the other hand, ΣT ' ΣB.

Cup Products on a Torus

I B is the suspension of S0 ∨ S0 ∨ S1

I Follows that cup products of positive elements vanish

I Homology and cohomology of B and T are identical asvector spaces

I Cup product shows them to be distinct as unstablehomotopy types. On the other hand, ΣT ' ΣB.

Cup Products and Path Components

I For a space X , H0(X ) is a vector space with dimensionequal to the number of path components of X .

I The path components determines a basis for the vectorspace H0(X )

I Homology alone does not permit us to identify the basis

I H0(−) does not determine π0(−) as a set-valued functor

Cup Products and Path Components

I For a space X , H0(X ) is a vector space with dimensionequal to the number of path components of X .

I The path components determines a basis for the vectorspace H0(X )

I Homology alone does not permit us to identify the basis

I H0(−) does not determine π0(−) as a set-valued functor

Cup Products and Path Components

I For a space X , H0(X ) is a vector space with dimensionequal to the number of path components of X .

I The path components determines a basis for the vectorspace H0(X )

I Homology alone does not permit us to identify the basis

I H0(−) does not determine π0(−) as a set-valued functor

Cup Products and Path Components

I For a space X , H0(X ) is a vector space with dimensionequal to the number of path components of X .

I The path components determines a basis for the vectorspace H0(X )

I Homology alone does not permit us to identify the basis

I H0(−) does not determine π0(−) as a set-valued functor

Cup Products and Path Components

I H0(X ) is a ring under cup product

I H0(X ) is isomorphic to the k-algebra of k-valued functionson π0(X ) under pointwise addition and multiplication

I The set of k-algebra homomorphisms H0(X )→ k is in oneto one correspondence with the elements of π0(X )

I Means that we can recover π0 from the k-algebra valuedfunctor H0(−)

Cup Products and Path Components

I H0(X ) is a ring under cup product

I H0(X ) is isomorphic to the k-algebra of k-valued functionson π0(X ) under pointwise addition and multiplication

I The set of k-algebra homomorphisms H0(X )→ k is in oneto one correspondence with the elements of π0(X )

I Means that we can recover π0 from the k-algebra valuedfunctor H0(−)

Cup Products and Path Components

I H0(X ) is a ring under cup product

I H0(X ) is isomorphic to the k-algebra of k-valued functionson π0(X ) under pointwise addition and multiplication

I The set of k-algebra homomorphisms H0(X )→ k is in oneto one correspondence with the elements of π0(X )

I Means that we can recover π0 from the k-algebra valuedfunctor H0(−)

Cup Products and Path Components

I H0(X ) is a ring under cup product

I H0(X ) is isomorphic to the k-algebra of k-valued functionson π0(X ) under pointwise addition and multiplication

I The set of k-algebra homomorphisms H0(X )→ k is in oneto one correspondence with the elements of π0(X )

I Means that we can recover π0 from the k-algebra valuedfunctor H0(−)

Cup Products of Complements

I The various duality theorems allow us to understand thehomology of the complements, including H0.

I Is it possible to use the same methods to recover cupproducts of the complement, and consequently the set π0

applied to the complement?

I This can be done, using the fact that cup products areinduced by a map, namely the diagonal map, and afunctoriality result for the duality theorems

Cup Products of Complements

I The various duality theorems allow us to understand thehomology of the complements, including H0.

I Is it possible to use the same methods to recover cupproducts of the complement, and consequently the set π0

applied to the complement?

I This can be done, using the fact that cup products areinduced by a map, namely the diagonal map, and afunctoriality result for the duality theorems

Cup Products of Complements

I The various duality theorems allow us to understand thehomology of the complements, including H0.

I Is it possible to use the same methods to recover cupproducts of the complement, and consequently the set π0

applied to the complement?

I This can be done, using the fact that cup products areinduced by a map, namely the diagonal map, and afunctoriality result for the duality theorems

Functoriality of Alexander Duality

All the Alexander Duality isomorphism for X = Sn describedabove is functorial, in the sense that for an inclusionY0 ⊆ Y1 ⊆ X , the diagram

H i (X − Y0) Hn−i−1(Y0)

H i (X − Y1) Hn−i−1(Y1)

-

? ?-

commutes. Note that we have X − Y1 ↪→ X − Y0. There areanalogous statements for X = Rn or more general.

Cup Products of Complements

I We assume that we are given a compact subset A ⊆ Rn,and let CA = Rn − A

I Hi (CA) ∼= Hn−i−1(A) by Alexander Duality Theorem

I Let C∆A denote the complement of A ⊆ A× A→ C∆A

I Let C (A× A) denote the complement of A× A in R2n

I H0(A× A) ∼= H2n−1(C (A× A)) and H0(A) ∼= H2n−1(C∆A)

I Obtain map C (A× A)→ C∆A inducing the cup producton H0(A) via functoriality of Alexander Duality

Cup Products of Complements

I We assume that we are given a compact subset A ⊆ Rn,and let CA = Rn − A

I Hi (CA) ∼= Hn−i−1(A) by Alexander Duality Theorem

I Let C∆A denote the complement of A ⊆ A× A→ C∆A

I Let C (A× A) denote the complement of A× A in R2n

I H0(A× A) ∼= H2n−1(C (A× A)) and H0(A) ∼= H2n−1(C∆A)

I Obtain map C (A× A)→ C∆A inducing the cup producton H0(A) via functoriality of Alexander Duality

Cup Products of Complements

I We assume that we are given a compact subset A ⊆ Rn,and let CA = Rn − A

I Hi (CA) ∼= Hn−i−1(A) by Alexander Duality Theorem

I Let C∆A denote the complement of A ⊆ A× A→ C∆A

I Let C (A× A) denote the complement of A× A in R2n

I H0(A× A) ∼= H2n−1(C (A× A)) and H0(A) ∼= H2n−1(C∆A)

I Obtain map C (A× A)→ C∆A inducing the cup producton H0(A) via functoriality of Alexander Duality

Cup Products of Complements

I We assume that we are given a compact subset A ⊆ Rn,and let CA = Rn − A

I Hi (CA) ∼= Hn−i−1(A) by Alexander Duality Theorem

I Let C∆A denote the complement of A ⊆ A× A→ C∆A

I Let C (A× A) denote the complement of A× A in R2n

I H0(A× A) ∼= H2n−1(C (A× A)) and H0(A) ∼= H2n−1(C∆A)

I Obtain map C (A× A)→ C∆A inducing the cup producton H0(A) via functoriality of Alexander Duality

Cup Products of Complements

I We assume that we are given a compact subset A ⊆ Rn,and let CA = Rn − A

I Hi (CA) ∼= Hn−i−1(A) by Alexander Duality Theorem

I Let C∆A denote the complement of A ⊆ A× A→ C∆A

I Let C (A× A) denote the complement of A× A in R2n

I H0(A× A) ∼= H2n−1(C (A× A)) and H0(A) ∼= H2n−1(C∆A)

I Obtain map C (A× A)→ C∆A inducing the cup producton H0(A) via functoriality of Alexander Duality

Cup Products of Complements

I We assume that we are given a compact subset A ⊆ Rn,and let CA = Rn − A

I Hi (CA) ∼= Hn−i−1(A) by Alexander Duality Theorem

I Let C∆A denote the complement of A ⊆ A× A→ C∆A

I Let C (A× A) denote the complement of A× A in R2n

I H0(A× A) ∼= H2n−1(C (A× A)) and H0(A) ∼= H2n−1(C∆A)

I Obtain map C (A× A)→ C∆A inducing the cup producton H0(A) via functoriality of Alexander Duality

Time Varying Robotics and Sensor Net Problems

I We may have a motion planning problem or a situation inwhich we have moving sensors

I In this case we want to have path in the ambient spacewhich avoids the obstacles at every point in time

I Formulate the problem in terms of spaces over a base

I Ambient space will now be X = [0, 1]× Rn, consisting of aposition and a time

I π : X → [0, 1] is the projection

I The time varying obstacles will now consist of a subspaceY ⊆ X

Time Varying Robotics and Sensor Net Problems

I We may have a motion planning problem or a situation inwhich we have moving sensors

I In this case we want to have path in the ambient spacewhich avoids the obstacles at every point in time

I Formulate the problem in terms of spaces over a base

I Ambient space will now be X = [0, 1]× Rn, consisting of aposition and a time

I π : X → [0, 1] is the projection

I The time varying obstacles will now consist of a subspaceY ⊆ X

Time Varying Robotics and Sensor Net Problems

I We may have a motion planning problem or a situation inwhich we have moving sensors

I In this case we want to have path in the ambient spacewhich avoids the obstacles at every point in time

I Formulate the problem in terms of spaces over a base

I Ambient space will now be X = [0, 1]× Rn, consisting of aposition and a time

I π : X → [0, 1] is the projection

I The time varying obstacles will now consist of a subspaceY ⊆ X

Time Varying Robotics and Sensor Net Problems

I We may have a motion planning problem or a situation inwhich we have moving sensors

I In this case we want to have path in the ambient spacewhich avoids the obstacles at every point in time

I Formulate the problem in terms of spaces over a base

I Ambient space will now be X = [0, 1]× Rn, consisting of aposition and a time

I π : X → [0, 1] is the projection

I The time varying obstacles will now consist of a subspaceY ⊆ X

Time Varying Robotics and Sensor Net Problems

I We may have a motion planning problem or a situation inwhich we have moving sensors

I In this case we want to have path in the ambient spacewhich avoids the obstacles at every point in time

I Formulate the problem in terms of spaces over a base

I Ambient space will now be X = [0, 1]× Rn, consisting of aposition and a time

I π : X → [0, 1] is the projection

I The time varying obstacles will now consist of a subspaceY ⊆ X

Time Varying Robotics and Sensor Net Problems

I We may have a motion planning problem or a situation inwhich we have moving sensors

I In this case we want to have path in the ambient spacewhich avoids the obstacles at every point in time

I Formulate the problem in terms of spaces over a base

I Ambient space will now be X = [0, 1]× Rn, consisting of aposition and a time

I π : X → [0, 1] is the projection

I The time varying obstacles will now consist of a subspaceY ⊆ X

Time Varying Robotics and Sensor Net Problems

I By an admissible path, we’ll mean a section

σ : [0, 1]→ X − Y

i.e. a continuous map σ : [0, 1]→ Y so that π ◦ s = id[0,1]

I Many questions can be asked about such sections

I Does one exist?

I How many homotopy classes are there?

I What is the structure of the space of sections?

Time Varying Robotics and Sensor Net Problems

I By an admissible path, we’ll mean a section

σ : [0, 1]→ X − Y

i.e. a continuous map σ : [0, 1]→ Y so that π ◦ s = id[0,1]

I Many questions can be asked about such sections

I Does one exist?

I How many homotopy classes are there?

I What is the structure of the space of sections?

Time Varying Robotics and Sensor Net Problems

I By an admissible path, we’ll mean a section

σ : [0, 1]→ X − Y

i.e. a continuous map σ : [0, 1]→ Y so that π ◦ s = id[0,1]

I Many questions can be asked about such sections

I Does one exist?

I How many homotopy classes are there?

I What is the structure of the space of sections?

Time Varying Robotics and Sensor Net Problems

I By an admissible path, we’ll mean a section

σ : [0, 1]→ X − Y

i.e. a continuous map σ : [0, 1]→ Y so that π ◦ s = id[0,1]

I Many questions can be asked about such sections

I Does one exist?

I How many homotopy classes are there?

I What is the structure of the space of sections?

Time Varying Robotics and Sensor Net Problems

I By an admissible path, we’ll mean a section

σ : [0, 1]→ X − Y

i.e. a continuous map σ : [0, 1]→ Y so that π ◦ s = id[0,1]

I Many questions can be asked about such sections

I Does one exist?

I How many homotopy classes are there?

I What is the structure of the space of sections?

Time Varying Robotics and Sensor Net Problems

I Interesting work done on the first problem byGhrist-DeSilva, Adams-C., and Ghrist-Krishnan

I Second and third problems not addressed

I Second and third problems potentially useful as startingpoints for finding optimal paths

Time Varying Robotics and Sensor Net Problems

I Interesting work done on the first problem byGhrist-DeSilva, Adams-C., and Ghrist-Krishnan

I Second and third problems not addressed

I Second and third problems potentially useful as startingpoints for finding optimal paths

Time Varying Robotics and Sensor Net Problems

I Interesting work done on the first problem byGhrist-DeSilva, Adams-C., and Ghrist-Krishnan

I Second and third problems not addressed

I Second and third problems potentially useful as startingpoints for finding optimal paths

Time Varying Robotics and Sensor Net Problems: Zig-ZagApproach

I Break up [0, 1] into closed intervals I0, I1, . . . , In thatintersect in endpoints only

I Construct the spaces π−1(Is) and π−1(Is ∩ Is+1), andcreate a zig-zag diagram of spaces

I Clear that if there is a section, then zig-zag barcode willhave along bar

I Converse is not true

Time Varying Robotics and Sensor Net Problems: Zig-ZagApproach

I Break up [0, 1] into closed intervals I0, I1, . . . , In thatintersect in endpoints only

I Construct the spaces π−1(Is) and π−1(Is ∩ Is+1), andcreate a zig-zag diagram of spaces

I Clear that if there is a section, then zig-zag barcode willhave along bar

I Converse is not true

Time Varying Robotics and Sensor Net Problems: Zig-ZagApproach

I Break up [0, 1] into closed intervals I0, I1, . . . , In thatintersect in endpoints only

I Construct the spaces π−1(Is) and π−1(Is ∩ Is+1), andcreate a zig-zag diagram of spaces

I Clear that if there is a section, then zig-zag barcode willhave along bar

I Converse is not true

Time Varying Robotics and Sensor Net Problems: Zig-ZagApproach

I Break up [0, 1] into closed intervals I0, I1, . . . , In thatintersect in endpoints only

I Construct the spaces π−1(Is) and π−1(Is ∩ Is+1), andcreate a zig-zag diagram of spaces

I Clear that if there is a section, then zig-zag barcode willhave along bar

I Converse is not true

Time Varying Robotics and Sensor Net Problems: Zig-ZagApproach

Analogue of two distinct knots in this setting

Time Varying Robotics and Sensor Net Problems: Sheavesand Cosheaves

I Zig-zag approach relies on choice of covering of [0, 1] byintervals

I Need a more intrinsic invariant

I WARNING: Remainder is speculation - joint work with B.Filippenko

I Should rely on notion of sheaves and cosheaves

I Related notion of parametrized homology, S. Kalisnik

Time Varying Robotics and Sensor Net Problems: Sheavesand Cosheaves

I Zig-zag approach relies on choice of covering of [0, 1] byintervals

I Need a more intrinsic invariant

I WARNING: Remainder is speculation - joint work with B.Filippenko

I Should rely on notion of sheaves and cosheaves

I Related notion of parametrized homology, S. Kalisnik

Time Varying Robotics and Sensor Net Problems: Sheavesand Cosheaves

I Zig-zag approach relies on choice of covering of [0, 1] byintervals

I Need a more intrinsic invariant

I WARNING: Remainder is speculation - joint work with B.Filippenko

I Should rely on notion of sheaves and cosheaves

I Related notion of parametrized homology, S. Kalisnik

Time Varying Robotics and Sensor Net Problems: Sheavesand Cosheaves

I Zig-zag approach relies on choice of covering of [0, 1] byintervals

I Need a more intrinsic invariant

I WARNING: Remainder is speculation - joint work with B.Filippenko

I Should rely on notion of sheaves and cosheaves

I Related notion of parametrized homology, S. Kalisnik

Time Varying Robotics and Sensor Net Problems: Sheavesand Cosheaves

I Zig-zag approach relies on choice of covering of [0, 1] byintervals

I Need a more intrinsic invariant

I WARNING: Remainder is speculation - joint work with B.Filippenko

I Should rely on notion of sheaves and cosheaves

I Related notion of parametrized homology, S. Kalisnik

Sheaves

A sheaf on a topological space with values in a category C iscontravariant functor F from the category of open subsets of Xto C so that for any two open sets U,V ⊆ X , the diagram

F (U ∪ V ) F (U)

F (V ) F (U ∩ V )?

-

?-

is a pullback or an equalizer.

Sheaves - Examples

I Functions on X with values in R creates a sheaf ofR-vector spaces

I Cohomology can be sheafified to create a cohomology sheaf

I Maps from X to a topological space Y is a sheaf of sets

Sheaves - Examples

I Functions on X with values in R creates a sheaf ofR-vector spaces

I Cohomology can be sheafified to create a cohomology sheaf

I Maps from X to a topological space Y is a sheaf of sets

Sheaves - Examples

I Functions on X with values in R creates a sheaf ofR-vector spaces

I Cohomology can be sheafified to create a cohomology sheaf

I Maps from X to a topological space Y is a sheaf of sets

Sheaves

A sheaf on a topological space with values in a category C iscovariant functor F from the category of open subsets of X toC so that for any two open sets U,V ⊆ X , the diagram

F (U ∪ V ) F (U)

F (V ) F (U ∩ V )?

-

?-

is a pushout or a coequalizer.

Cosheaves - Examples

I Given π : E → X , the functor F (U) = π−1(U) is a cosheafof spaces.

I π0 can be cosheafified to create a cosheaf of sets.

I Hi can be cosheafified to create a homology sheaf of vectorspaces.

Cosheaves - Examples

I Given π : E → X , the functor F (U) = π−1(U) is a cosheafof spaces.

I π0 can be cosheafified to create a cosheaf of sets.

I Hi can be cosheafified to create a homology sheaf of vectorspaces.

Cosheaves - Examples

I Given π : E → X , the functor F (U) = π−1(U) is a cosheafof spaces.

I π0 can be cosheafified to create a cosheaf of sets.

I Hi can be cosheafified to create a homology sheaf of vectorspaces.

Sheaf Theoretic Approach

I Above approach to constructing components using cupproducts should work in this “sheafy setting”

I S. Kalisnik has shown that Alexander duality holds incontext of parametrized homology

I Should permit constructing H0 of complement of Y interms of the homology cosheaves of Y

I H0 creates a sheaf of k-algebras.

I Should be able to construct the cosheaf π0 as the cosheafof algebra homomorphisms from H0 to the constant sheafk .

Sheaf Theoretic Approach

I Above approach to constructing components using cupproducts should work in this “sheafy setting”

I S. Kalisnik has shown that Alexander duality holds incontext of parametrized homology

I Should permit constructing H0 of complement of Y interms of the homology cosheaves of Y

I H0 creates a sheaf of k-algebras.

I Should be able to construct the cosheaf π0 as the cosheafof algebra homomorphisms from H0 to the constant sheafk .

Sheaf Theoretic Approach

I Above approach to constructing components using cupproducts should work in this “sheafy setting”

I S. Kalisnik has shown that Alexander duality holds incontext of parametrized homology

I Should permit constructing H0 of complement of Y interms of the homology cosheaves of Y

I H0 creates a sheaf of k-algebras.

I Should be able to construct the cosheaf π0 as the cosheafof algebra homomorphisms from H0 to the constant sheafk .

Sheaf Theoretic Approach

I Above approach to constructing components using cupproducts should work in this “sheafy setting”

I S. Kalisnik has shown that Alexander duality holds incontext of parametrized homology

I Should permit constructing H0 of complement of Y interms of the homology cosheaves of Y

I H0 creates a sheaf of k-algebras.

I Should be able to construct the cosheaf π0 as the cosheafof algebra homomorphisms from H0 to the constant sheafk .

Sheaf Theoretic Approach

I Above approach to constructing components using cupproducts should work in this “sheafy setting”

I S. Kalisnik has shown that Alexander duality holds incontext of parametrized homology

I Should permit constructing H0 of complement of Y interms of the homology cosheaves of Y

I H0 creates a sheaf of k-algebras.

I Should be able to construct the cosheaf π0 as the cosheafof algebra homomorphisms from H0 to the constant sheafk .

Sheaf Theoretic Approach

I Maps from the constant cosheaf of sets with value the onepoint set to the cosheaf π0 should be the set ofcomponents of the space of sections of π : X → [0, 1]

I Therefore this yields an approach to Problem 2 above.

I Unstable Adams spectral sequence takes cohomologyinformation (including cup products and cohomologyoperations and reconstructs mapping spaces

I Analogue of U.A.s.s. for the setting of spaces over [0, 1]could address Problem 3 above.

I Embedding calculus is a technique for parametrizing theembeddings of one manifold in another. A. Jin and G.Arone working on applying it in the setting of spaces over[0, 1].

Sheaf Theoretic Approach

I Maps from the constant cosheaf of sets with value the onepoint set to the cosheaf π0 should be the set ofcomponents of the space of sections of π : X → [0, 1]

I Therefore this yields an approach to Problem 2 above.

I Unstable Adams spectral sequence takes cohomologyinformation (including cup products and cohomologyoperations and reconstructs mapping spaces

I Analogue of U.A.s.s. for the setting of spaces over [0, 1]could address Problem 3 above.

I Embedding calculus is a technique for parametrizing theembeddings of one manifold in another. A. Jin and G.Arone working on applying it in the setting of spaces over[0, 1].

Sheaf Theoretic Approach

I Maps from the constant cosheaf of sets with value the onepoint set to the cosheaf π0 should be the set ofcomponents of the space of sections of π : X → [0, 1]

I Therefore this yields an approach to Problem 2 above.

I Unstable Adams spectral sequence takes cohomologyinformation (including cup products and cohomologyoperations and reconstructs mapping spaces

I Analogue of U.A.s.s. for the setting of spaces over [0, 1]could address Problem 3 above.

I Embedding calculus is a technique for parametrizing theembeddings of one manifold in another. A. Jin and G.Arone working on applying it in the setting of spaces over[0, 1].

Sheaf Theoretic Approach

I Maps from the constant cosheaf of sets with value the onepoint set to the cosheaf π0 should be the set ofcomponents of the space of sections of π : X → [0, 1]

I Therefore this yields an approach to Problem 2 above.

I Unstable Adams spectral sequence takes cohomologyinformation (including cup products and cohomologyoperations and reconstructs mapping spaces

I Analogue of U.A.s.s. for the setting of spaces over [0, 1]could address Problem 3 above.

I Embedding calculus is a technique for parametrizing theembeddings of one manifold in another. A. Jin and G.Arone working on applying it in the setting of spaces over[0, 1].

Sheaf Theoretic Approach

I Maps from the constant cosheaf of sets with value the onepoint set to the cosheaf π0 should be the set ofcomponents of the space of sections of π : X → [0, 1]

I Therefore this yields an approach to Problem 2 above.

I Unstable Adams spectral sequence takes cohomologyinformation (including cup products and cohomologyoperations and reconstructs mapping spaces

I Analogue of U.A.s.s. for the setting of spaces over [0, 1]could address Problem 3 above.

I Embedding calculus is a technique for parametrizing theembeddings of one manifold in another. A. Jin and G.Arone working on applying it in the setting of spaces over[0, 1].