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

learn more… | top users | synonyms (1)

3
votes
1answer
16 views

Equivalence of relative and (reduced) homology for arbitrary pairs

I could not find my mistake in the following argument, though I know it is wrong. This is more like a "Q&A", since there is nothing to "prove" in the positive sense. Here it goes: For an ...
0
votes
0answers
25 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
3
votes
1answer
28 views

$\tilde{H}_i(S^n-X)$, $X$ a Finite Graph

I came across this question. Prove that $\tilde{H}_i(S^n-X)\cong H_{n-i-1}(X)$ if $X$ is a finite connected graph embedded in $S^n$. By Alexander Duality, this is true if the group on the right is a ...
2
votes
2answers
30 views

Expressing homotopy groups of spaces of (unpointed) maps $S^1\to M$ in terms of homotopy groups of spaces of pointed maps.

I came across the following problem while studying for a topology exam: Let $M$ be a topological space, let $\Lambda(M)=M^{S^1}$, the space of continuous maps $S^1\to M$ with the compact-open ...
1
vote
0answers
20 views

Coboundary map problem

I have to show that if $A$ and $B$ are compact connected subsets in the plane such that $A\cap B$ is not connected (and not empty), then $\Bbb R^2\setminus(A\cup B)$ is not connected. The tool I must ...
13
votes
1answer
258 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
0
votes
1answer
29 views

Suspension of a CW complex

I want to prove that the suspension $\Sigma X$ of a CW-complex $X$ is a CW-complex, buy I'm starting with CW-complexes and I don't have a clue of how start, so I'd appreciate any help. Thanks. ...
1
vote
1answer
40 views

Why notion of fundamental group is defined only over a connected scheme?

I went to different references on fundamental group on schemes. It is quite strange for me that the notion of fundamental group is only defined on connected scheme. Does anybody know why?
1
vote
1answer
29 views

Is $H^0(S^0;G)\simeq G\oplus G$ or $G$?

In the article on topospaces for the (co)homology of spheres, it says $H^0(S^n,G)\simeq H^n(S^n,G)\simeq G$. Is this true when $n=0$? I think not, for if we view $S^0$ as the union of two ...
0
votes
1answer
17 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 ...
0
votes
1answer
41 views

Why is $H^2(\mathbb{R}P^2,\mathbb{Z})\simeq\mathbb{Z}_2$?

Why is the second cohomology group of $X=\mathbb{R}P^2$ with $\mathbb{Z}$-coefficients $\mathbb{Z}_2$? We can put the usual $\Delta$-structure on $X$ with two vertices, three $1$-simplices, say $a$, ...
6
votes
1answer
121 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
0
votes
1answer
43 views

Equivalent statements to fixed-point theorem

I'm trying to show that they're equivalent statements: 1) $1_{S^1}$ is not homotopic to a constant map. 2) $S^1$ is not a retract of $D^2$ ($D^2$ is the closed unit ball). 3) Every continuous map ...
2
votes
1answer
24 views

Fixed point equivalence [duplicate]

Let $\Bbb D^n$ be n-dimensional ball and $S^{n-1}$ the $n-1$ dimensional sphere realized as boundary of $\Bbb D^n$. Prove that following are equivalent. There is no retraction $\Bbb D^n\to S^{n-1}$ ...
0
votes
0answers
28 views

Fixed point of simplicial map is subcomplex

If $s:|K|\to |K|$ is a simplicial map, prove that the set of fixed points of s is the polyhedron of a subcomplex of $K^1$, though not necessarily a subcomplex of $K$. I know s at least fixes ...
3
votes
1answer
212 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
0
votes
0answers
33 views

How to compute the Lefschetz number

Given a continuous function $f: X \to X$ how do you compute: $$ \Lambda_f = \sum_{k \geq 0} (-1)^k \mathrm{Tr}(f_*|H_k(X,\mathbb{Q})) $$ which is known as the Lefschetz number. For instance let $X: ...
2
votes
2answers
43 views

Is there an isomorphism $\mathrm{Hom}(H_1(X),G)\simeq\mathrm{Hom}(\pi_1(X),G)$ when $X$ is path connected?

In Hatcher 3.1.5 on pg. 205, one proves that if $\varphi\in C^1(X;G)$ is a cocycle, where $X$ a space and $G$ an abelian group, then for paths $f$ and $g$ one has various properties $\varphi(f\cdot ...
14
votes
1answer
344 views

What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these ...
3
votes
2answers
41 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 ...
0
votes
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
81 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
78 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
25 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 $$ H^1(S \setminus X, \mathbb{C}) \simeq H^1(S,X,\mathbb{C}) $$
2
votes
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 ...
0
votes
2answers
35 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
97 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 ...
1
vote
0answers
40 views

simplicial approximation of $f(x)=4x^2-1$ [on hold]

I have a question on practice sheet: Find simplicial approximations to $f:|K|\to |L|$ and $f:|K|\to |M|$ where $f(x)=4x^2-1$ I doubt this question. Do you have any idea to start with.
2
votes
0answers
27 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
28 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
13 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 ...
0
votes
0answers
46 views

Ring structure of $K(X)$ - definition of multiplication

Maybe it's a silly question, but I can't find a satisfactory answer to it. Hatcher defines the multiplication of two arbitrary elements in $K(X)$ as $$(E_1-E_1')(E_2-E_2') := E_1 \otimes E_2 - E_1 ...
4
votes
0answers
53 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
17 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
50 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
431 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
15 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
70 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
126 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
43 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
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 ...
2
votes
1answer
46 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
66 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
35 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
54 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
45 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
24 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
19 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 ...