Notes and reflections on the things I’ve learned while Doing Scholarship the last two three weeks (i.e. studying math).
The past three weeks were busier than usual so I had slower progress this time but here it is:
Chapter 6 continued: Sard’s theorem
Tubes! I might have thought the fact that you can embed manifolds in RN might have been one of those theorems whose main values are conceptual but not that useful in practice (such as Cayley’s theorem and the famous quote that the fact that any group is a subgroup of a symmetry group never actually made the task of studying & classifying groups easier) - but that is not the case, due to tubular neighborhoods.
The main issue with trying to solve problems about smooth maps between manifolds by embedding the codomain manifold in RN is that the resulting construction may not lie in the original manifold. Tubular neighborhoods address this by showing that it is always possible, given an embedding of a manifold in RN, to come up with a tube-like open neighborhood of the manifold, equipped with a retraction map, i.e. a map from this neighborhood to itself such that it is an identity when restricted to the manifold.
So conceptually, tubular neighborhoods let us reason about manifold problems by solving them in RN, and bringing it back to manifolds.
Applications are massive! Might be my favorite concept so far.
Whitney approximation theorem (any continuous map is homotopic to a smooth one. If the original continuous map is smooth on a closed subset, the homotopy can be taken relative to it)
Proof: Prove it for the case when the codomain is Rm, and just compose it with the tubular neighborhood retraction.
If a compact manifold has a nonzero vector field, then there exists a map to itself without fixed points
Proof: Embed everything (the manifold, its tangent spaces) in R^m, let the map be the one that takes a point to the direction indicated by the nonzero vector field at some small magnitude ϵ. This will take things outside of the manifold, so compose with the tubular neighborhood retraction (which is allowed if ϵ is small enough).
Transversality was also introduced in this chapter (though I am already somewhat familiar with this from my difftop class long time ago).
In a sense, it generalizes regular values. Preimage of regular values form embedded submanifolds, preimage of submanifolds transversal to the map leads to embedded submanifolds.
Two submanifolds X, Y \subseteq M being transverse formalize the notion of them intersecting in a “generic manner.”
Transversality Homotopy Theorem: Given any embedded submanifold X \subseteq M, any smooth map is homotopic to another smooth map which is transverse to X.
This shows that transversality is generic. Very important for the study of stable property of smooth maps.
Chapter 8: Vector Fields
Vector fields a section X:TM→M of the projection map of the tangent bundle π:TM→M. Very elegant definition! Also lets me better appreciate the smooth structure defined over the tangent bundle, which automatically implies that a vector field is smooth iff their coordinate functions are smooth, which is what should morally be true.
Chapter 10: Vector Bundles
Vector bundles are objects that locally look like product spaces / vector-space “fibers” attached to each point of a manifold. Comb-like picture.
Chapter 11: Cotangent Bundle
I finally understand what covectors are and what’s their point. Cotangent bundle is just the dual of the tangent bundle. Covector takes a tangent vector and returns a number.
Applications:
Gradient of a smooth map, defined as a vector field where (under a given coordinate chart) the components of a function’s partial derivatives, is not invariant under change of coordinate chart, so it’s ill-defined. But when defined as a covector field, it is well-defined. This makes sense, since a “gradient of a function at a point” is morally an object that you take the inner product with a direction to return a scalar (directional derivative), thus it must be a covector that takes a tangent vector (“direction vector”) and returns a number.
Chapter 13: Riemannian Manifold
Smooth manifolds are metrizable. Why?
Broke: Manifold is locally compact Hausdorff second countable. Locally compact Hausdorff ⇒ completely regular. By Urysohn metrization theorem, completely regular & second countable ⇒ metrizability.
Woke: Use the distance metric on the manifold induced by the Riemannian structure, which always exists on manifold. This is much more intuitive.
Then I read some Bredon for Algebraic Topology.
Turns out, tubular neighborhoods are also useful for algebraic topology: How to show that a sphere Sn is not a retract of a disc Dn+1?
Assume such a retract f:Dn+1→Sn exists. Scale it and compose it appropriately to make it a radial projection (z↦z||z||) near the boundary Sn, and f but rescaled in the inside of the disc (easy to do). Smooth out the map in the inside via the Whitney approximation theorem. By Sard’s theorem, there is a point z∈Sn that is a regular value of f. Its preimage f−1(z), then, is a 1-dimensional manifold with boundary, where z is the only boundary (via radial projection). This contradicts the classification theorem of compact 1-manifolds which in particular says they have even number of boundary points.
From this follows Brouwer’s fixed point theorem (Dn−>Dn has fixed point), the fact that the sphere Sn is not contractible, etc.
I should more easily skip sections that I don’t understand. I struggled a bit with the first sub-section of the Fundamental Group chapter because it introduced the general notion of [SX;Y]∗ (the basepoint-preserving homotopy class of pointed maps SX→Y where SX refers to the reduced suspension X×I/(X×∂I∪{x0}×I)), for which the nth homotopy group becomes a special case [SSn−1;Y]∗=[Sn;Y]∗, for which the fundamental group is a special case, which is the only thing that is really needed until like very late in the book.
But thanks to muddling through, I think I much better understand this construction and motivation for constructions like the reduced suspension.
Fundamental group & Covering spaces & Lifting theorems & Deck Transformation
Mostly a review. Functors are just really natural objects (no wonder why they came from algebraic topology).
Specifically, the functor that transforms Y to [SX;Y]∗ and g:Y→Z to g#:[SX;Y]∗→[SX;Z]∗ where f↦g∘f. Homotopy groups (including the fundamental group) are a special case of this.
Covering spaces are an important tool for calculating fundamental groups. Also covering spaces have a general lifting theorem characterizing when maps can be lifted by the covering map.
Deck transformation and fundamental group are resp right / left actions on the fiber of the covering map, and they are commuting actions.
(Singular) Homology
This is new. Has the advantage of not caring about base-points. Weaker than homotopy.
Hatcher and Bredon are great complements. I saw a lot of anti-recommendations for Hatcher, but I think they’re great together.
Hatcher provides great intuition (geometric and conceptual) that Bredon just never really talked about. eg geometrically visualizing a deformation retract of a shape onto its skeleton by extruding the map in 3d mirrors the mapping cylinder construction, which is something I learned from Bredon but never (so far) learned the motivation for.
Many such examples and expositional niceness that helped reorganize my ontology (eg motivation-of-concept-wise, deformation retract comes prior to mapping cylinder, which comes prior (and motivates) the more general notion of homotopy and retracts. From this, it becomes intuitively clear why eg deformation retract should imply homotopy equivalence. Bredon taught the concepts the opposite way I think.)
Becoming Stronger™ (Oct 13 - Nov 2)
Notes and reflections on the things I’ve learned while Doing Scholarship the last
twothree weeks (i.e. studying math).The past three weeks were busier than usual so I had slower progress this time but here it is:
Chapter 6 continued: Sard’s theorem
Tubes! I might have thought the fact that you can embed manifolds in RN might have been one of those theorems whose main values are conceptual but not that useful in practice (such as Cayley’s theorem and the famous quote that the fact that any group is a subgroup of a symmetry group never actually made the task of studying & classifying groups easier) - but that is not the case, due to tubular neighborhoods.
The main issue with trying to solve problems about smooth maps between manifolds by embedding the codomain manifold in RN is that the resulting construction may not lie in the original manifold. Tubular neighborhoods address this by showing that it is always possible, given an embedding of a manifold in RN, to come up with a tube-like open neighborhood of the manifold, equipped with a retraction map, i.e. a map from this neighborhood to itself such that it is an identity when restricted to the manifold.
So conceptually, tubular neighborhoods let us reason about manifold problems by solving them in RN, and bringing it back to manifolds.
Applications are massive! Might be my favorite concept so far.
Whitney approximation theorem (any continuous map is homotopic to a smooth one. If the original continuous map is smooth on a closed subset, the homotopy can be taken relative to it)
Proof: Prove it for the case when the codomain is Rm, and just compose it with the tubular neighborhood retraction.
If a compact manifold has a nonzero vector field, then there exists a map to itself without fixed points
Proof: Embed everything (the manifold, its tangent spaces) in R^m, let the map be the one that takes a point to the direction indicated by the nonzero vector field at some small magnitude ϵ. This will take things outside of the manifold, so compose with the tubular neighborhood retraction (which is allowed if ϵ is small enough).
Transversality was also introduced in this chapter (though I am already somewhat familiar with this from my difftop class long time ago).
In a sense, it generalizes regular values. Preimage of regular values form embedded submanifolds, preimage of submanifolds transversal to the map leads to embedded submanifolds.
Two submanifolds X, Y \subseteq M being transverse formalize the notion of them intersecting in a “generic manner.”
Transversality Homotopy Theorem: Given any embedded submanifold X \subseteq M, any smooth map is homotopic to another smooth map which is transverse to X.
This shows that transversality is generic. Very important for the study of stable property of smooth maps.
Chapter 8: Vector Fields
Vector fields a section X:TM→M of the projection map of the tangent bundle π:TM→M. Very elegant definition! Also lets me better appreciate the smooth structure defined over the tangent bundle, which automatically implies that a vector field is smooth iff their coordinate functions are smooth, which is what should morally be true.
Chapter 10: Vector Bundles
Vector bundles are objects that locally look like product spaces / vector-space “fibers” attached to each point of a manifold. Comb-like picture.
Chapter 11: Cotangent Bundle
I finally understand what covectors are and what’s their point. Cotangent bundle is just the dual of the tangent bundle. Covector takes a tangent vector and returns a number.
Applications:
Gradient of a smooth map, defined as a vector field where (under a given coordinate chart) the components of a function’s partial derivatives, is not invariant under change of coordinate chart, so it’s ill-defined. But when defined as a covector field, it is well-defined. This makes sense, since a “gradient of a function at a point” is morally an object that you take the inner product with a direction to return a scalar (directional derivative), thus it must be a covector that takes a tangent vector (“direction vector”) and returns a number.
Chapter 13: Riemannian Manifold
Smooth manifolds are metrizable. Why?
Broke: Manifold is locally compact Hausdorff second countable. Locally compact Hausdorff ⇒ completely regular. By Urysohn metrization theorem, completely regular & second countable ⇒ metrizability.
Woke: Use the distance metric on the manifold induced by the Riemannian structure, which always exists on manifold. This is much more intuitive.
Then I read some Bredon for Algebraic Topology.
Turns out, tubular neighborhoods are also useful for algebraic topology: How to show that a sphere Sn is not a retract of a disc Dn+1?
Assume such a retract f:Dn+1→Sn exists. Scale it and compose it appropriately to make it a radial projection (z↦z||z||) near the boundary Sn, and f but rescaled in the inside of the disc (easy to do). Smooth out the map in the inside via the Whitney approximation theorem. By Sard’s theorem, there is a point z∈Sn that is a regular value of f. Its preimage f−1(z), then, is a 1-dimensional manifold with boundary, where z is the only boundary (via radial projection). This contradicts the classification theorem of compact 1-manifolds which in particular says they have even number of boundary points.
From this follows Brouwer’s fixed point theorem (Dn−>Dn has fixed point), the fact that the sphere Sn is not contractible, etc.
I should more easily skip sections that I don’t understand. I struggled a bit with the first sub-section of the Fundamental Group chapter because it introduced the general notion of [SX;Y]∗ (the basepoint-preserving homotopy class of pointed maps SX→Y where SX refers to the reduced suspension X×I/(X×∂I∪{x0}×I)), for which the nth homotopy group becomes a special case [SSn−1;Y]∗=[Sn;Y]∗, for which the fundamental group is a special case, which is the only thing that is really needed until like very late in the book.
But thanks to muddling through, I think I much better understand this construction and motivation for constructions like the reduced suspension.
Fundamental group & Covering spaces & Lifting theorems & Deck Transformation
Mostly a review. Functors are just really natural objects (no wonder why they came from algebraic topology).
Specifically, the functor that transforms Y to [SX;Y]∗ and g:Y→Z to g#:[SX;Y]∗→[SX;Z]∗ where f↦g∘f. Homotopy groups (including the fundamental group) are a special case of this.
Covering spaces are an important tool for calculating fundamental groups. Also covering spaces have a general lifting theorem characterizing when maps can be lifted by the covering map.
Deck transformation and fundamental group are resp right / left actions on the fiber of the covering map, and they are commuting actions.
(Singular) Homology
This is new. Has the advantage of not caring about base-points. Weaker than homotopy.
Hatcher and Bredon are great complements. I saw a lot of anti-recommendations for Hatcher, but I think they’re great together.
Hatcher provides great intuition (geometric and conceptual) that Bredon just never really talked about. eg geometrically visualizing a deformation retract of a shape onto its skeleton by extruding the map in 3d mirrors the mapping cylinder construction, which is something I learned from Bredon but never (so far) learned the motivation for.
Many such examples and expositional niceness that helped reorganize my ontology (eg motivation-of-concept-wise, deformation retract comes prior to mapping cylinder, which comes prior (and motivates) the more general notion of homotopy and retracts. From this, it becomes intuitively clear why eg deformation retract should imply homotopy equivalence. Bredon taught the concepts the opposite way I think.)