Science topics: Geometry and TopologyAlgebraic Topology

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# Algebraic Topology - Science topic

Explore the latest questions and answers in Algebraic Topology, and find Algebraic Topology experts.

Questions related to Algebraic Topology

I noticed that there is a structural similarity between the syntactic operations of Bealer's logic (see my paper "Bealer's Intensional Logic" that I uploaded to Researchgate for my interpretation of these operations) and the notion of non-symmetric operad. However for the correspondence to be complete I need a diagonalisation operation.

Consider an operad P with P(n) the set of functions from the cartesian product X^n to X.

Then I need operations Dij : P(n) -> P(n-1) which identify variables xi and xj.

Has this been considered in the literature ?

I am starting a new area of research that is Algebraic topology. Kindly suggest some latest problems and related publications

How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?

Is the Poincaré Conjecture relatively easy to prove for the case of simply connected real analytic 3- manifolds ?

How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?

A fascinating question in theoretical physics is whether it is possible to extend Einstein's ideas beyond gravitation to all aspects of physics. The energy-momentum tensor is usually defined extrinsically over the space-time manifold. But could it rather be derived from the geometry alone ? Likewise our local subjective notion of time is given by a local orientation which need not be globally consistent as in Gödel's famous model.

It has been proposed that space-time may have a foam- or sponge-like fine-grained structure (possible involving extra dimensions) which explains energy and matter and the other fundamental forces in a Kaluza-Klein style. That is, "microlocally" the topology of the space-time manifold is highly complex and there may be even a direct relationship between mass, energy and cohomology complexes in an appropriate derived category. At this fine scale there may even be non-local wormholes that connect distant regions of space-time and explain quantum entanglement.

But why not consider the universe as a Thom-Mather stratified space (one can think of this as a smooth version of analytic spaces or algebraic varieties) rather than a manifold ? In this case "singularities" would be "natural" structures not pathologies as in black holes. It is difficult not to think of matter (or localised energy) as corresponding to a singular region of this stratified space. Has this approach been considered in the literature ?

It is well known the theoretical applications of generalized open sets in topological spaces, for example we can by them define various forms of continuous maps, compact spaces, separation axioms, etc. My question is: what the practical (reality) applications of generalized open sets such as semi-open and pre-open sets?.

Are these sets used to modeling some phenomena or problems?

If I wanted to link algebra and topology in order to specialize in algebraic topology (mathematics), what researches would you recommend me to start reading with?

The version I came across gives the upper and lower bounds in terms of simple binomial coefficients expressed in terms of the degree of the Conway polynomial and the exponent of z. Any other known references for bounds on the Conway (or other skein) polynomial coefficients related to braid classes would also be of interest.

In general in mathematics we work with groups that includ numbers, generally we work with constants.

So for exempel: For any

**x**from**R**we can choose any element from**R**we will find it a constant, the same thing for any complex number, for any**z**from**C**we will find it a constant.My question say: Can we find a group or groups that includ variabels not constants for exemple, we named

**G**a group for any**x**from**G**we will find an**x'**it's not a constant it's another variabel ?and if it exisit can you give me exempels ?

**Thank you !**

Hamilton defined the Ricci flow in 1982 to prove the famous three-dimensional sphere theorem. How singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture?

How did Thurston's geometrization conjecture eventually lead to the proof of the Poincaré conjecture?

The problem is:

Let Z denote the set of all integers.

Consider Z/nZ = Z

_{n}as trivial G-module.Show that there is a isomorphism between the First Homology Group H

_{1}(G, Z_{n}) and the factor G/G'G^{n};Where G' is the commutator (derived) subgroup G'=[G,G] and G

^{n}is the subgroup of G generated by all its n-th Powers.- With this isomorphism proved, we want to conclude that if G is a
*finite p-Group,*then H_{1}(G,**F**_{p}) is a**F**_{p}-module (vector space) with dimension equal the number of least generators of G.

Let Z denote the set of all integers, and let G be a finite cyclic Group.

For every ZG-module A, and n=1,2,3...

Show that:

H

_{n}(G,A) is isomorphic to H^{n+1}(G,A).Please I would like your help on getting an e-copy of the book entitled "The theory of fixed point classes" by Tsai-Han Kiang.

Best regards,

M.S. Abdullahi

There are many different forms of vortexes in the physical world. Here are two common examples:

**Tornado vortex**. These are common atmospheric disturbance in the Midwest in North America (see, e.g., the attached image).

**Seashell vortex**. Seashell sometimes are vortex-shaped (see, e.g., the attached image).

I am hoping that followers of this thread can post or point to other examples of vortexes in the physical world.

Here are several more examples more examples:

**Photon vortex and electromagnetic vortexes**.

This interest in vortexes is related to photon vortexes and electromagnetic vortexes.

**Shape Vortexes [Geometry and Algebraic Topology]**.

From a shape theory perspective, it seems that many physical shapes are non-concentric, nesting cycles. In homology theory, the analogue of a physical vortexes is a collection of non-concentric, nesting 1-cycles. A 1-cycle is a collection of connected edges or surfaces that form a connected path so that one can find a path between any pair of vertices in the 1-cycle.

A compact topological model which relates displacement and rotation isotropically leads to diverging geodesics. A volume of points which individually follow these geodesics becomes expanded during the transport.

The volume expansion red shifts electromagnetic energy transported with the volume. Receiving the enlarged volume after a long transport leads to the impression of a time dilatation.

What is the impact of this volume expansion on particles or matter transported along cosmic distances? Is there a loss of kinetic energy? Does this mean that a very weak static gravitational potential exists, which extracts energy from movements over cosmic distances?

Attachment: Description of the model

The model relates an orientation O=(x/R, y/R, z/R) to a point P=(x,y,z) and also relates a rotation vector V=(dx/R,dy/R,dz/R) to a displacement D=(dx,dy,dz).The axes of the rotation operation is perpendicular to the orientation O and the rotation vector V. R is in the range of about 14 billion light years.

The model has the following properties:

The coupling properties of the three axis rotation operations cause the divergence of the geodetic lines. The rotation operation takes place in an image space. A displacement operation in the native space includes a transformation to the image space, a rotation and a back transformation. The Quaternion formalism allows finding the diverging properties of the geodesics in a straight forward way.

Most of the modern texts use category theory for algebraic topology rather than set theory. What are the pros and cons of both the set theory and the category theory in this formulation.

Moreover, is it necessary to use Category theory or set theory suffices for all the concepts.

Thanks in advance!!!

I am looking for an algorithm that constructs Cech Complex for point clouds.

I am specifically looking for some thing in Mathematica but any resource would be very helpful.

I am looking for speeded up implementations as the traditional approach is very computationally intensive.

Regards and Thanks in advance!!!

Zubair

Some easy to understand references from the literature will be much appreciated. Thanks in advance for all the answer.

Regards,

Zubair

What is quantitative topology and what quality of a topological space does it aim to quantify?

Thank you for all the answers in advance

Kindly suggest good books and other references to the shape theory using a notion of fibrations.

Good reviews are also welcome.

I thank all of you in advance.

**Reference links:**

The

**Inscribed Square Problem**(**ISP**) is difficult because the general**Jordan curve (J)**may take countless forms. So we should seek out its invariant properties and apply our knowledge of analytic geometry (equations of parallel and perpendicular lines and Pythagoras’s Theorem) and Euclidean geometry (similar or almost similar triangles with common diagonals, Law of sines and the Law of cosines, angular rotation about a point and translation from a point to another point).We believe the first step is to inscribe the general

**Jordan curve**(**J**) in a**square, S (please see 'Jordan Curve (2).docx' attached file below for some details of S),**with a diagonal whose endpoints are the two points on**J**that are farthest apart. And we define the**Center Point**(**C**) of_{p}**J**as the intersection of the two diagonals of**S**. And the work of analysis can begin there… And we believe the result should be affirmative!Let

**J’**be the complement object of**J**such that**S = J’ + J**.Let’s fill our

**square**(**S**) with**unit squares**(**s**), and let’s assume our exceptional inscribed_{1}**Jordan curve**(**J**) does not have four points on it that form an inscribed square of any size from**s**to_{1}**s**less than the size of_{n}**S = J’ + J**. We should expect a mix of square sizes, orientations, and shapes in and about**J**without violating our prime assumption.Moreover, we need to formulate boundary conditions between

**J**and**J’**and search for a contradiction to our assumption about**J**. It could be a limiting process or some kind of process of elimination. We are not sure what will work initially. But we have hope in our formulation of the boundary conditions between**J**and**J’**that we shall discover the correct process or procedure and solve our problem affirmatively.**Notes**: We can arrange and rearrange (translate or rotate) our

**unit**

**squares**(

**s**) appropriately without affecting our general exceptional

_{1}**Jordan Curve**(

**J**).

**"Jordan Curve Theorem:**A simple closed curve,

**J**, partitions the plane into exactly two faces, each having

**J**as boundary." -- (

**http://www.ti.inf.ethz.ch/ew/lehre/GT06/lectures/PDF/lecture13.pdf**).

Furthermore, we believe we can have a tentative proof of

**ISP**in less than a week. Wish ourselves luck! :-)I have triangle mesh and calculate normal of triangles then calculate vertex normal and do some calculations on it and want to calculate vertex coordinates from this vertex normal after do calculations.

We call local homeomorphism of R a function f:U--->R with: U open neighborhood of 0 \in R, f(0)=0, f injective continuous and open.

We identify two local homeomorphisms of R if and only if they coincide in a suitable neighborhood of 0.

We denote by LH the set of all equivalence classes of local homeomorphisms of R.

LH becomes a group when endowed with the operation of composition of functions.

I am interested in results describing the structure of the group LH.

Fix a positive integer n.

We call local homeomorphism of R^n a function f:U--->R^n with: U open neighborhood of 0 \in R^n, f(0)=0, f injective continuous and open.

We identify two local homeomorphisms of R^n if and only if they coincide in a suitable neighborhood of 0.

We denote by LO the set of all equivalence classes of local homeomorphisms of R^n.

We call local diffeomorphism of R^n a local homeomorphism of R^n which is smooth and has a smooth inverse.

We identify two local homeomorphisms of R^n if and only if they coincide in a suitable neighborhood of 0.

We denote by LD the set of all equivalence classes of local diffeomorphisms of R^n. LO is a group with the compiosition of functions, and LD is a subgroup of LO.

Is LD a normal subgroup of LO? If yes then what is known about the quotient group LO/LD? References are very welcome.

It is known that homeomorphism between topological spaces implies homotopy but the converse is not true. And if I'm not mistaken, some topological properties such as separablility are not preserved under homotopy. My question is:

Let

*X*be separable, completely metrizable topological space and let*Y*be a topological space. If*X*and*Y*are homotopic, does that mean that*Y*is also separable, and completely metrizable, necessarily? I searched the basic algebraic topology books but they just repeat the same piece of information and I couldn't find explicit answer.The string detained between two points can sustain harmonic motion. At the string midpoint, the potential and kinetic energy are inverse, so that when the string is in the mid-line, the kinetic energy is 1 and the potential energy is 0. When the string is at the boundary the string stops for an instant, so the kinetic energy goes to zero while the potential energy is 1.

A node is defined as a point on a string where there is no movement possible, so that both the kinetic and potential energy are zero. The fixed-point theorem says if pitch is a real function defined on [0, 1], then there must be a fixed point on the interval.

Since the kinetic and potential energy is just the result of basic trig functions sin and cos, it seems clear to me then that wave reflection cannot occur at the string endpoints. The endpoints are fixed points which are in effect

*fulcrums*with a fixed-point position so that length L = 1 is a bound variable. The fulcrum allows the fundamental in the monochord to drive the string on the other side of a node, but the condition for wave reflection does not exist at the node.When a sin wave crosses zero there is no requirement that the point is fixed, so the boundary cannot simply be added to the wave function arbitrarily without changing the nature of Fourier analysis.

If the waves reflect at endpoints, do they also reflect at nodes that are not enpoints? Of course not! But then, in the 1/3 mode, what makes the middle wave where there are 4 nodes and 3 waves? Is the middle wave the reflection of 2 traveling waves between the two non-endpoint nodes?

The boundary condition for traveling wave reflection is 1, 0 , 1 which is clearly a false statement.

Significantly, the frequency and the wave length are bound by the string and not free variables subject to real analysis as continuous variables. Nodes and waves cannot add at the same point.

If physicists really think there are two traveling waves on a string moving in opposite directions that make a standing wave, but no one can see or demonstrate these waves, then maybe they have action-at-a-distance wrong, too. After all, the basic error is assuming frequency is continuous, and then using functions with free variables adjoined with arbitrary integers.

I want a way to avoid the Mersenne Theory notion that the wavelength of the fundamental is 2L, not L. The problem is that assuming 2L means the fundamental is not the lowest mode in the system, which of course is absurd.

So in a previous question I asked if there is a curve-lifting map that sends the interval 2L to L.

The reference I am using here is

*Algebraic Topology*by CRF Maunder, Dover 1996. Page 84, Example 3.3.21"Consider the real projective space

*RP*again. By Proposition (ref) this space is S^{3}^{1}U E^{2}, where*f:(cos x, sin x) = (cos 2x, sin 2x)"*where E is Euclidean space and I replaced the angle theta with x for typing purposes.There follows a figure showing alpha as the generator of the group Z

_{2}.This seems to describe the boundary of the string vibration modes which have the boundary condition (0, 1, 0) for the fundamental n = 1, (0, 1, 0, 1, 0) for the first overtone. Each mode has an n x n square matrix where the determinant is 1 because the matrix values are all zero except 1 where the wave is on the diagonal.

The string is the union of waves and nodes so it is Z

_{2}. Then there is one generator, not 2 (as in the torus).This does not result in the same model of the string as Mersenne.

Bounded sets are defined on general topological vector spaces, topological modules, topological rings and topological groups. But, I could not find a suitable definition of a bounded set in a topological field.

Any nice material regarding the question will be appreciated.

In "A Geometry of Music" by Tymoczko the author describes a lattice that "lives in what mathematicians call 'the interior of a twisted triangular two-torus,' otherwise known as a triangular doughnut."

Clearly the torus is isomorphic to a closed, bound rectangle in a plane. I am assuming that the twisted torus is a torus that has a composition function that forms a line that winds around the torus.

It seems to me then that the twisting torus is really a projection of a sphere in two dimensions. The winding number implies there is a curve lifting function.

My assertion is the torus T is an approximation of the sphere because the inclusion map i: S

^{n}→ (R^{3}– 0) induces i: S^{n}/T→ RP^{3}and the map f:(R^{n+1}– 0)→ S^{n}induces f:RP^{n}→ S^{n}/T.This seems to be what the author implies anyway but does not say: "Pitch-class space is formed out of pitch space when we choose to ignore, or abstract away from, octave information. As a result, many properties of linear space are transferred to circular pitch-class space."

Doesn't this last quote indicate that the pitch line is automatically a circle in RP

^{3}? Then the manifold is a sphere and not a torus, right?My goal here is to show that music spaces are projective but have an affine cover with at single face in R

^{1}.The torus is absurd: it cannot reduce to a point.

The terminology in this question is taken from "Algebraic Topology" by CRF Maunder. See Problem 9 on Page 59.

"Let H be the abstract 1-dimensional simplicial complex with vertices a

^{0}, a^{1}, a^{2}, a^{3}, a^{4}, a^{5}, each pair of vertices being an abstract 1-simplex. Show H has no realization in R^{2}."Note that I have added a

^{5}to the author's text.The reason I ask is this. We have the guitar tuning as the union of 6 string intervals defined by a 6-tuple representing a point such as (0, 5, 5, 5, 4, 5) or EADGBE, so each string is itself a 1-simplex. Note that because each tuning interval is already defined on the system fundamental the secondary string spectrum is already inside the system fundamental spectrum. When the strings are subsets of the system fundamental we say the guitar is "in tune".

If the vertices are the fundamentals of six guitar strings and the each string is defined by an interval between the fundamental state of system and the string, so the interval is always a known as a whole prime number defined on the system fundamental, then does not the solution to the above problem show that guitar music cannot be realized in R

^{2}?I would like to prove in general the structure of music is 3-fold and not 2-fold as Euler thought.

We like to think that music is mathematical but according to Wikipedia there is no axiomatic basis for music. How are musical sets constructed using basic set theoretic tools of union, intersection, and complimentation?

I propose using sequences in tablature music for guitar to study how polyphonic objects are constructed by adding point-wise limits to the continuous function of pitch. Tablature music is a rich algebraic language that has substantial archival, educational, cultural, and economic significance but no mathematic theory of tablature exists.

I have mastered reading and writing tablature. Anyone familiar with my study of multiple parathyroid tumors in the Journal of Theoretic Biology back in 1985 can see I have substantial training in mathematics.

I need a mathematician who knows model theory or algebraic topology to review my work. Please see the attached manuscript if you are interested in making a modern mathematic model for music theory.

The basic problem: Are music sets constructible?

I need to implement various kind of topological chain. Arbitrary dimension and efficient traveling in the data structure is a necessity, space efficiency is less crucial. Construction of complexes from other complexes is appreciated (suspension, product, quotient, etc.). I am mostly interested in the combinatorial aspects, not in the embedding in an euclidean space.

Let (X,d) be a metric space, f: X->X be Borel measurable. Let (K(X),d_H) be a space of compact subsets of X with Hausdorff metric. Is f:K(X)->K(X), where f(A)=\cup_{x\in A}{f(x)} Borel measurable? Is there needed separability of X?

Dear Professor,

I have some questions about the labelled configuration space in homotopy theory. I post my questions at: http://mathoverflow.net/questions/225990/questions-about-configuration-spaces-in-homotopy-theory.

When I use the definition of smash product (i.e. the quotient space of Cartesian product by identifying the wedge product), I always get something wrong, different from the equations. Could you give a help?

Thanks!

Best regards,

Ren Shiquan

Omega groupoids are equivalent to certain crossed complexes in Algebraic Topology. Is there a publication that defines the 3- or n- cube groupoids, not the n-groupoids as a particular case of

n-category ?

As the title suggests, how do i see that for any n, the covering map S^{2n} → RP^{2n} induces 0 in integral homology and cohomology, except in dimension 0?

Specifically, let M be a connected closed orientable n-manifold, n > 1, such that there is a map f : S^n → M of nonzero degree, i.e. for which the image of the generator of H_n(S^n) is equal to a nonzero multiple of the generator of H_n(M). How do I see that π_1(M) is finite?

I want to accelerate my research in topological quantum field theories, I know that in N=1 super Yang-Mills there is conformal twist, but how to get to obtain topological twist in this case is actually my problem, does somebody have any idea?

Please explain digitalbtopology, digital manifold , and digital algebraic topolog.

Condition (7) of Theorem 4.54 in the book ``Axiom of Choice" by Horst Herrlich is the sentence: Each second countable topological space is separable.

Theorem 4.54 of this book says that (7) is equivalent to CC(R) where CC(R) is the axiom of countable choice for the real line R. I am sure that this is not true in a model for ZF because I can prove that (7) is equivalent to CC where CC is the axiom of countable choice. I believe that I can prove that CC(R) is equivalent in ZF to the statement that every second countable T_0-space is separable. I would be grateful if you could tell me whether someone else has ever noticed this mistake of H. Herrlich and published it somewhere. If the answer is YES, please, tell me where and by whom the correction of this Theorem 4.54 was published.

What is the Euler-Poincare principle and how to apply it to (1.4) and get (1.6)?

A polyhedron mesh consists convex polyhedrons. Is this formula right: F+1=E+K, F=number of the interior faces, E=number of the interior edges, K=number of the polyhedrons in the mesh? If yes answer, references?

Many thanks.

how can i prove the following statement? In other words, how can i prove existence of the following norm?

Let K be any totally disconnected local field. Then there is an integer q=p

^{r}, where p is a fixed prime element of K and r is a positive integer, and a norm ∣⋅∣ on K such that for all x∈K we have ∣x∣≥0 and for each x∈K other than 0, we get ∣x∣=q^{k}for some integer k.Let

*C*be closed Jordan curve in R^{ 2}and for arbitrary circle*L*from property that intersection C ∩L contains 3 points follows, that C ∩L contains no less then 4 points. Is it true, that*C*also be a circle?Suppose G is a discrete group (may be infinite). Does there exist a topological space, say W, with finite dimension as a CW complex, such that G is the fundamental group of W?

Suppose X is a simplicial complex and A its subcomplex. So, we can consider relative cocycles, in particular, relative coboundaries in dimension 2, for concreteness. Also, we can consider

*absolute*cocycles in dimension 2, i.e. such that are not obliged to vanish on A.And if we factor the space of these absolute cocycles by the space of those relative coboundaries - what is the generally accepted name for the resulting space?

Assume that a Lie group G acts on a manifold M, effectively. So the Lie algebra g of G is embedded in $\chi^{\infty}(M)$ in a natural way. (effective action: if x.g=x for all x then g=e)

Under what dynamical conditions this embedding is an "Ideal embedding"?

That is : The image of g is an ideal in the Lie algebra of smooth vector fields on M.

By dynamical conditions I mean the dynamical properties arising from the action of G on M.

Let f and g be continuous self-maps on a compact metric space X, and the product system of f and g be mean sensitive. Is f or g mean sensitive?

Let f and g be continuous self-maps on a compact metric space X, and the product system of f and g be snydetically sensitive. Is f or g snydetically sensitive?

What is an example of a simple $C^{*}$ algebra which is acted by $\mathbb{Z}$ but this action can not be extended to an $\mathbb{R}$-action?\Are there such examples for compact operators as nonunital example and reduced C* algebra of F2, as unital example?

The motivation for this question is the following question in dynamical system:

"What diffeomorphisms of a compact manifold can be considered as time-one flow map of a vector field"?

In this classic case the orientation reversing is the first obstruction. Now my question is that "What type of non commutative obstruction can be introduced in the context of NC C* algebras?

In the semialgebraic context, Delfs and Knebusch defined in 1985 their "locally semialgebraic spaces" and later (only Knebusch) "weakly semialgebraic spaces" as some infinite gluings of semialgebraic spaces. But the majority of model theory seems to be carried out in M

^{n}, where (M,...) is a structure (a kind of "affine" situation).Do model theorists need to pass to infinite gluings from time to time?

Note that a k- mean on a topological space $X$ is a continuous function $f:X^{k}\to X$ which is identity on the diagonal and is invariant under permutations.

In the literature is there an appropriate analogy of concept "mean" in the context of commutative Banach algebra? That is a morphism $\alpha:A\to A\otimes A$ which is invariant under flip-operator and its composition with mutiplicative operator would be identity. (with respect to an appropriate norm tensor product). (And the generalization to k-fold tensor product). Please see the attached file for a topological mean.

Let smooth maps f,g:N→S, where N and S are compact manifolds of the same dimension, be such that the generic point of S has an even number of preimages for f and an odd number for g. It can be shown that if S is a

*sphere,*then f(x)=g(x) for some x, for arbitrary N. What other pairs (N,S) have this property? For what manifolds S may N be arbitrary?One would like a good definition of etale cohomology for non-commutative rings A with corresponding Chern characters from Higher Algebraic K-theory (Quillen type) of A. In particular, one would like a non-commutative analogue of Soule's definition of etale cohomology for rings of integers in a number field with Chern characters from the K-theory of such rings. A possibly accessible setting is to define such a theory for maximal orders in semi-simple algebras over number fields and then extend this to arbitrary orders in semi-simple algebras over number fields. The goal in this case is to be able to understand such theories for non-commutative integral group-rings i.e group-rings of finite non-abelian groups over integers in number fields.

REMARKS: Geometrically, Soule's construction translates into etale cohomology of affine and related schemes and so the envisaged construction should translate into etale cohomology for a suitably defined 'non-commutative' scheme.

If polynomial of degree n with zeros z

_{1},z_{2},...z_{n}assumes maximum at w on |z|=1 and a_{k}=1/|w-z_{k}|. Is the following inequality truea

_{1}+a_{2}+....+a_{n}<=n.How to determine and generate all different topologies of full binary trees? By different topology is meant that the tree cannot be obtained by some transformation that is composed by switch of subtrees of particular node

I saw this sentence "The spin structure of the Fe sites is of the GxAyFz type in Bertaut's notation" in the literature(DOI: 10.1038/NMAT2469). Does anybody has the illustration of GxAyFz ??

In GRP, as in any category, isomorphisms are bimorphisms. How can I prove the converse, using the fact of being in GRP?

I found in the book of Kashiwara-Schapira a precise description of this construction but I want to know some illustrative applications maybe a clearer motivation. Thanks a lot!

I am looking for a homomorphism in a topological group that which uses by a single and fixed point. I mean for a fixed element of G, how we can construct a group homomorphism. For a fixed g, I tried the definition x \to gx, by this is not a group homomorphism. How we can do it?

We can add some hypothesis to G, like being locally compact or connected.

Let G be a topological group and U be a fixed neighborhood of identity. For a fixed element g, consider gU=\{gu, u\in U\}. Do we have the following property in an Abelian topological group.

For each neighborhood V of identity there is a positive integer n such that gU is a subset of V^n?

Closed 1-forms are well studied in foliation topology, algebraic geometry, and theory of manifolds. What are their applications in physics?

Let $F=g\circ H$, where $H:\mathbb{C}^n\to \mathbb{C}^n$ is a homeomorphism such that $H(tz)=tH(z)$ for $t>0$ and $g$ is a homogeneous polynomial of degree $k$. Let $L$ be a complex line such that $(g|_H(L))^{−1}(0)=0$. Is it true that $F|_{L\setminus\{0\}}:L\setminus\{0\}\to \mathbb{C}\setminus\{0\}$ has topological degree $r$, such that $|r|≤k$?

For example, this is true when H is $\mathbb{R}$-linear!

For example Ivan Kolar worked on this issue but on functeurs with general fibers.

i.e., Is a convex set in R^n with the fixed point property always compact ?

(A) Let X\neq\emptyset and \tau\in P(x), where P(X) is the power set of X. Then if we have that:

i) X,\emptyset\in\tau

ii) i\in I, A_{i}\in\tau \Rightarrow \cup_{i\in I}{A_{i}} \in \Tau

iii) j\in J, J finite, A_{j}\in \tau \Rightarrow \cap_{j\in J}{A_{j}}\in \tau

We say that the collection \tau is a Topology in X and that (X,\tau) is a topological space.

(B) The minimum requirement for every algebraic structure is closure under the defined binary operation.

What is the essential difference between Algebra & Topology?

Both of them are guided by the concept of closure.

So, why have we defined two branches that are almost of the same philosophy?

In thermodynamics (like other areas of physics), work is defined as a function that assigns a real number to each thermodynamic process. Some of the latter (called quasi static) are modeled as continuous curves on certain smooth manifold. I want to know what conditions must work (as a function) to satisfy in order to be represented by the integral of certain 1-form along a quasi static process.

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to this very characteristic, Clifford or geometric algebra is believed to be a reinterpretation of differential geometry as suggested mainly by Hestenes and Doran.

But as far as I know, many manifold-related theorems depend on the topology of the manifold such as connectedness, compactness, boundaryless or not. I want to know how Clifford algebra behaves in different topologies?

During the review of Eilenberg-Montgomery fixed point theorem, I have faced a term acyclic space, map which needs a background in algebraic topology. Can any one simply tell what are those terms and if a contractible set valued map is an acyclic map?

I am interested in manifolds with finitely generated homotopy groups. As I remember I saw somewhere that the homotopy groups of a compact manifold are finitely generated, but I am not sure if this is true.

Is it too hard to prove that for every compact smooth manifold with boundary M, there exists an imbedding of the double of M, 2M, in euclidean space E=R^{2n + 2} (n = dim M), such that under this imbedding, the intersection of 2M with the hyperplane \partial E_+ = {x \in E | x_{2n + 2} = 0 } is precisely the boundary of M ?

I need to check the proof for my work. Any references will be greatly appreciated.