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

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1answer
63 views

Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
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1answer
28 views

Identification of points versus line drawn between points

I have a question regarding fundamental groups. If I take a sphere and union a line between it's poles, is that the same space as the sphere with those poles identified? I am trying to find the ...
0
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1answer
16 views

homeomorphism classes of compact surfaces with addition operation is a monoid

This is essentially pg 6 of serge lang's algebra's discussion about an interesting example. Homeomorphism classes of compact surfaces with the addition operation defined as following. Say M and $M'$ ...
2
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3answers
118 views

More elementary proof that $\pi_n(S^n) \cong \mathbb{Z}$

The proof I know that $\pi_n(S^n) \cong \mathbb{Z}$ is based on the Hurewicz theorem (which implies that $\pi_n(S^n) \cong H_n(S^n)$). I'm looking for a more elementary argument - preferably ...
1
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1answer
30 views

Étalé space for sheaf of sections of a fiber bundle

Let $X$ be a topological space, $\pi:E\to X$ a fiber bundle over $X$ with fiber $F$ and structure group $G$. Let $\mathcal{F}$ denote the sheaf of continuous sections of the bundle. I probably want to ...
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1answer
20 views

Complex bundles on $S^{2n+1}$

We know that the complex $K$ theory of spheres are 2 periodic. On the other hand every complex bundle on $S^{1}$ is trivial. So $K(S^{2n+1})=0$. So this is a motivation to ask: Is there a ...
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1answer
48 views

Real vector bundles on $S^{7}$

Is it true that $\pi_{6}(O(n))=0$ for all n? Equivalently, are all real bundles on $S^{7}$ trivial?
4
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1answer
47 views

map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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1answer
28 views

How to find a sequence in discrete group

Let $\Gamma$ be a discrete group. Can we find an increasing sequence $F_{n}\subset \Gamma$ of finite subsets, such that $\cup F_{n}=\Gamma$?
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1answer
33 views

A question about winding numbers.

This is a question from Needham's "Visual Complex Analysis". Kindly refer to the photo below. Let $K$ be a line moving downwards. The book says that if we move a point $r$ from the left to the ...
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4answers
174 views

Are there nontrivial continuous maps between complex projective spaces?

Are there maps $f: \Bbb{CP}^n \rightarrow \Bbb{CP}^m$, with $n>m$, that are not null-homotopic? In particular, is there some non-null-homotopic map $\Bbb{CP}^n \rightarrow S^2$ for $n>1$? Can we ...
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1answer
36 views

Converse to the Jordan-Brouwer separation theorem

By the Jordan curve theorem, if $C \subset S^2$ is (the image of) a simple closed curve, then $S^2 \setminus C$ has precisely two connected components. This statement admits the following "converse". ...
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3answers
161 views

The existence of a 1-1 continuous map between two topological spaces.

Show that there is no one-to-one continuous map $f$ from $\mathbb{R}^n$ to $\mathbb{R}^2$ for $n\gt 2$ with $f(0)=0$. I tried using the hint: consider $f:\mathbb{R}^n-\{0\}\rightarrow ...
5
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1answer
56 views

Homotopical classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$

Which are homotopy classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$ for $n < m$? In real case, even for any cellular complex $X$ with $\dim X<m$ homotopy classes of mappings $X \to ...
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0answers
26 views

A question from May's notes on Algebraic Topology.

I have a question regarding the following diagram from May's notes on Algebraic Topology. On pg. 7, the following diagram is given as a proof of the fact that $[f^{-1}.f]=[c_x]$. Here $f$ is a path ...
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1answer
45 views

How to compute homotopy groups of torus?

The results are listed here: http://topospaces.subwiki.org/wiki/Homotopy_of_torus Is there an intuitive way to understand these results? In particular, why would the higher homotopy group be the ...
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2answers
31 views

Fixed point property in topology

I have a few questions concerning relating the fixed point property for a space $X$ (every continuous map from $X$ to $X$ has at least one fixed point) to some concepts in topology. a). I know that a ...
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0answers
58 views

How to prove a direct sum?

$X=X_1\cup X_2$, $X_i, i=1,2$ are closed and disjoint. $i_{2_*}: H_k(X_2)\rightarrow H_k(X),$ induced by the inclusion $i_2: X_2\rightarrow X$ and $j_{1_*}: H_k(X)\rightarrow H_k(X,X_1)$ induced by ...
3
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0answers
29 views

Can one prove that $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$ without invoking the long exact sequence of a fibration?

In trying to remind myself why $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$, the natural argument that comes to mind is to take the long exact sequence associated to the fibration $S^1 \rightarrow S^\infty ...
0
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1answer
67 views

Direct Sum on Homology

I have a big problem and i don't know how to solve it i have no idea So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is ...
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1answer
27 views

smash product of Eilenberg-Maclane spaces

Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)
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0answers
73 views

Question about Property of Homology

I have this theorem, with the proof, but i dont understand, why they prove that $i_{1_*}, i_{2_*}$ are injective, we have that $i_{j_*},j=1,2$ are induced by an inclusion it is injective, so they are ...
2
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1answer
45 views

Is the homology theory given by Eilenberg Maclane spectrum equal to ordinary homology?

(I think I'm missing something very simple). Let $R$ be a ring and $HR$ the associated Eilenberg-Maclane spectrum, defined by $$[\Sigma^\infty_+ X,HR]_{-*}={H}^*(X; R)$$ for any CW-complex $X$, and ...
2
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0answers
35 views

Universal property of tensor product of vector bundles

To define the tensor product of vector bundles $\xi_1$ and $\xi_2$ over base $B$, Milnor-Stasheff's Characteristic Classes takes the space $\sqcup_{b \in B} F_b(\xi_1) \otimes F_b(\xi_2)$ and ...
2
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3answers
52 views

homotopy groups of wedge sum

Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book, $$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$ Is it true in general $$\pi_{n}(\bigvee_{\alpha} ...
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0answers
23 views

Verlinde formula, moduli space vector bundle on genus 2,3 curves.

I'd like to prove "by hands" the Verlinde formula for moduli space of rank two semistable vector bundles with fixed determinant on a curve of genus two and three. For a curve of genus two and even ...
1
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2answers
31 views

Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let ...
0
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1answer
36 views

Homology group of space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$

I have to calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$ and then determine to which of the following spaces it is homeomorphic: $S^2, ...
5
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1answer
80 views

Definition of multiplication in Grothendieck ring

Let $X$ be a smooth variety over an algebraically closed field $k$ of dimension $n$. Consider the Grothendieck Group $K(X)$ of coherent sheaves on $X$, i.e. the free abelian group generated by ...
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3answers
64 views

Algebraic topology - angle function

I just know "if $f$ is continuous, $f: [a, b]\rightarrow S^1$, then there is a continuous function $g: [a, b] \rightarrow \mathbb R$, such that $f (x) = e ^ {g (x)}$, for all $x$ in $[a, b]$" I want ...
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0answers
11 views

Cup product against holomorphic volume form.

Let $σ$ be a holomorphic volume form. Then I can not see why cup product against $σ$ defines a linear isomorphism $$H^1(X, TX)\to H^{n−1,1}(X,\mathbb C)$$
2
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0answers
37 views

Basic question about lifting maps to covering spaces

Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers). The space $\tilde X_1$ is certainly ...
1
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2answers
46 views

$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
0
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3answers
51 views

A proof of compactness, connectedness of real projective space

I need a reference for a complete proof of the below theorem: Let $RP^n$ be $n$-dimensional real projective space. Then $RP^n$ is a compact, connected manifold. (Consider the standard topology over ...
1
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1answer
50 views

Is the constant sheaf $\mathbb{Q}$ injective?

Let $X$ be a topological space, and let $\mathbb{Q}$ be the constant sheaf of abelian groups on $X$ associated to the group of rational numbers under addition. Is $\mathbb{Q}$ an injective object in ...
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0answers
26 views

lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
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1answer
31 views

Finding generators of homology groups

Take the simplicial complex with vertices {a,b,c} and edges {ab, ac, bc}. In other words, a circle. If I build a chain complex, and make the matrix of my differential, I get that the kernel of ...
1
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1answer
28 views

spin structure definition

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map ...
2
votes
1answer
24 views

Homotopy colimit of a 3x3

Hi I am wondering how you calculate homotopy colimits of a 3x3 diagram. In particular if we have (sorry not sure how to Tex these) Top/bottom row: * <-- * --> * Middle row: * <-- X --> ...
6
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1answer
105 views

Deck transformations on $S^1\times \mathbb{R}P^2$

I'm studying for qualifying exams and stuck on the following problem: Suppose that $S^1\times \mathbb{R}P^2$ covers a space, and let $h$ be a deck transformation of the covering. Show that the ...
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0answers
29 views

Meyer-vietoris sequence to compute the compact cohomology for Möbius strip

How do you use Meyer-vietoris sequence to compute the compact cohomology for Möbius strip without the bounding edge? Please give detail math. In particular explain how inclusion map is used. On page ...
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1answer
34 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
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0answers
25 views

boundary in homology group

f is a reflection on a sphere $S^{n}$, $\sigma_{1}$ is a diffeomorphism from $D^{n}\subset \mathbb{R}^{n}$ to one of the two caps of the sphere, separated by the plane of the reflection and ...
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0answers
75 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if ...
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2answers
31 views

Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
2
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1answer
45 views

Quotient map not nullhomotopic

I have the following qual problem: Let $M$ be a connected closed surface, not necessarily orientable, with an embedded closed disk $D$. Let $Q$ be the quotient space of $M$ by $\overline{M\setminus ...
2
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1answer
144 views

Homotopy invariance in homology

i have this from Hatcher's book "Algebric topology" And i don't understand why we have $i-1$ in $(-1)^{i-1}$ and strict inequality in $P\partial(\sigma)$ ? Please. Thank you.
2
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0answers
32 views

An exercise in homology computation / What is the geometric fixed points of an Eilenberg Maclane Spectrum?

The question I want to ask has a reasonably elementary formulation and I think there is a good chance it can be answered in this form (by someone more computationally skilled than me, or perhaps by ...
2
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0answers
67 views

$C(X)$ is finite dimensional iff $X$ is finite [duplicate]

If $X$ is compact Hausdorff space and $C(X)$ is the set of all continuous complex valued functions on $X$,then prove that $C(X)$ is finite dimensional if and only if $X$ is finite. My problem:If we ...
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1answer
31 views

How does one triangulate the mapping cylinder of a diffeomorphism?

The question is fairly self-explanatory. In particular, I would like to know how to triangulate the mapping cylinder arising from applying a Dehn twist to the torus. The reason is that I am thinking ...