Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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6
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2answers
136 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 ...
2
votes
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 ...
-2
votes
1answer
76 views

Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
3
votes
1answer
51 views

For compact surface $M$ and loop $f$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such that $f \notin \ker(\phi)$

Why is this sentence true? For every not nullhomologous loop $f$ without selfintersections on orientable compact surface $M$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such ...
6
votes
1answer
59 views

Haar Measure for Algebraic Number Theory: What Should I Know?

I recently taught myself some algebraic number theory and am preparing to take a course in class field theory this fall. I understand the notion of a Haar measure on a locally compact topological ...
0
votes
1answer
36 views

Empty set in a simplicial complex

Should the empty set be considered a simplex in a simplicial complex? Which justifications exist for the answer? I guess it is somewhat comparable to $1$ not being a prime number.
3
votes
1answer
25 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
3
votes
1answer
40 views

Does a pseudo-Anosov homeomorphism of a punctured surface possess infinitely many periodic points?

In A Primer on Mapping Class Groups by Farb and Margalit theorem 14.19 implies that every pseudo-Anosov homeomorphism $f:S \rightarrow S$ on a compact surface $S$ possesses infinitely many periodic ...
1
vote
1answer
21 views

Topological Tensor Product is a Topological Ring Independent of the Choice of Basis

Let $A, B$ be commutative rings containing a field $k$, with $B$ a finite dimensional $k$-module, $w_1, ... , w_N$ a basis. If $w_iw_j = \sum\limits_{n=1}^N c_{ijn}w_n$, then we can define $C$ to be ...
1
vote
0answers
32 views

Intersection of Two Homeomorphic Copies of $\mathbb{R}^2$

There is an exercise in Bredon's "Topology and Geometry" in the Van Kampen section that supposes that $\mathbb{R}P^2=U_1\cup\cdots\cup U_n$ where each $U_i$ is open and homeomorphic to $\mathbb{R}^2$. ...
3
votes
2answers
75 views

Topological/homotopical classification for 1-dim CW-complexes?

It's a common exercise to classify a collection of 1-dim objects, say the figures of 0-9, or A-Z, up to homeomorphism or homotopy equivalence. I suddenly raise a question in general: Is there any ...
2
votes
0answers
26 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
4
votes
1answer
70 views

Fundamental Group of a Hexagon with Edge Identifications

What's the easiest way to compute this thing's fundamental group? I've been playing with it for a little while, and I'm getting $\mathbb{Z}+\mathbb{Z}$. After making the ID's I think the 1-cells ...
0
votes
1answer
55 views

Explain a sentence in Hatcher's “Vector Bundles and K-Theory”

I was looking for a proof of this statement For any space $X$ the set $[X,SO(n)]$ has a natural group structure coming from the group structure in SO(n). Namely, the product of two maps $f,g:X ...
2
votes
1answer
33 views

Is the boundary of a triangulated manifold a subtriangulation?

Let $K$ be a finite simplicial complex with the underlying topological space $|K|=\cup K$. If $|K|$ is also a topological manifold with boundary, does it hold that some subcomplex of $K$ triangulates ...
3
votes
1answer
39 views

Effect of the degree of a map $S^n\to M$ on lower homology groups

I'm looking through some old algebraic topology problems to study for an exam, and I came across the following: Let $M$ be a compact, orientable $n$-manifold, and let $f:S^n\to M$ be a map of degree ...
0
votes
1answer
58 views

Show That $\dim H_m(\partial M;\mathbb{R})$ is Even

A student asked me this. Suppose that $M$ is a compact, orientable $n$-manifold with boundary. It is a fact that for each $k$ with $0\leq k\leq n$ the vector spaces $H_k(M;\mathbb{R})$ and ...
0
votes
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 ...
1
vote
1answer
47 views

Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let ...
4
votes
0answers
40 views

Holomorphically simply connected implies simply connected

In my book on complex analysis a "Holomorphically simply connected" set is defined as a set where for any holomorphic function $f $ and any closed path $\gamma _1 $ we have that $\int_{\gamma _1 } ...
4
votes
1answer
62 views

Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
1
vote
0answers
27 views

Showing this Null homotopic composite factors through a Null homotopic map

I was having some trouble with this concept which makes sense to me intuitively but the understanding of which is not yet fully clear. Suppose $CS^n$ is the unreduced cone on the n-sphere $S^n$. By ...
3
votes
1answer
63 views

Signature of $S^2 \times D^2$

Every closed connected oriented $4$-manifold has a signature, defined via a cohomological intersection form. In Turaev's book Quantum Invariants of Knots and 3-Manifolds the definition of a certain ...
4
votes
3answers
122 views

Do freely homotopic maps induce the same homomorphism on fundamental groups?

Let $f,g\colon X\to Y$ be two continuous maps that are freely homotopic, such that there is some $x_0\in X$ with $f(x_0)=g(x_0)$. Is it true that the induced homomorphisms $f_*,g_*\colon ...
6
votes
1answer
114 views

Singular $\simeq$ Cellular homology?

Given an arbitrary CW-complex, are the singular chain complex $S_\ast(X)$ and cellular chain complex $C_\ast(X)$ homotopy equivalent or just quasi-isomorphic (some chain map induces isomorphisms on ...
0
votes
0answers
15 views

Interesting observation WRT 1,2,3-dimensional convex polytopes and higher dimensional ones as counterpart

When I experimenting with qhull utility and Quickhull Algorithm, I found that in $\mathbb{R}^d, d \in \{1,2,3\}$ space the number $F$ of convex hull's $(d - ...
2
votes
1answer
46 views

Universal cover of the pinched sphere?

Consider the sphere $S^2$ and identify its north and south poles to get a "pinched" sphere. What is the universal cover of this space?
3
votes
1answer
81 views

Proof that the Euler characteristic is additive

I'm reading through a set of notes which assumes that the Euler characteristic is additive, but doesn't give a proof, so I would like to understand why this is. Let $A_n$ be a finitely generated ...
1
vote
0answers
47 views

Show That a Lift Always Exists

I've been considering this problem: Suppose that $X$ is a topological space and that $H_1(X)$ is a finite group of odd order. Show that if $p:\tilde{Y}\rightarrow Y$ is a covering space of index ...
2
votes
2answers
93 views

Covering through group action and corresponding deck transformations

I'm having a bit of trouble with the following exercise: Let $G$ be a group acting properly discontinuous and continuous on a topological space $E$. Then $p:E\to G\backslash E$ is a covering. Let ...
1
vote
0answers
30 views

Proof the Associative Property of H-group

I would like to prove the following proposition: Let $X$ and $Y$ be based topological spaces and let $[X,Y]$ be the set of homotopy classes of based maps $X\to Y$. If for every $X$, $[X,Y]$ is a ...
3
votes
1answer
53 views

Deletion of disc

Let $M$ be an closed manifold, $D$ a disc inside it. As far as I understand, in orientable case the only difference between the homology (over a given field) of $M$ and $M \setminus D$ is one more ...
1
vote
1answer
40 views

Question from Munkres algebraic topology section 58: retractions

This is question 7 on page 366 from section 58 of Munkres Topology: Let $A$ be a subspace of $X$, let $j: A \to X$ the inclusion map, $f:X \to A$ continuous. Suppose there is a homotopy $H$ between ...
2
votes
1answer
72 views

Homotopy groups of $n$-torus with a point removed.

Is there a simple way how to compute and present homotopy groups of $T^n=S^1\times \ldots\times S^1$ with a point (or several points) removed?
4
votes
2answers
76 views

Funky Fundamental Group Question

Let $D$ be a closed disk (w/ boundary $C$) and let $D_a$, $D_b$ be two disjoint closed disks in the interior of $D$ (w/ boundaries $C_a$ and $C_b$, resp.) . Now remove the interiors of $D_a$ and ...
0
votes
2answers
30 views

Cardinality of fibers of covering map and the fundamental group of E

Can anyone provide a source or a proof of the following fact: If $p : E \to B$ is a covering map and $E$ is simply connected, then each fiber of $p$ has the cardinality of that of $\pi_1(B)$?
2
votes
1answer
99 views

Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
0
votes
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 ...
2
votes
1answer
45 views

Triangulations, PL-triangulations and related conecpts

I'm confused about various definitions of triangulations and piece-wise linearity. I read, for example, on wikipedia "..the question of whether all topological manifolds have triangulations is an ...
2
votes
2answers
62 views

A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
4
votes
0answers
52 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
3
votes
0answers
49 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
2
votes
1answer
86 views

$\mathbb{R}^2$ is a Retract of $\mathbb{R}^5$

Looking through some old qualifying exams, I noticed the question: Let $X\subseteq \mathbb{R}^5$ be homeomorphic to $\mathbb{R}^2$. Prove that $X$ is a retract of $\mathbb{R}^5$. I'm not even sure ...
6
votes
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) ...
3
votes
0answers
41 views

When is a graded ring the cohomology ring of a CW-complex?

Let $A^*$ be a graded-commutative ring with $A^n = 0$ for sufficiently large $n$ and each $A^n$ finitely generated. When does there exist a finite CW-complex $X$ with $H^*(X) \cong A$ as graded rings? ...
1
vote
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)= ...
0
votes
0answers
25 views

The relative homology of a pair $H_n(S_n, A)$.

At page 136 of Hatcher's book he says: By excision, the central term $H_n(S^n, S^n - f^{-1}(y))$ in the preceding diagram is the direct sum of the groups $H_n(U_i, U_i-x_i) \approx \mathbb{Z}$. ...
0
votes
0answers
25 views

Group of 3-chains of tetrahedron

The textbook I use is the book "A First Course in Abstract Algebra", John B. Fraleigh, 7 edition, Section 41. The group $C_n(X)$ of $n$-chains of $X$ is defined by the free abelian group generated by ...
0
votes
1answer
39 views

a topological space is the union of its irreducible components

if we define irreducible component of a topological space $X$ as the maximal closed irreducible subset of $X$,prove that we can write $X$ as the union of its irreducible components. how to approach ...
2
votes
0answers
49 views

Proof of Hurewicz' thm. in Hatcher

I'm having trouble understanding the last part of Hatcher's proof of Hurewicz' theorem. (It's on page 367, thm. 4.32). We want to show, that a cellular boundary map: $d:H_{n+1}(X^{n+1},X^n) ...