Use this tag if your question involves some type of homology, including (but not limited to) simplicial homology, singular homology, or group homology.

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Why Must the Degree of this Map be 0? [on hold]

Let $f:S^3 \times S^1\times S^1\rightarrow S^3 \times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$.
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1answer
36 views

Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
3
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2answers
57 views

How do we get a simplicial homology functor?

The $n$-th simplicial homology group $H_n(A)$ of an abstract simplicial complex $A$ depends on the choice of an orientation for $A$ (but for different orientations, the homology groups are ...
2
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0answers
26 views

Homology of connected sum of $\mathbb{R}P^4$ and $\mathbb{C}P^2$.

I'm studying for my topology qualifying exam, and I'm having trouble computing the homology of the connected sum of $\mathbb{R}P^4$ and $\mathbb{C}P^2$. I tried using a relative long exact sequence ...
4
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0answers
32 views

Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
3
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0answers
44 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
3
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1answer
38 views

Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
2
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2answers
50 views

Find a CW complex with prescribed homology groups

A past qual question asks to construct a CW-complex $X$ with $H_0(X) = \mathbb{Z}$, $H_5(X) = \mathbb{Z} \oplus \mathbb{Z}_6$, and $H_n(X) = 0$ for $n\not= 0, 5$. One can build a CW-complex $Y$ by ...
2
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1answer
29 views

Homology groups of a simplicial complex

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all ...
6
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2answers
106 views

Is this Space Homotopy Equivalent to $S^2$

Let $X$ be the space $S^1$ with two $2$-cells attached via maps of relatively prime degrees. This space is simply connected and has the homology of $S^2$, but is it homotopy equivalent to $S^2$?
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2answers
50 views

$H_n(\mathbb{R}P^4 \times S^1)$

I have been trying to compute the homology of $\mathbb{R}P^4 \times S^1$ by using cellular homology. Nevertheless, I cannot see what the attaching maps are.
2
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2answers
69 views

Homology groups equal when $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) \rightarrow 0 \rightarrow \cdots$

I'm reading a set of notes but I don't understand the following concept. We have a long exact sequence $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) ...
0
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1answer
21 views

Degree of an induced map on $\mathbb{CP}^n$

Let $r :\mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1} $ be the map $r(z_0, z_1,\ldots, z_n)=(-z_0, z_1,\ldots, z_n)$. $r$ induces a map $\bar r : \mathbb{CP}^n \rightarrow \mathbb{CP}^n $. What is the ...
3
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2answers
42 views

How to show that $f_* (\sigma)=\sigma$ where $f$ is mapping between projective spaces $\mathbb{R}\text{P}^3$

Suppose that $f:\mathbb{R}\text{P}^3 \to \mathbb{R}\text{P}^3$ is continuous mapping without fix points and let $\sigma$ be (some) generator of group $H_3(\mathbb{R}\text{P}^3)$. Prove that ...
2
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0answers
23 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
6
votes
2answers
132 views

How do Homology Groups work

How do homology groups work? Looking at the wikipedia article, it lists, for example, $H_k(S^1) = \mathbb Z$ for $k = 0,1$ and ${0}$ otherwise. It also says that $H_k(X)$ is the k-dimensional holes in ...
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1answer
23 views

What is the meaning of “Homotopy of Little disc Operads”

I want to understand what means the homotopy of the little discs operad. I'm starting to research in this area and I have some questions. 1) I don't understand why little discs operad is a ...
2
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1answer
50 views

Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
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1answer
61 views

Triviality of $H_3(G,\mathbb{Z})$

We know that the triviality of the Schur multiplier means that projective representations can be lifted to ordinary ones. The Schur multiplier is also a measure of the failure of how the commutator ...
0
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1answer
37 views

Is the composition of a homeomorphism with itself orientation-preserving?

Just a short question about the degree of a homeomorphism. So, I understand that in the continuous setting we define the degree of a map $\ f: M \rightarrow M$ on a connected orientable manifold as ...
0
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1answer
23 views

0-cycles with null augmentation are boundaries in convex spaces?

Well as the title says I would like to know if given a convex space $ X $, a $0$-cycle (equivalently any $0$-chain, right?), such that its augmentation is null is a boundary? All this in singular ...
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0answers
21 views

Enunciation of $\partial$ as the boundary map

How is $\partial$ typically pronounced when it is used as the boundary map in homology theory? The answer to this question provides some good information on the enunciation of $\partial$, but more ...
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0answers
88 views

Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.

So I have shown that the natural projection $\pi: \mathbb{CP^n}\rightarrow \mathbb{CP^n/CP^k}$ induces a monomorphism $\pi^*:H^*(\mathbb{CP^n/CP^k},\mathbb Z)\rightarrow H^*(\mathbb{CP^n},\mathbb Z) ...
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1answer
39 views

Basic Simplicial Homology Question

Let K be a 4-dimensional simplicial complex which has 8 0-simplices, 12 1-simplices, 9 2-simplices, 10 3-simplices and 6 4-simplices. Suppose that $H_0(K)= \mathbb{Z}, H_1(K)= ...
6
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2answers
149 views

Can we simultaneously realize arbitrary homotopy groups and arbitrary homology groups?

Let's keep our groups finitely presented for the time being. All spaces in this post are path connected. Background: By a standard construction (e.g., on p. 365 of Hatcher), there exists a $K(\pi, ...
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1answer
49 views

Triangulate the triangle with edges identified

Consider the closed triangle (simplex) $ \Delta = [(0,1), (0,0) , (1,0)] \subset \mathbb{R}^2 $, ie, $\Delta$ is the convex hull of the points $v_1 = (0,1)$, $v_0 = (0,0)$ and $v_2 = (1,0)$ in ...
0
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1answer
17 views

not complete solution in computation of singular cohomology

I know how to compute singular cohomology of single point. Then I want to compute cohomology of $\mathbb{Z}$. Consider $C_{k}(\mathbb{Z})=\oplus_{i\in Z} C_{k}(point)$. $$\text{Hom}(A\oplus B, G)= ...
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0answers
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Comparison between Eilenberg-Steenrod excision and Brown representability excisive

One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that $H^n(X,A)\cong H^n(X\setminus U,A\setminus U)$, for good subspaces such as when $\overset{\circ}U\subseteq ...
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1answer
83 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
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56 views

If two maps induce the same homomorphism, then they are homotopic

If two chain maps $f,g:\mathcal{X} \rightarrow \mathcal{Y}$, where $\mathcal{X},\mathcal{Y}$ are chain complexes with free modules $X_p$ and $Y_p$ over a PID, $R$, induce the same homomorphism in the ...
3
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2answers
36 views

Generator of singular homology of n-sphere

I am learning singular homology theory right now. The homology of n-sphere is computed by Mayer-Vietoris argument. Intuitively, for example the class represented by a loop is the generator of ...
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1answer
26 views

Obtaining Chain Complex from a Cochain Complex

In this question: Constructing a cochain complex out of a chain complex , palio asked how to construct a co-chain complex when given a chain complex as well as how to go in the opposite direction, ...
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0answers
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Equivariant version of Morse theory [closed]

Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology? Any reference/idea would be greatly appreciated. Crossposted on MathOverflow.
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1answer
79 views

what is th homology group of $\mathbb{Q}$?

what is the 0'th homology group of $\mathbb{Q}$ I mean $ H_{0}(\mathbb{Q})$?as the 0'th homology group is counting the path component of the space so it should be infinite direct sum of copies of ...
4
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1answer
35 views

an arc is not a retraction of the Klein bottle

I want to use homology to solve the following problem: Prove that the circle represented by the blue arc in the picture is not a retract of the Klein bottle. (See the attached picture of the Klein ...
4
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1answer
75 views

Homology and Fundamental group of $\mathbb{R}^4\setminus S^1$

I was attempting to help someone with this problem and realized I could not solve it myself. Let $S^1=\{(x,y,0,0):x^2+y^2=1\}$ be the unit circle in $\mathbb{R}^4$ and consider $M=\mathbb{R}^4 ...
2
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1answer
62 views

Can you compute relative homology using simplicial chain complex?

So I had my Algebraic Topology exam yesterday and one of the questions asked to compute the homology groups $H_*(M)$, $H_*(\partial M)$ and $H_*(M, \partial M)$ where $M$ is the Möbius strip and ...
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58 views

Cofiber Sequences in Reduced Homology Theory

While going through axiomatic treatments of homology theories I got a bit stuck on this problem. Consider given a reduced homology theory, i.e. functors $(\tilde{E}_q:Top_* \to Ab)_{q \in ...
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39 views

what is the homology groups some quotient space of torus

what is the homology group for The quotient space of $S^1 \times S^1$ obtained by identifying points in the circle $S1 \times\{x_0\} $ that differ by $\frac{2 \pi}{m}$ rotation and identifying points ...
5
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1answer
58 views

Computing Homology using Mayer-Vietoris

(This is exercise 2.2.28 from Hatcher) Consider the space obtained from a torus $T^2$, by attaching a Mobius band $M$ via a homeomorphism from the boundary circle of the Mobius band to the circle ...
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35 views

Homology Theory based on “linear symmetric differences”

Homology Theory is based on the idea that if you have a sequence $A \rightarrow B \rightarrow C$ of homomorphisms $f \colon A \rightarrow B$ and $g \colon B \rightarrow C$ such that their composition ...
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what is $H_i(S^n-X)$ when $X$ is a subspace of $S^n$ homeomorphic to $S^k \vee S^l$ or to $S^k \sqcup S^l$.

Compute $H_i(S^n-X)$ when $X$ is a subspace of $S^n$ homeomorphic to $S^k \vee S^l$ or to $S^k \sqcup S^l$. it seems that I should use this:when $h:S^k \rightarrow S^n$ is an embedding then we have ...
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0answers
28 views

Degree One Map induces Surjections on Homology

Is the following statement true: If $f:M\to N$ is a degree one map of compact closed manifolds, then $f$ induces surjections $f^*:H_q(M)\to H_q(N)$. I found this claimed on ...
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0answers
54 views

Degree of a map from $ T^2 \rightarrow T^2$

Given a map $f:T^2 \rightarrow T^2$, the degree of $f$ is given by the induced homomorphism $f^*:H_2(T^2) \rightarrow H_2(T^2)$. If I know the induced homomorphism $f^{**}:H_1(T^2) \rightarrow ...
2
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2answers
41 views

What is the point of triangulating topological spaces?

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra? ...
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1answer
26 views

Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
2
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1answer
36 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...
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11 views

some intuition about the degree of a map

Consider a map $$ f: \Sigma \to X/\sigma,$$ where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution, $\sigma:X\to X$ is an antiholomorphic involution of some ...
2
votes
1answer
142 views

Why do these geometric assumptions imply these statements about relative homology?

I'm reading the paper Coverage in sensor networks via persistent homology. As in the paper, let $\mathcal{D}$ be a bounded domain in $\mathbf{R}^d$. We make the following assumptions: A5 The ...
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46 views

In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...