This is a very nice and concise 8 minute introduction to cohomology. Very clear and tells you the gist of cohomology.
Tag: Topology
Recommended Books for Spectral Sequences
Best Spectral Sequence Book
So far the most comprehensive book looks like McCleary’s book: A User’s Guide to Spectral Sequences. It is also suitable for those interested in the algebraic viewpoint. W.S. Massey wrote a very positive review to this book.
A User’s Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)
Another book is Rotman’s An Introduction to Homological Algebra (Universitext). This book is from a homological algebra viewpoint. Rotman has a nice easy-going style, that made his books very popular to read.
The classic book may be MacLane’s Homology (Classics in Mathematics). This may be harder to read (though to be honest all books on spectral sequences are hard).
***Update: I found another book that gives a very nice presentation of certain spectral sequences, for instance the Bockstein spectral sequence. The book is Algebraic Methods in Unstable Homotopy Theory (New Mathematical Monographs) by Joseph Neisendorfer.
Topology application to Physics
Source: https://www.scientificamerican.com/article/the-strange-topology-that-is-reshaping-physics/?W
The Strange Topology That Is Reshaping Physics
Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing
Charles Kane never thought he would be cavorting with topologists. “I don’t think like a mathematician,” admits Kane, a theoretical physicist who has tended to focus on tangible problems about solid materials. He is not alone. Physicists have typically paid little attention to topology—the mathematical study of shapes and their arrangement in space. But now Kane and other physicists are flocking to the field.
In the past decade, they have found that topology provides unique insight into the physics of materials, such as how some insulators can sneakily conduct electricity along a single-atom layer on their surfaces.
Some of these topological effects were uncovered in the 1980s, but only in the past few years have researchers begun to realize that they could be much more prevalent and bizarre than anyone expected. Topological materials have been “sitting in plain sight, and people didn’t think to look for them”, says Kane, who is at the University of Pennsylvania in Philadelphia.
Now, topological physics is truly exploding: it seems increasingly rare to see a paper on solid-state physics that doesn’t have the word topology in the title. And experimentalists are about to get even busier. A study on page 298 of this week’s Nature unveils an atlas of materials that might host topological effects, giving physicists many more places to go looking for bizarre states of matter such as Weyl fermions or quantum-spin liquids.
Read more at: https://www.scientificamerican.com/article/the-strange-topology-that-is-reshaping-physics/?WT.mc_id=SA_WR_20170726
Summary: Shapes, radius functions and persistent homology
This is a summary of a talk by Professor Herbert Edelsbrunner, IST Austria. The PDF slides can be found here: persistent homology slides.
Biogeometry (2:51 in video)
We can think of proteins as a geometric object by replacing every atom by a sphere (possibly different radii). Protein is viewed as union of balls in .
Decompose into Voronoi domains , and take the nerve (Delaunay complex).
Inclusion-Exclusion Theorem:
Volume of protein is alternating sum over all simplices
in Delaunay complex.
Nerve Theorem: Union of sets have same homotopy type as nerve (stronger than having isomorphic homology groups).
Wrap (14:04 in video)
Collapses: 01 collapse means 0 dimensional and 1 dimensional simplices disappear (something like deformation retract).
Interval: Simplices that are removed in a collapse (always a skeleton of a cube in appropriate dimension)
Generalised Discrete Morse Function (Forman 1998): Generalised discrete vector field partition into intervals (for acyclic case only)
Critical simplex: The only simplex in an interval (when a critical simplex is added, the homotopy type changes)
Lower set of critical simplex: all the nodes that lead up to the critical simplex.
Wrap complex is the union of lower sets.
Persistence (38:00 in video)
Betti numbers in :
components,
loops,
voids.
Incremental Algorithm to compute Betti numbers (40:50 in video). [Deffimado, E., 1995]. Every time a simplex is added, either a Betti number goes up (birth) or goes down (death).
is born when it is not in image of previous homology group.
Stability of persistence: small change in position of points leads to similar persistence diagram.
Bottleneck distance between two diagrams is length of longest edge in minimizing matching. Theorem: [Cohen-Steiner, E., Hares 2007]. One of the most important theorems in persistent homology.
Expectation (51:30 in video)
Poisson point process: Like uniform distribution but over entire space. Number of points in region is proportional to size of region. Proportionality constant is density .
Paper: Expectations in . [E., Nikitenko, Reitones, 2016]
Reduces to question (Three points in circle): Given three points in a circle, what is the probability that the triangle (with the 3 points as vertices) contains the center of the circle? Ans: 1/4 [Wendel 1963].
Brain has 11 dimensions
One of the possible applications of algebraic topology is in studying the brain, which is known to be very complicated.
Site: https://www.wired.com/story/the-mind-boggling-math-that-maybe-mapped-the-brain-in-11-dimensions/
If you can call understanding the dynamics of a virtual rat brain a real-world problem. In a multimillion-dollar supercomputer in a building on the same campus where Hess has spent 25 years stretching and shrinking geometric objects in her mind, lives one of the most detailed digital reconstructions of brain tissue ever built. Representing 55 distinct types of neurons and 36 million synapses all firing in a space the size of pinhead, the simulation is the brainchild of Henry Markram.
Markram and Hess met through a mutual researcher friend 12 years ago, right around the time Markram was launching Blue Brain—the Swiss institute’s ambitious bid to build a complete, simulated brain, starting with the rat. Over the next decade, as Markram began feeding terabytes of data into an IBM supercomputer and reconstructing a collection of neurons in the sensory cortex, he and Hess continued to meet and discuss how they might use her specialized knowledge to understand what he was creating. “It became clearer and clearer algebraic topology could help you see things you just can’t see with flat mathematics,” says Markram. But Hess didn’t officially join the project until 2015, when it met (and some would say failed) its first big public test.
In October of that year, Markram led an international team of neuroscientists in unveiling the first Blue Brain results: a simulation of 31,000 connected rat neurons that responded with waves of coordinated electricity in response to an artificial stimulus. The long awaited, 36-page paper published in Cell was not greeted as the unequivocal success Markram expected. Instead, it further polarized a research community already divided by the audacity of his prophesizing and the insane amount of money behind the project.
Two years before, the European Union had awarded Markram $1.3 billion to spend the next decade building a computerized human brain. But not long after, hundreds of EU scientists revolted against that initiative, the Human Brain Project. In the summer of 2015, they penned an open letter questioning the scientific value of the project and threatening to boycott unless it was reformed. Two independent reviews agreed with the critics, and the Human Brain Project downgraded Markram’s involvement. It was into this turbulent atmosphere that Blue Brain announced its modest progress on its bit of simulated rat cortex.
Read more at the link above.
Guide to Starting Javaplex (With Matlab)
Guide to Starting Javaplex (With Matlab)
Step 1)
Visit https://appliedtopology.github.io/javaplex/ and download the Persistent Homology and Topological Data Analysis Library
2)
Download the tutorial at http://www.math.colostate.edu/~adams/research/javaplex_tutorial.pdf and jump to section 1.3. Installation for Matlab.
3)
In Matlab, change Matlab’s “Current Folder” to the directory matlab examples that you just extracted from the zip file.
(See https://www.mathworks.com/help/matlab/ref/cd.html to change current folder)
Type this in Matlab: cd /…/matlab_examples
Where … depends on where you put the folder
4) In the tutorial (from the link given in step 2), proceed to follow the instructions starting from “In Matlab, change Matlab’s “Current Folder” to the directory matlab examples that you just extracted from the zip file. In the Matlab command window, run the load javaplex.m file.”.
5) Test: Run example 3.2 (House example) by typing in the code (following the tutorial)
How the Staircase Diagram changes when we pass to derived couple (Spectral Sequence)
Set and
. The diagram then has the following form:
When we pass to the derived couple, each group is replaced by a subgroup
. The differentials
go two units to the right, and we replace the term
by the term
, where the
‘s refer to the
‘s leaving and entering
respectively.
The maps now go diagonally upward because of the formula
. The maps
and
still go vertically and horizontally,
being a restriction of
and
being induced by
.
Relative Homology Groups
Given a space and a subspace
, define
. Since the boundary map
takes
to
, it induces a quotient boundary map
.
We have a chain complex where
holds. The relative homology groups
are the homology groups
of this chain complex.
Relative cycles
Elements of are represented by relative cycles:
– chains
such that
.
Relative boundary
A relative cycle is trivial in
iff it is a relative boundary:
for some
and
.
Long Exact Sequence (Relative Homology)
There is a long exact sequence of homology groups:
The boundary map is as follows: If a class
is represented by a relative cycle
, then
is the class of the cycle
in
.
Exact sequence (Quotient space)
Exact sequence (Quotient space)
If is a space and
is a nonempty closed subspace that is a deformation retract of some neighborhood in
, then there is an exact sequence
where is the inclusion
and
is the quotient map
.
Reduced homology of spheres (Proof)
and
for
.
For take
so that
. The terms
in the long exact sequence are zero since
is contractible.
Exactness of the sequence then implies that the maps are isomorphisms for
and that
. Starting with
,
for
, the result follows by induction on
.
Reduced Homology
Define the reduced homology groups to be the homology groups of the augmented chain complex
where
. We require
to be nonempty, to avoid having a nontrivial homology group in dimension -1.
Relation between and
Since ,
vanishes on
and hence induces a map
with
. So
. Clearly,
for
.
Klein Bottle as Gluing of Two Mobius Bands
Mayer-Vietoris Sequence applied to Spheres
Mayer-Vietoris Sequence
For a pair of subspaces such that
, the exact MV sequence has the form
Example:
Let with
and
the northern and southern hemispheres, so that
. Then in the reduced Mayer-Vietoris sequence the terms
are zero. So from the reduced Mayer-Vietoris sequence
we get the exact sequence
We obtain isomorphisms .
Spectral Sequence
Spectral Sequence is one of the advanced tools in Algebraic Topology. The following definition is from Hatcher’s 5th chapter on Spectral Sequences. The staircase diagram looks particularly impressive and intimidating at the same time.
Unfortunately, my LaTeX to WordPress Converter app can’t handle commutative diagrams well, so I will upload a printscreen instead.
Echelon Form Lemma (Column Echelon vs Smith Normal Form)
The pivots in column-echelon form are the same as the diagonal elements in (Smith) normal form. Moreover, the degree of the basis elements on pivot rows is the same in both forms.
Proof:
Due to the initial sort, the degree of row basis elements is monotonically decreasing from the top row down. For each fixed column
,
is a constant. We have,
. Hence, the degree of the elements in each column is monotonically increasing with row. That is, for fixed
,
is monotonically increasing as
increases.
We may then eliminate non-zero elements below pivots using row operations that do not change the pivot elements or the degrees of the row basis elements. Finally, we place the matrix in (Smith) normal form with row and column swaps.
Persistent Homology Algorithm
Algorithm for Fields
In this section we describe an algorithm for computing persistent homology over a field.
We use the small filtration as an example and compute over , although the algorithm works for any field.
A filtered simplicial complex with new simplices added at each stage. The integers on the bottom row corresponds to the degrees of the simplices of the filtration as homogenous elements of the persistence module.
The persistence module corresponds to a -module by the correspondence in previous Theorem. In this section we use
and
to denote homogeneous bases for
and
respectively.
We have since we are computing over
. Then the representation matrix for
is
In general, any representation of
has the following basic property:
provided
.
We need to represent relative to the standard basis for
and a homogenous basis for
. We then reduce the matrix according to the reduction algorithm described previously.
We compute the representations inductively in dimension. Since ,
hence the standard basis may be used to represent
. Now, suppose we have a matrix representation
of
relative to the standard basis
for
and a homogeneous basis
for
.
For the inductive step, we need to compute a homogeneous basis for and represent
relative to
and the homogeneous basis for
. We first sort the basis
in reverse degree order. Next, we make
into the column-echelon form
by Gaussian elimination on the columns, using elementary column operations. From linear algebra, we know that
is the number of pivots in the echelon form. The basis elements corresponding to non-pivot columns form the desired basis for
.
Source: “Computing Persistent Homology” by Zomorodian & Carlsson
De Rham Cohomology
De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined.
Also, do check out our presentation on the relation between De Rham Cohomology and physics: De Rham Cohomology.
Definition:
A differential form on a manifold
is said to be closed if
, and exact if
for some
of degree one less.
Corollary:
Since , every exact form is closed.
Definition:
Let be the vector space of all closed
-forms on
.
Let be the vector space of all exact
-forms on
.
Since every exact form is closed, hence .
The de Rham cohomology of in degree
is defined as the quotient vector space
The quotient vector space construction induces an equivalence relation on :
in
iff
iff
for some exact form
.
The equivalence class of a closed form is called its cohomology class and denoted by
.
Singular Homology
A singular -simplex in a space
is a map
. Let
be the free abelian group with basis the set of singular
-simplices in
. Elements of
, called singular
-chains, are finite formal sums
for
and
. A boundary map
is defined by
The singular homology group is defined as .
Mapping Cone Theorem
Mapping cone
Let be a map in
. We construct the mapping cone
, where
is identified with
for all
.
Proposition:
For any map we have
if and only if
has an extension
to
.
Proof:
By an earlier proposition (2.32 in \cite{Switzer2002}), iff
has an extension
.
() If
, define
by
,
. Note that
. Since
induces a map
which satisfies
. That is
.
() If
has an extension
, then define
by
. We have
. Then
That is,
.
Tangent space (Derivation definition)
Let be a smooth manifold, and let
. A linear map
is called a derivation at
if it satisfies
The tangent space to at
, denoted by
, is defined as the set of all derivations of
at
.
Homology Group of some Common Spaces
Homology of Circle
Homology of Torus
Homology of Real Projective Plane
Homology of Klein Bottle
Summary of Persistent Homology
We summarize the work so far and relate it to previous results. Our input is a filtered complex and we wish to find its
th homology
. In each dimension the homology of complex
becomes a vector space over a field, described fully by its rank
. (Over a field
,
is a
-module which is a vector space.)
We need to choose compatible bases across the filtration (compatible bases for and
) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module
corresponding to
, which is a direct sum of these vector spaces (
). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.
Specifically, each -interval
describes a basis element for the homology vector spaces starting at time
until time
. This element is a
-cycle
that is completed at time
, forming a new homology class. It also remains non-bounding until time
, at which time it joins the boundary group
.
A natural question is to ask when is a basis element for the persistent groups
. Recall the equation
Since
for all
, hence
for
. The three inequalities
define a triangular region in the index-persistence plane, as shown in Figure below.
The triangular region gives us the values for which the -cycle
is a basis element for
. This is known as the
-triangle Lemma:
Let be the set of triangles defined by
-intervals for the
-dimensional persistence module. The rank
of
is the number of triangles in
containing the point
.
Hence, computing persistent homology over a field is equivalent to finding the corresponding set of -intervals.
Source: “Computing Persistent Homology” by Zomorodian and Carlsson
Structure Theorem for finitely generated (graded) modules over a PID
If is a PID, then every finitely generated module
over
is isomorphic to a direct sum of cyclic
-modules. That is, there is a unique decreasing sequence of proper ideals
such that
where
, and
.
Similarly, every graded module over a graded PID
decomposes uniquely into the form
where
are homogenous elements such that
,
, and
denotes an
-shift upward in grading.
Persistence Interval
Next, we want to parametrize the isomorphism classes of the -modules by suitable objects.
A -interval is an ordered pair
with
.
We may associate a graded -module to a set
of
-intervals via a bijection
. We define
for a
-interval
. When
, we have
.
For a set of -intervals
, we define
We may now restate the correspondence as follows.
The correspondence defines a bijection between the finite sets of
-intervals and the finitely generated graded modules over the graded ring
.
Hence, the isomorphism classes of persistence modules of finite type over are in bijective correspondence with the finite sets of
-intervals.
Homogenous / Graded Ideal
Let be a graded ring. An ideal
is homogenous (also called graded) if for every element
, its homogenous components also belong to
.
An ideal in a graded ring is homogenous if and only if it is a graded submodule. The intersections of a homogenous ideal with the
are called the homogenous parts of
. A homogenous ideal
is the direct sum of its homogenous parts, that is,
Persistence module and Graded Module
We show that the persistent homology of a filtered simplicial complex is the standard homology of a particular graded module over a polynomial ring.
First we review some definitions.
A graded ring is a ring (a direct sum of abelian groups
) such that
for all
,
.
A graded ring is called non-negatively graded if
for all
. Elements of any factor
of the decomposition are called homogenous elements of degree
.
Polynomial ring with standard grading:
We may grade the polynomial ring non-negatively with the standard grading
for all
.
Graded module:
A graded module is a left module over a graded ring
such that
and
.
Let be a commutative ring with unity. Let
be a persistence module over
.
We now equip with the standard grading and define a graded module over
by
where the
-module structure is the sum of the structures on the individual components. That is, for all
,
The action of is given by
That is, shifts elements of the module up in the gradation.
Source: “Computing Persistent Homology” by Zomorodian and Carlsson.
Persistence module and Finite type
A persistence module is a family of
-modules
, together with homomorphisms
.
For example, the homology of a persistence complex is a persistence module, where maps a homology class to the one that contains it.
A persistence complex (resp.\ persistence module
) is of finite type if each component complex (resp.\ module) is a finitely generated
-module, and if the maps
(resp.\
) are isomorphisms for
for some integer
.
If is a finite filtered simplicial complex, then it generates a persistence complex
of finite type, whose homology is a persistence module
of finite type.
Homotopy for Maps vs Paths
Homotopy (of maps)
A homotopy is a family of maps ,
, such that the associated map
given by
is continuous. Two maps
are called homotopic, denoted
, if there exists a homotopy
connecting them.
Homotopy of paths
A homotopy of paths in a space is a family
,
, such that
(i) The endpoints and
are independent of
.
(ii) The associated map defined by
is continuous.
When two paths and
are connected in this way by a homotopy
, they are said to be homotopic. The notation for this is
.
The above two definitions are related, since a path is a special kind of map .
Universal Property of Quotient Groups (Hungerford)
If is a homomorphism and
is a normal subgroup of
contained in the kernel of
, then
“factors through” the quotient
uniquely.
This can be used to prove the following proposition:
A chain map between chain complexes
and
induces homomorphisms between the homology groups of the two complexes.
Proof:
The relation implies that
takes cycles to cycles since
implies
. Also
takes boundaries to boundaries since
. Hence
induces a homomorphism
, by universal property of quotient groups.
For , we have
. Therefore
.
Some Homology Definitions
Chain Complex
A sequence of homomorphisms of abelian groups with
for each
.
th Homology Group
is the free abelian group with basis the open
-simplices
of
.
-chains
Elements of , called
-chains, can be written as finite formal sums
with coefficients
.
Introduction to Persistent Homology (Cech and Vietoris-Rips complex)
Motivation
Data is commonly represented as an unordered sequence of points in the Euclidean space . The global `shape’ of the data may provide important information about the underlying phenomena of the data.
For data points in , determining the global structure is not difficult, but for data in higher dimensions, a planar projection can be hard to decipher.
From point cloud data to simplicial complexes
To convert a collection of points in a metric space into a global object, one can use the points as the vertices of a graph whose edges are determined by proximity (vertices within some chosen distance
). Then, one completes the graph to a simplicial complex. Two of the most natural methods for doing so are as follows:
Given a set of points in Euclidean space
, the Cech complex (also known as the nerve),
, is the abstract simplicial complex where a set of
vertices spans a
-simplex whenever the
corresponding closed
-ball neighborhoods have nonempty intersection.
Given a set of points in Euclidean space
, the Vietoris-Rips complex,
, is the abstract simplicial complex where a set
of
vertices spans a
-simplex whenever the distance between any pair of points in
is at most
.
Top left: A fixed set of points. Top right: Closed balls of radius centered at the points. Bottom left: Cech complex has the homotopy type of the
cover (
) Bottom right: Vietoris-Rips complex has a different homotopy type (
). Image from R. Ghrist, 2008, Barcodes: The Persistent Topology of Data.
Natural Equivalence relating Suspension and Loop Space
Theorem:
If ,
,
,
,
Hausdorff and
locally compact, then there is a natural equivalence
defined by
, where if
is a map then
is given by
.
We need the following two propositions in order to prove the theorem.
Proposition
\label{prop13}
The exponential function induces a continuous function
which is a homeomorphism if
and
are Hausdorff and
is locally compact\footnote{every point of
has a compact neighborhood}.
Proposition
\label{prop8}
If is an equivalence relation on a topological space
and
is a homotopy such that each stage
factors through
, i.e.\
, then
induces a homotopy
such that
.
Proof of Theorem
i) is surjective: Let
. From Proposition \ref{prop13} we have that
is a homeomorphism. Hence the function
defined by
is continuous since
and thus
. By the universal property of the quotient,
defines a map
such that
. Thus
, so that
.
ii) is injective: Suppose
are two maps such that
, i.e.\
. Let
be the homotopy rel
. By Proposition \ref{prop13} the function
defined by
is continuous. This is because
so that
, thus
where
is a homeomorphism. For each
we have
. This is because if
, then
or
. If
, then
. If
,
as
is the homotopy rel
. Then by Proposition \ref{prop8} there is a homotopy
rel
such that
. Thus
and similarly
. Thus
via the homotopy
.
Loop space
If , we define the loop space
of
to be the function space
with the constant loop
(
for all
) as base point.
Suspension
If , we define the suspension
of
to be the smash product
of
with the 1-sphere.
Corollary (Natural Equivalence relating and
)
If ,
and
is Hausdorff, then there is a natural equivalence
Fundamental Group of S^n is trivial if n>=2
if
We need the following lemma:
If a space is the union of a collection of path-connected open sets
each containing the basepoint
and if each intersection
is path-connected, then every loop in
at
is homotopic to a product of loops each of which is contained in a single
.
Proof:
Take and
to be the complements of two antipodal points in
. Then
is the union of two open sets
and
, each homeomorphic to
such that
is homeomorphic to
.
Choose a basepoint in
. If
then
is path-connected. By the lemma, every loop in
based at
is homotopic to a product of loops in
or
. Both
and
are zero since
and
are homeomorphic to
. Hence every loop in
is nullhomotopic.
Functors, Homotopy Sets and Groups
Functors
Definition:
A functor from a category
to a category
is a function which
– For each object , we have an object
.
– For each , we have a morphism
Furthermore, is required to satisfy the two axioms:
– For each object , we have
. That is,
maps the identity morphism on
to the identity morphism on
.
– For ,
we have
That is, functors must preserve composition of morphisms.
Definition:
A cofunctor (also called contravariant functor) from a category
to a category
is a function which
– For each object , we have an object
.
– For each we have a morphism
satisfying the two axioms:
– For each object we have
. That is,
preserves identity morphisms.
– For each and
we have
Note that cofunctors reverse the direction of composition.
Example
Given a fixed pointed space , we define a functor
as follows: for each
we assign
. Given
in
we define
by
for every
.
We can check the two axioms:
– for every
.
– For ,
we have
for every
.
Similarly, we can define a cofunctor by taking
and for
in
we define
for every
.
Note that if rel
, then
and similarly
. Therefore
(resp.\
) can also be regarded as defining a functor (resp.\ cofunctor)
.
Homotopy Sets and Groups
Theorem:
If ,
,
,
,
Hausdorff and
locally compact, then there is a natural equivalence
defined by
, where if
is a map then
is given by
.
We need the following two propositions in order to prove the theorem.
Proposition 1:
The exponential function induces a continuous function
which is a homeomorphism if
and
are Hausdorff and
is locally compact\footnote{every point of
has a compact neighborhood}.
Proposition 2:
If is an equivalence relation on a topological space
and
is a homotopy such that each stage
factors through
, i.e.\
, then
induces a homotopy
such that
.
Algebraic Topology: Fundamental Group
Homotopy of paths
A homotopy of paths in a space is a family
,
, such that
(i) The endpoints and
are independent of
.
(ii) The associated map defined by
is continuous.
When two paths and
are connected in this way by a homotopy
, they are said to be homotopic. The notation for this is
.
Example: Linear Homotopies
Any two paths and
in
having the same endpoints
and
are homotopic via the homotopy
Simply-connected
A space is called simply-connected if it is path-connected and has trivial fundamental group.
A space is simply-connected iff there is a unique homotopy class of paths connecting any two parts in
.
Path-connectedness is the existence of paths connecting every pair of points, so we need to be concerned only with the uniqueness of connecting paths.
() Suppose
. If
and
are two paths from
to
, then
since the loops
and
are each homotopic to constant loops, due to
.
() Conversely, if there is only one homotopy class of paths connecting a basepoint
to itself, then all loops at
are homotopic to the constant loop and
.
is isomorphic to
if
and
are path-connected.
A basic property of the product topology is that a map is continuous iff the maps
and
defined by
are both continuous.
Hence a loop in
based at
is equivalent to a pair of loops
in
and
in
based at
and
respectively.
Similarly, a homotopy of a loop in
is equivalent to a pair of homotopies
and
of the corresponding loops in
and
.
Thus we obtain a bijection ,
. This is clearly a group homomorphism, and hence an isomorphism.
Note: The condition that and
are path-connected implies that
,
.
Topological Monster: Alexander horned sphere
Very interesting object indeed. Also see this previous video on How to Unlock Interlocked Fingers Topologically?
The horned sphere, together with its inside, is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. The exterior is not simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere. (Wikipedia)
Hatcher 2.1.6
Wonderful Topology Notes for Beginners
Recently found a wonderful topology notes, suitable for beginners at: http://mathcircle.berkeley.edu/archivedocs/2010_2011/lectures/1011lecturespdf/bmc_topology_manifolds.pdf
It starts by pondering the shape of the earth, then generalizes to other surfaces. It also has a nice section on Fundamental Polygons and cutting and gluing, which was what I was looking for at first.
I have backed up a copy on Mathtuition88.com, in case the original site goes down in the future: bmc_topology_manifolds
From Triangle to Mobius Strip
Covering map is an open map
We prove a lemma that the covering map is an open map.
Let be open in
. Let
, then
has an evenly covered open neighborhood
, such that
, where the
are disjoint open sets in
, and
is a homeomorphism.
is open in
, and open in
, so
is open in
, thus open in
.
There exists such that
. Thus
so
for some
. Thus
and thus
. This shows
is an interior point of
. Hence
is open, thus
is an open map.
Topology Puzzle
Assume you are a superman who is very elastic, after making linked rings with your index fingers and thumbs, could you move your hands apart without separating the joined fingertips?
In other words, is it possible to go from (a) to (b) without “breaking” the figure above?
Figure taken from Intuitive Topology (Mathematical World, Vol 4).
The answer is yes!
This is an animation of the solution: https://vk.com/video-9666747_142799479
Topology book for General Audience
Previously I blogged about the book The Shape of Space, here is another book that is also about topology and suitable for a general audience (high school and above)!
Effective Homotopy Method
The main idea of the effective homotopy method is the following: given some Kan simplicial sets , a topological constructor
produces a new simplicial set
. If solutions for the homotopical problems of the spaces
are known, then one should be able to build a solution for the homotopical problem of
, and this construction would allow us to compute the homotopy groups
.
Representing map of x
Proposition:
(Representing map of ). Let
be a simplicial set and let
. There exists a unique simplicial map
such that
.
Proof:
Let .
can be written as iterated compositions of faces and degeneracies of
, i.e.
where
is an iterated composition of faces
and degeneracies (
). Then
This defines a unique simplicial map such that
.
Minimal Simplicial Sets
Let be a space. We can have a fibrant simplicial set, namely the singular simplicial set
, where
is the set of all continuous maps from the
-simplex to
. However
seems too large as there are uncountably many elements in each
. On the other hand, we need the fibrant assumption to have simplicial homotopy groups. This means that the simplicial model for a given space cannot be too small. We wish to have a smallest fibrant simplicial set which will be the idea behind minimal simplicial sets.
Let be a fibrant simplicial set. For
we say that
if the representing maps
and
are homotopic relative to
.
A fibrant simplicial set is said to be minimal if it has the property that implies
.
Let be a fibrant simplicial set.
is minimal iff for any
,
such that
for all
implies
.
In other words, it means that a fibrant simplicial set is minimal iff for any two elements with all faces but one the same, then the missed face must be the same.
Higher Homotopy Groups
We can generalise the idea to higher homotopy groups as follows.
Let be a pointed fibrant simplicial set. The fundamental group
is the quotient set of the spherical elements in
subject to the relation generated by
if there exists
such that
,
and
for
.
The product structure in is given by:
, where
such that
,
and
for
. Furthermore, the map
preserves the product structure.
Associativity and Path Inverse for Fundamental Groupoids
Continued from Path product and fundamental groupoids
(Associativity). Let be a fibrant simplicial set and let
,
and
be paths in
such that
and
. Then
Let be a point. Denote
as the constant simplicial map
for
.
(Path Inverse). Let be a fibrant simplicial set and let
be a path in
. Then there exists a path
such that
.
Path product and fundamental groupoids
Let . A path is a simplicial map
. Since
, the paths are in one-to-one correspondence to the elements in
via the function
. The initial point of
is
, and the end point of
is
.
Let and
be two paths such that
. Then
and thus the elements
and
have matching faces (with respect to 1).
Homotopy Groups
Let be a pointed fibrant simplicial set. The homotopy group
, as a set, is defined by
, i.e. the set of the pointed homotopy classes of all pointed simplicial maps from
to
.
as sets.
An element is said to be spherical if
for all
.
Given a spherical element , then its representing map
factors through the simplicial quotient set
. Conversely, any simplicial map
gives a spherical element
, where
is the nondegenerate element in
. This gives a one-to-one correspondence from the set of spherical elements in
to the set of simplicial maps
.
Path product and fundamental groupoids
Let . A path is a simplicial map
.
Geometrical Meaning of Matching Faces
Let be a simplicial set. The elements
are said to be matching faces with respect to
if
for
and
.
Geometrically, matching faces are faces that “match” along lower-dimensional faces. In other words, they are “adjacent”.
In the 2-simplex, let ,
,
. Then
are matching faces with respect to 1, since
.
In the 3-simplex, let ,
,
,
. Then
,
,
are matching faces with respect to 1, since the following hold:
Fibrant Simplicial Set
Let be a simplicial set. Then
is fibrant if and only if every simplicial map
has an extension for each
.
Assume that is fibrant. Let
. The elements
are matching faces with respect to
. This is because for
and
,
Thus, since is fibrant, there exists an element
such that
for
. Then, the representing map
,
, is an extension of
.
Conversely let be any elements that are matching faces with respect to
. Then the representing maps
for
defines a simplicial map
such that the diagram
commutes for each .
By the assumption, there exists an extension such that
. Let
. Then
for
. Thus
is fibrant.