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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9 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 $$[X,HR]_{-*}=\tilde{H}^*(X; R)$$ for any pointed CW-complex $X$, and ...
0
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0answers
103 views

Elements of $\operatorname{SL}(2,\mathbb{Z})$ and their negative multiples acting with the mapping class group of the two-torus $T^2$

Given some element A of $\operatorname{SL}(2,\mathbb{Z})$ which is associated with a particular orientation preserving homeomorphism up to isotopy (by the isomorphism $\phi$), what orientation ...
2
votes
0answers
25 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
34 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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2answers
22 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 ...
1
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0answers
13 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 ...
0
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1answer
27 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, ...
4
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1answer
54 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 ...
4
votes
2answers
194 views

Reference request for “Hodge Theorem”

I have been told about a theorem (it was called Hodge Theorem), which states the following isomorphism: $H^q(X, E) \simeq Ker(\Delta^q).$ Where $X$ is a Kähler Manifold, $E$ an Hermitian vector ...
5
votes
1answer
60 views

Second Stiefel-Whitney Class of a Five Manifold

There is a unique rank 4 nontrivial orientable vector bundle over the 2-torus, denote this by $p:E\rightarrow T^2$. Denote the associated sphere bundle by $S(E)$. Then since $S(E)$ is orientable, the ...
0
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2answers
43 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 ...
1
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1answer
31 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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0answers
10 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 $$H^1(X, TX)\to H^{n−1,1}(X,\mathbb C)$$
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3answers
42 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 ...
2
votes
0answers
35 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 ...
0
votes
1answer
28 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 ...
2
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0answers
68 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 ...
32
votes
13answers
2k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
1
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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
vote
1answer
40 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 ...
0
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0answers
24 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 ...
6
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1answer
104 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
40 views

A homotopy of functions which isn't a path homotopy [on hold]

The definitions are: A homotopy between two continuous functions $f$ and $g$ from the topological space $[0,1]$ to a topological space X is defined to be a continuous function $H : [0,1] \times [0,1] ...
1
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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 ...
1
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0answers
34 views

How to make the orbit space $T/G$ of torus $T$ homeomorphic to the Klein bottle?

Actually it is one of the exercises of Munkres. $G$ is a group of homeomorphisms of the torus having order $2$. How do I get $G$ in order to make $T/G$ homeomorphic to the Klein bottle? Can anybody ...
3
votes
0answers
36 views

Cycles non-homologous but with the same winding number at each point outside?

Problem (Fulton's Algebraic Topology: A First Course, Problem 6.28) Can we find a closed set $X\subseteq\mathbb R^2$, such that there's a cycle $\gamma$ in $X$ not homologous to zero but for each ...
1
vote
1answer
26 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 ...
1
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1answer
22 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 --> ...
1
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0answers
26 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 ...
8
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1answer
105 views

What are necessary and sufficient conditions for the product of spheres to be paralellizable?

Okay, so I found the result that the tangent-bundle of any product of spheres is parallizable, given that some element of the product is either $S^1$, $S^3$, or $S^7$. I prove this as follows, first ...
4
votes
1answer
84 views

Cohomology of wedge equals direct sum of cohomologies

I have seen the following fact used somewhere (for example to show that $\mathbb{RP}^3$ is not homotopy equivalent to $S^3\vee\mathbb{RP}^2$): Let $X,Y$ be two path connected pointed spaces such ...
2
votes
1answer
139 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.
3
votes
2answers
24 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!
7
votes
1answer
193 views

Can a division algebra over $\mathbb{R}^3$ be used to construct a counterexample to the hairy ball theorem?

Suppose (for contradiction) that there is a (if necessary associative and/or normed) division algebra over $\mathbb{R}^3$. Is there a simple way to use this to construct a nonvanishing continuous ...
2
votes
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 ...
1
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0answers
46 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
0
votes
1answer
44 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
2
votes
1answer
26 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 ...
2
votes
1answer
64 views

Isomorphism of Fundamental Groups (arcwise connected)

In an arcwise connected topological space $X$, we can show that the two groups $\pi(X,x)$ and $\pi(X,y)$ are isomorphic for $x,y \in X$ by defining a mapping $u: \pi(X,x) \to \pi(X,y)$ by $\alpha ...
2
votes
0answers
23 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
votes
0answers
66 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 ...
3
votes
0answers
36 views

Some questions about homology with local coefficients.

If $F:\pi_1(X)\rightarrow Ab$ is a system of local coefficiens, on the topological space $X$, then we can define the homology of $X$ with coefficients $F$ by taking the homology of the chain complex ...
3
votes
0answers
28 views

Group bundle over a topological space

Suppose $p:\tilde X\rightarrow X$ is the universal cover of $X$. Take $G$ a group where $\pi_1(X,x)$ acts by isomorphisms. I read that if we consider $X\times G$ ($G$ with the discrete topology) and ...
2
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1answer
53 views

Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
1
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1answer
42 views

Property of Homology: group isomorphism

I have this proposition, but I don't understand how they use the axiom 5, since in the axiom 5; $f,g: (X,A)\rightarrow (Y,B)$ and in the theorem we have $f:(X,A)\rightarrow (Y,B)$, $g:(Y,B)\rightarrow ...
0
votes
1answer
39 views

A short exact sequence

I have this proposition, and I don't understand how to do to obtain the short exact sequence: where axiom 4 is:
4
votes
2answers
101 views

Property of homology

I have this proposition, and I have two questions: 1) Why $H_k=\text{Im} i_*\oplus \ker r_*$ ? 2) Why $j_*: \ker r_*\rightarrow H_k(X,A)$ ? Edit: For the second, I try the 1th theorem of ...
6
votes
3answers
176 views

Klein bottle as two Möbius strips.

I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle. The Möbius strip comes in two versions that are mirrored ...
2
votes
2answers
29 views

Homology of a finite graph follows from Mayer-Vietoris sequence?

Problem (Fulton's Algebraic Topology: A First Course, Exercise 10.15) If $X$ is a finite graph with $v$ vertices and $e$ edges, and $X$ has $k$ connected components, show that $H_1X$ is a free ...
3
votes
1answer
45 views

A generalization of Jordan curve theorem to connected open sets in the plane

Problem (Fulton's Algebraic Topology: A First Course, Problem 5.23) Let $U\subseteq\mathbb R^2$ be any connected open set in the plane. If $X\subseteq U$ is homeomorphic to $[0,1]$, then ...