quickconverts.org
☰

Homology Of Wedge Sum

Image related to homology-of-wedge-sum

The Homology of Wedge Sums: A Deep Dive



The study of topological spaces and their properties often hinges on understanding how different spaces relate to each other. One crucial construction for combining spaces is the wedge sum, denoted by ∨. This article delves into the homology of wedge sums, a critical concept in algebraic topology that allows us to compute the homology groups of a complex space by leveraging the known homology groups of its simpler constituent parts. We will explore the fundamental theorem governing this relationship and illustrate its application through concrete examples.

1. Understanding Wedge Sums



Before diving into homology, let's clarify the definition of a wedge sum. Given two topological spaces X and Y, and points x₀ ∈ X and y₀ ∈ Y, the wedge sum X ∨ Y is formed by taking the disjoint union of X and Y, and then identifying the points x₀ and y₀. Intuitively, we are "gluing" X and Y together at these base points. This process can be extended to any finite number of spaces.

For instance, consider two circles, S¹ and S¹. If we choose a point on each circle (say, (1,0) in both cases using the standard representation), the wedge sum S¹ ∨ S¹ is a figure-eight space—two circles joined at a single point. Similarly, the wedge sum of n circles, S¹ ∨ S¹ ∨ ... ∨ S¹, is a space with n loops joined at a single point.

2. The Mayer-Vietoris Sequence and its Application to Wedge Sums



The key to computing the homology of a wedge sum lies in the Mayer-Vietoris sequence. This powerful tool relates the homology groups of a space to the homology groups of its open subsets and their intersection. When applied to a wedge sum, it provides a remarkably efficient way to calculate the homology groups.

Let's consider X ∨ Y, where X and Y are path-connected spaces. We can choose open sets U and V such that U contains X (excluding the base point) and V contains Y (excluding the base point), with U ∩ V being empty except at the identified base point. The Mayer-Vietoris sequence then yields:

... → Hₙ(U ∩ V) → Hₙ(U) ⊕ Hₙ(V) → Hₙ(X ∨ Y) → Hₙ₋₁(U ∩ V) → ...

Since U ∩ V is contractible to a point (homotopically equivalent to a single point), its homology groups Hₙ(U ∩ V) are trivial for n > 0 and H₀(U ∩ V) ≈ ℤ. This simplification significantly reduces the complexity of the sequence.

3. Calculating Homology Groups of Wedge Sums



Exploiting the simplified Mayer-Vietoris sequence, we can determine the homology groups of X ∨ Y. For n > 0:

Hₙ(X ∨ Y) ≅ Hₙ(X) ⊕ Hₙ(Y)

This means that for dimensions greater than 0, the homology groups of the wedge sum are simply the direct sums of the homology groups of the individual spaces.

The 0th homology group, H₀(X ∨ Y), requires a slightly different treatment. Since X and Y are path-connected, H₀(X) ≅ ℤ and H₀(Y) ≅ ℤ. However, the gluing process identifies the base points, resulting in:

H₀(X ∨ Y) ≅ ℤ

This reflects the single path-connected component of the wedge sum.


4. Example: Homology of the Figure-Eight



Let's calculate the homology groups of the figure-eight space, S¹ ∨ S¹. We know that the homology groups of a circle S¹ are:

H₀(S¹) ≅ ℤ
H₁(S¹) ≅ ℤ
Hₙ(S¹) ≅ 0 for n > 1

Using the formula derived above:

H₀(S¹ ∨ S¹) ≅ ℤ
H₁(S¹ ∨ S¹) ≅ H₁(S¹) ⊕ H₁(S¹) ≅ ℤ ⊕ ℤ ≅ ℤ²
Hₙ(S¹ ∨ S¹) ≅ 0 for n > 1

Therefore, the figure-eight has one generator for its 0th homology group and two generators for its first homology group, reflecting the two independent loops.

5. Conclusion



The homology of wedge sums, elegantly described by the Mayer-Vietoris sequence, provides a powerful tool for computing the homology groups of complex spaces by breaking them down into simpler components. Understanding this relationship is crucial for advanced studies in algebraic topology and its applications in various fields. The key takeaway is that the homology groups, for dimensions greater than zero, of the wedge sum of two path-connected spaces are the direct sums of their respective homology groups.


FAQs:



1. What if the spaces aren't path-connected? The formula for Hₙ(X ∨ Y) (n>0) still holds, but you need to consider the number of path components in each space. H₀ becomes more complex, reflecting the total number of path components.

2. Can we use this for infinite wedge sums? The Mayer-Vietoris sequence is primarily designed for finite sums. Infinite wedge sums require more sophisticated techniques.

3. What is the geometric intuition behind the direct sum in the homology groups? Each generator in the direct sum corresponds to a distinct homology class within each individual space. The direct sum reflects the independence of these classes in the wedge sum.

4. Are there other methods for computing the homology of wedge sums? While the Mayer-Vietoris sequence is a powerful approach, other techniques, like cellular homology, can also be effective depending on the specific spaces involved.

5. How does this relate to other topological invariants? Homology groups are just one type of topological invariant. The results obtained from the homology of wedge sums influence other invariants, providing a richer understanding of the space's topological properties.

Links:

Converter Tool

Conversion Result:

=

Note: Conversion is based on the latest values and formulas.

Formatted Text:

85 inch to feet
56 liters to gallons
how much is five pounds of gold worth
350g to lbs
66 lbs to kg
106 ounces to pounds
2000 mins in hours
24 ft to meters
7 7 cm
900m to miles
240 c to fahrenheit
140 grams to oz
135ml to oz
how much is 200 milliliters
5 oz to tbsp

Search Results:

Solution e - Stanford University On the other hand, the homology of the wedge sum of gtori is: H i(∨ g(S1 ×S1)) ≈ Z i= 0 ⊕ 2gZ i= 1 ⊕ gZ i= 2 0 otherwise H 1 is generated by glongitudinal classes and glatitudinal classes, a 1,...,a g and b 1,...,b g respectively. H 2 is generated by the …

Equivalence of Simplicial and Singular Homology Wedge 2 - ETH Z of LEMMA 1 For a wedge sum YXa the inclusions is Xx byXa induce an isomorphism in Tp Xx Iply xx provided that the wedge sum is formed at basepoints Xat Xx such that the pairs Xa Xa are good Recall The wedge sum is a one point union of a family of topologicalspaces YXuxa 14xxx y Exay quotientofthedisjoint union oftheXa's by the equivalence relation that identifies all xp …

On Homotopy Types of Vietoris{Rips Complexes of Metric Gluings with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris{ Rips complexes. We also provide generalizations for when two metric spaces are glued together along a common isometric subset.

The Capacity of Wedge Sum of Spheres of Different Dimensions this paper, during computing the capacity of wedge sum of finitely many spheres of different dimensions and the complex projective plane, we give a negative answer to a question of Borsuk whether the capacity of a compactum determined by its homology properties. Keywords: Homotopy domination, Homotopy type, Moore space, Polyhedron,

On the Hurewicz theorem for wedge sum of spheres - SciELO In this work we use methods of Homological Algebra to provide an alter-native proof of the celebrated Hurewicz theorem in the case that the topo-logical space is a CW-complex. Formally speaking, we ̄rst show that if X0 μ X1 μ ¢ ¢ ¢ Xn¡1 μ Xn = X is the CW decomposition of X, then the Hurewicz homomorphism ¦n+1 (Xn+1; Xn) ¡!

Homework 6: Excision for homotopy groups - Harvard University In this homework, using the homotopy excision theorem, you will define the stable homotopy groups of spheres, and compute homotopy groups for quotients. 1. Wedge sum v. Disjoint union Recall that given two pointed spaces X and Y,theirwedge sum is the topological space X ∨Y := (X Y)/x. 0∼ y. given by gluing x. 0to y. 0.

HOMOLOGY OF THE UNIVERSAL COVERING OF A CO-H … the homotopy type of a one-point-sum of a wedge sum of circles and a simply connected space? It is known by S. Eilenberg and T. Ganea [5] that if a co-H-space X is paracom-pact and normal, Bˇ 1(X) has the homotopy type of a wedge sum of circles, sayB. Thus we have two mappings i: B ! X and j: X ! B,whereiinduces an isomor-

Lecture 19: Hurewicz Theorem - GitHub Pages The homology of Sn and Hurewicz Theorem implies that ˇk(Sn) = {0 if k < n Z if k = n: In particular, the degree of a map f : Sn → Sn can be described by either homotopy or homology.....

Homework 4: Mayer-Vietoris Sequence and CW complexes 2. Homology of wedges Let X and Y be spaces, and choose a point x 0 ∈ X, y 0 ∈ Y in each space. The wedge sum of the two spaces is defined to be the space X ∨Y := X x0∼y0 Y ∼= (X Y)/x 0 ∼ y 0. This is the space obtained by gluing x 0 to y 0. For instance, if X and Y are both homeomorphic to S1, their wedge sum is a figure ...

EULER CHARACTERISTICS OF DIGITAL WEDGE SUMS AND … In pure mathematics, homology groups are used for introducing the notion of Euler characteristic which naturally has the product property, bration property, homotopy invariance and so forth.

HOMEWORK #6 - University of Wisconsin–Madison 4. For a wedge sum W X , the inclusions i : X ,! W X induce an isomorphism i : H~ n(X ) !H~ n(_ X ); provided that the wedge sum is formed at basepoints x 2X such that the pairs (X ;x ) are good. 5. Show that S 1 S and S1 _S _S2 have isomorphic homology groups in all dimensions. Are these spaces homeomorphic? 6. Show that the quotient map S 1 S ...

COMPLEXITY OF SIMPLICIAL HOMOLOGY AND … For a co-chordal graph Gthe complex Cl(G) is homotopy equivalent to a wedge of spheres (Corollary 2.2), therefore He (Cl(G)) = 0 is equivalent to Cl(G) being contractible.

Spectra and Homology Theories - uni-bonn.de As we saw for cohomology in the last talk, it is natural to ask if every homology theory comes from a spectrum. This happens to be true if we replace the wedge sum axiom by the Direct limit axiom, that states that for any CW-complex X, there is an isomorphism h i(X) ∼=lim−→h i(X j), where {X j

Wedge Sum, Merge and Inconsistency - Springer In this paper, we directly construct consistent logical theories which describe wedge sums, and inconsistent theories which extend them. Then we utilize the technique Merge to show how to obtain inconsistent theories of the wedge sum in a different way.

Homology and Cohomology - IISER Pune Wedge sum Given spaces X and Y with chosen points x 0 2X and y 0 2Y, then the wedge sum X_Y is the quotient of the disjoint union X ‘ Y obtained by identifying x 0 and y 0 to a single point. Smash product Inside a product space X Ythere are copies of Xand Y, namely Xf y 0g and fx 0g Y for points x 0 2Xy 0 2Y. These two copies intersect only ...

Homology, Brouwer’s Fixed-Point Theorem, and Invariance of Doma Homology is a powerful tool from algebraic topology that is useful not only for characterizing topological spaces, but also for proving some important theorems that themselves have lots of applications.

arXiv:1712.06224v5 [math.MG] 12 Aug 2019 We study Vietoris{Rips complexes of metric wedge sums and metric gluings. We show that the Vietoris{Rips complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris{Rips complexes. We also provide generalizations for when two metric spaces are glued together along a common isometric subset. As ...

Whitehead, CW complexes, homology, cohomology - University … the n-th homology of this complex. For n= 1 we de ne C 1(X) = Z and the map d 0 takes the sum of the coe cients. All other negative n’s are 0. We’re going to skip verifying axiom (2). For (3), note that for n6= 0 the generators satisfy the conditions given, and for a cell in X the characteristic map only touches cells in X . Thus (3) works.

On the Hurewicz theorem for wedge sum of spheres - UNAL In this work we use methods of Homological Algebra to provide an alter native proof of the celebrated Hurewicz theorem in the case that the topo logical space is a CW-complex.