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

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0
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1answer
35 views

Composition of simplicial approximation is again simplicial approximation

If $s:|K^m|\to |L|$ is a simplicial approximation to $f:|K^m|\to |L|$ and $t:|L^n|\to |M|$ is a simplicial approximation to $g:|L^n|\to |M|$. Is $t\circ s: |K^{m+n}|\to |M|$ necessarily a simplicial ...
5
votes
2answers
86 views

Explicit expression for homeomorphism and homotopy equivalence

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy ...
4
votes
1answer
79 views

Topological degree of a complex valued map defined over a circle

Given a continuous map $f \colon S^n \to S^n$, it induces a map $f_{*} \colon \tilde{H}_n(S^n) \to \tilde{H}_n(S^n)$ of the form $f_{*}(z)=k*z$, where $k$ is an integer. Define the degree of $f$ as ...
0
votes
0answers
31 views

Homeomorphic homogeneous 3-simplicial complexes

I have a simple question on homogeneous (i.e. made only of tetrahedron) 3-simplicial complexes. Suppose we have an homogeneous 3-simplicial complexes. Suppose we choose any couple of simplices having ...
0
votes
1answer
40 views

Relative cohomology versus cohomology.

Let $S$ be a closed oriented surface, $X$ a finite set of points on $S$. Is it true that $$ H^1(S \setminus X, \mathbb{C}) \simeq H^1(S,X,\mathbb{C}) $$
3
votes
0answers
27 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 ...
0
votes
2answers
36 views

Null homotopic by simplicial approximation

If $m<n$ use the simplicial approximation theorem to prove that any map $f:S^m\to S^n$ is null homotopic. Deduce that $\pi_1(S^n)$ is trivial if $n>1$. we have not covered lot on simplicial ...
1
vote
1answer
101 views

How an empty set is collapsed to a point?

In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that ...
2
votes
0answers
31 views

Filling the details of a construction via clutching function of a Vector Bundle

Let $(E,\pi,X)$ a complex vector bundle over X (which we assume to be Compact-Hausdorff) Let $$f \colon E \times S^1 \to E \times S^1$$ an automorphism of the product bundle $E \times S^1$. We define: ...
1
vote
2answers
32 views

Fundamental Group of the special Euclidean matrix group of the plane

How do you do this? Compute the fundamental group of the special Euclidean group of the plane, that is, all matrices of the form: $ \left( \begin{array}{ccc} \cos(z) & \sin(z) & x \\ ...
2
votes
0answers
15 views

Euler class of quotient bundle of real projective space

Let $\gamma$ be the real tautological line bundle over the real projective space $\mathbb{R}P^n$, $V$ be the trivial bundle of rank $n+1$, and $Q$ be the quotient bundle $V/\gamma$. What is the Euler ...
4
votes
0answers
73 views

A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
0
votes
1answer
21 views

Isotropy groups of tetrahedron after identifying its sides

If we identify the 4 sides of a regular tetrahedron in $\mathbb{R}^3$ by letting the group of all isometries of the tetrahedron act on it, what would the resulting space look like? The resulting ...
2
votes
1answer
61 views

Determining the induced map on homology $\tilde{H}_n(\mathbb{R}^n-\{0\})$ of $f\colon \mathbb{R}^n\to\mathbb{R}^n$ based on sign of $\det(f)$.

I'm having difficulty understanding the following. It appears as Exercise 7, p. 155 in Hatcher's Algebraic Topology: (this is not homework, by the way) For an invertible linear transformation ...
1
vote
1answer
440 views

Group action and covering spaces

Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition ...
1
vote
1answer
17 views

An equivalence class is open in the basis of a covering map.

Let $p:\tilde X\to X$ be a covering. Define an equivalence class on $X$ as follows $$x\sim z \Leftrightarrow |p^{-1}(x)|=|p^{-1}(z)| $$ where $||$ means the cardinality of the fiber. Take any $x\in ...
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 ...
6
votes
2answers
140 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
36 views

Calculating fundamental group of adjunction space with linear transformation.

$X = D^{2} \times S^{1} \cup_{f} S^{1} \times D^{2}$, where $f : S^{1} \times S^{1} \to S^{1} \times S^{1}$ is a map induced by the linear map on $\mathbf{R}^{2}$ given by the matrix $$\left( ...
1
vote
0answers
44 views

A problem I met when reading Griffiths'Periods of Integrals on Algebraic Manifolds I

I am reading Griffiths' paper Periods of Integrals on Algebraic Manifolds I, and in section 2 I met some problems. I wish that I could get some help here. My problem is that I cannot understand ...
3
votes
1answer
41 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 ...
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 ...
0
votes
1answer
38 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.
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 ...
3
votes
1answer
58 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 ...
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 ...
1
vote
1answer
22 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 ...
9
votes
2answers
252 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
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 ...
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 $f:E\rightarrow E$ ...
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 ...
2
votes
1answer
343 views

homotopic between two maps imply the homotopy between their mapping cone

Recall the mapping cone of a map $f: X\rightarrow Y$ is defined as the space $C_f: X\times [0, 1]\dot{\cup} Y/\sim$, where $\sim$ is the equivalence relation given by $(x, 1)\sim f(x)$ and $(x, ...
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 $\mathbb{R}^2$. ...
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 ...
4
votes
1answer
71 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 ...
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
1answer
40 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 ...
2
votes
1answer
100 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
38 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
49 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 } ...
8
votes
1answer
268 views

What is known about $Aut(\mathbb{I}^n)$

Few months ago, I asked a related question: Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$? It was interesting for me to know that $Aut(\mathbb{I})\,$ has a ...
2
votes
2answers
95 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 ...
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 ...
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 ...
2
votes
1answer
47 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?
4
votes
3answers
124 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 ...