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

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1answer
54 views

Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?

A topological space $X$ which is an algebra over an $E_\infty$-operad $E$ consists of a sequence of maps $$ \mu_n':E(n)\times X^n\to X $$ with compatibility conditions. The spaces $E(n)$ are ...
3
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2answers
60 views

Example of an oriented manifold with cohomology not isomorphic to a homogeneus space

The question as in the title: Is there a simple example of a compact orientable smooth finite-dimensional manifold whose singular cohomology groups with integer coefficients are not isomorphic to ...
8
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0answers
174 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
4
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1answer
66 views

Local triviality condition on line bundles

We recall that a complex line bundle consists of a triple $(\pi,E,B)$ where $E,B$ are topological spaces, $\pi : E \to B$ a continuous map satisfying the following local triviality condition: ...
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0answers
84 views

When do equivariant quasi-isomorphisms of chain complexes induce a quasi isomorphism on the tensor product

Suppose I have chain complexes $A,B,C,D$ where $A$ and $C$ have right $R$-module structures and $B$ and $D$ have left $R$-module structures, and that I have maps $f:A\to C$ and $g:B\to D$ which ...
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1answer
58 views

Is a weakly contractible connected metric space, uniquely geodesic?

A topological space is weakly contractible if all the homotopy groups are trivial. It's connected if it's not the union of two disjoint nonempty open sets. A metric space $(X,d)$ is uniquely geodesic ...
2
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0answers
119 views

Prerequisites for 'Fibre Bundles' by 'Dale Husemoller'

I wish to study the book 'Fibre Bundles' by Dale Husemoller. How much Algebraic Topology is required for studying this book ? Would a knowledge of fundamental groups, covering spaces (say from second ...
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1answer
40 views

Is the image of a continuous idempotent necessarily homotopic to the original space?

Let $f$ be a continuous self-map of a topological space $X$ such that $f\circ f=f$. Is it true that $X$ is homotopic to its image $f(X)$?
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1answer
54 views

Why does a subcomplex of a cell complex being closed mean the characteristic map has an image in the subset?

I was trying to learn some Algebraic Topology though I haven't got very far yet so I would greatly appreciate it if you gave as simple answer as possible. On page 7 of Hatcher he says: Since $A$ ...
4
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0answers
59 views

Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman. Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and ...
5
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1answer
156 views

$f$ holomorphic from unit disc to itself. $f\left(\frac{1}{2}\right) = f\left(-\frac{1}{2}\right) = 0$. Show that $|f(0)| \le 1/3$.

I'm studying for a qual exam. I cannot solve the following problem: Let $f$ be holomorphic from the unit disc to itself. $f\left(\frac{1}{2}\right) = f\left(-\frac{1}{2}\right) = 0$. Show that ...
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0answers
85 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
5
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0answers
86 views

$\pi_2(G)$ for $G$ a Lie group. [duplicate]

It is well known that $\pi_2(G)$ is trivial for any Lie-group $G$. Is there an elementary proof of this, say, that can be understood with minimal homotopy theory? Also, who gave the first proof of ...
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1answer
76 views

Misprint in Switzer's Algebraic Topology?

I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can ...
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0answers
47 views

Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...
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1answer
85 views

Are these two definitions of $EG$ equivalent?

Let $G$ be a topological group with multiplication $\sigma:G\times G\to G$. The simplicial topological space $\mathcal{E}G$ defined by $$ \ldots \begin{array}{c}\to\\\to\\\to\\\to\end{array}G\times ...
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2answers
222 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
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0answers
71 views

Modified torus is homotopy equivalent to $S^2$

I need to prove that if we glue two unit discs along the parallel and the meridian of a torus (I'm not sure how this happens), the result will be homotopy equivalent to $S^2$. I also have this ...
4
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2answers
112 views

Examples of failure of excision for homotopy groups ($\pi_k(X, A)$ is not $\pi_k(X/A, *)$)

Let $A$ be a subcomplex of CW-complex $X$. The excision axiom for homology implies that $H_i(X, A)\cong H_i(X/A, *)$, and it is widely known that homotopy groups don't have this property. However, ...
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72 views

Prove $\mathbb{R}^3$ is not the product of two identical topological spaces

I can only prove this for $\mathbb{R}$: If $\mathbb{R}\cong T\times T$, then $T$ embeds in $\mathbb{R}$ as a closed subspace (e.g. $T\times pt$). Since $\mathbb{R}$ is connected, so is $T$. So $T$ ...
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0answers
93 views

find the connected covering space of $\mathbb{R}P^2 \lor \mathbb{R}P^2$

This is a problem on Hatcher' book. How to find all the connected covering spaces of $\mathbb{R}P^2 \lor \mathbb{R}P^2$? I don't know where to start. Is there a general way to construct the ...
2
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1answer
169 views

Degree and maps between closed orientable surfaces

Let $M_g, M_h$ be closed orientable surfaces of genus $g,h$ respectively. If $g>h$, we know there exists a map $M_g \rightarrow M_h$ of degree 1: just think of $M_g=M_h\#M_{g-h}$ and consider the ...
1
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1answer
119 views

Definition of attaching a cell to a manifold

I know "attaching a handle" to a manifold, but recently I faced "attaching a cell" and I don't know its definition in precise. It seems that the definition is very trivial (!) because my searches did ...
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0answers
120 views

Intersection of simply connected sets II

I read the following statement in the old question "Intersection of Simply-Connected Sets" (Intersection of Simply-Connected Sets): If $U$ and $V$ are simply connected and $U \cap V$ is path ...
4
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3answers
206 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
2
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1answer
184 views

Euler class of tangent bundle of the sphere

I am working through Milnor's Characteristic classes and am currently working problems on the topic of oriented bundles and euler class. I am having trouble computing the euler class of the tangent ...
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56 views

An h-cobordism problem

Im trying to understand the proof of Lemma 2.3 of Milnor and Kervaire: Groups of homotopy spheres I. Suppose we have a simply connected manifold $M$ which bounds a contractible manifold $W'$. Then ...
2
votes
1answer
81 views

self-intersection of lagrangian submanifold

Let's consider lagrangian submanifold $X$ in symplectic manifold $M$. Is it true that self-intersection index of $X$ is equal to the Euler characteristic $\chi(X)$? Can we construct (not canonical) ...
3
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0answers
33 views

How to give the coproduct of differential graded algebras explicitly?

Let $X$ and $Y$ be based spaces such that their respective loop spaces $\Omega X$ and $\Omega Y$ are connected. In the first paragraph of this article by Dula and Katz, it is given that ...
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0answers
285 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
7
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1answer
173 views

Why is there no compact manifold without boundary with the following homology groups?

I've been studying homology groups, and this question is stumping me: Prove there can be no compact manifold $X$ without boundary whose homology groups are $$H_i(X) = \left\{ \begin{array}{ll} ...
3
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2answers
137 views

On the quotient group $\pi_{1}(K)/N$ for the Klein bottle $K$

I know that the Klein bottle $K$ is obtained from the unit square by making identifications on the boundary with the appropriate directional arrows. Usually, what is done is that we identify the point ...
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2answers
123 views

determine which of the following spaces are contractible

Determine which of the following spaces are contractible? (a) Unit interval $I=[0,1]$ (b) $\mathbb{S^2}$$\setminus$ {$p$},where $\mathbb{S^2}$ is a $2$-sphere and $p$ is any point on $\mathbb{S^2}$ ...
2
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1answer
45 views

Proof and counter-example that a chain $c_{R, n} \ne \partial c$. Where is the error?

If $R > 0$ and $n \in \mathbb{Z}$ we can define the singular 1-cube $c_{R, n}\colon [0, 1] \rightarrow \mathbb{R}^2$ by $$c_{R, n}(t) = (R\cos(2\pi n t), R\sin(2\pi n t))$$ We know that $c_{R, n} ...
3
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3answers
216 views

Does there exist some relations between Functional Analysis and Algebraic Topology?

As the title: does there exist some relations between Functional Analysis and Algebraic Topology. As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially ...
4
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2answers
197 views

Normal bundle in tangent bundle

Let's consider the normal bundle $NM$ of zero section in $TM$. Is it true that $NM \cong TM$? There is exact sequence $$0 \rightarrow TM \rightarrow TE|_M \rightarrow NM \rightarrow 0$$ for the ...
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1answer
153 views

Covering space of $C \backslash \{0,1,\lambda \}$

Let $\lambda\in C \backslash \{0,1\}$, $E= \{(x,y) \in C^2 : y^2=x(x-1)(x-\lambda),\ x\neq 0,1,\lambda \}$. Prove that $E$ is a connected $2$-fold covering space of $C \backslash \{ 0,1,\lambda \}$ ...
8
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1answer
106 views

Existence of smooth elliptic curves with complex multiplication

this is my first question ever on a platform like this so please forgive me any kind of unintended misbehaving. In Kudla, Rapoport and Yang "On the derivative of an Eisenstein series of weight one" ...
1
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1answer
86 views

A question on Aut$(N)$ and Aut$(N/G)$

Let $N$ be a complex manifold and $G$ is a finite group freely acting on $N$. Define another complex manifold $M$ as $M=N/G$. I would like to study Aut$(M)$, the (holomorphic) automorphism group of ...
8
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2answers
156 views

Computing $\pi_3(\mathrm{Gr}_2(\mathbb{R}^4))$

How can one go about computing the 3rd homotopy group of the Grassmannian manifold of 2-planes through the origin in $\mathbb{R}^4$? I don't want to be more general in the question, because: 1) I ...
3
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0answers
93 views

The monodromy and cut planes.

I am trying to understand the following example from the book Riemann surface by Donaldson (p.48). This was given after introducing the notion of monodromy of the covering. Consider the Riemann ...
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1answer
102 views

A form problem between $S^3$ and $S^2$.

Let $\phi: S^3 \rightarrow S^2$ be an smooth map. a) Suppose that $\omega$ is a 2-form on $S^2$ with $\int_{S^2} \omega =1$. Show that there exists a 1-form $\alpha$ on $S^3$ with ...
3
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1answer
84 views

Surjective inclusions in Van Kampen's Theorem

Let $X = U \cup V$ be an open cover. Assume $U,V$ and $U \cap V$ path-connected. We have the inclusions $u : U \cap V \to U$ and $v : U \cap V \to V$ with induced maps $u_* : \pi_1(U \cap V) \to \pi_1 ...
4
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1answer
232 views

Request for companion of Mariano Suárez-Alvarez's proof.

Mariano Suárez-Alvarez's answer to Cohomology of projective plane seems very interesting. However, there are three pieces I could not stitch up for one of his proofs. Wonder if someone may help? ...
2
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1answer
222 views

Continuous maps from $S^1 \to X$ equivalent conditions

The following are equivalent for a topological space X according to a problem in Hatcher. $1$)Every continuous map $S^1 \to X$ is homotopic to a constant map. $2$)Every continuous map $S^1 \to X$ ...
4
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1answer
157 views

The homology of the torus.

I am reading "Riemann surface" by Donaldson. On page 68, the calculation of the first homology of the torus $T$ is given but there are several steps that I don't understand. Here is the calculation. ...
2
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3answers
144 views

$X$ is a connected space and $Y$ is a discrete space prove that the two maps $f,g\colon X\rightarrow Y$ are homotopic if and only if $f=g$.

$X$ is a connected space and $Y$ is a discrete space prove that the two maps $f,g\colon X\rightarrow Y$ are homotopic if and only if $f=g$. I am trying to solve few problems in algebraic topology, ...
2
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1answer
95 views

Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on ...
1
vote
1answer
255 views

Induced homomorphism between fundamental groups of a retract is surjective

I'm trying to understand why the induced map $i_*: \pi_1(A) \rightarrow \pi_1(X)$ is surjective, for $A$ being a retract of $X$ and $i: A \rightarrow X$ being the inclusion map? For homotopy retracts ...
5
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2answers
187 views

Sufficient conditions for $M(X \times Y, Z)$ to be homeomorphic to $M(X, M(Y, Z))$

Let $Y^X$ denote the set of all functions $f: X \to Y$. If $X$ and $Y$ are topological spaces, let $M(X,Y)$ denote the set of all continuous maps $f: X \to Y$, endowed with the compact-open topology. ...