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

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Cohomology of fiber bundle with a section

Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map $$ ...
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202 views

Homotopy equivalence in the category of arrows.

I'm reading Jeff Strom's book on Homotopy Theory and I am trying to make some sense of a certain exercise. On page 91, "Homotopy in Mapping Categories" we consider the category of arrows of ...
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678 views

Torsion in homology groups of a topological space

It seems as though "nice" spaces don't have torsion in their homology groups. What is the underlying characteristic of these nice spaces; that they can be embedded in $\mathbb{R}^3$? So what are some ...
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349 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
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660 views

Hatcher 1.3. problem 16

Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.
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104 views

Decomposition of vector bundles over a CW complex

Let $X$ be CW complex having only cells up to dimension $n$. I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a ...
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111 views

Fundamental group of the complement of a linear subspace

Let $m<n-1$ be two positive integers. Consider $\mathbb{R}^m$ as a subspace of $\mathbb{R}^n$ via $\mathbb{R}^m\times \{(0,0,...0)\}$. Any suggestions on how to compute ...
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313 views

Definition of Cobordism

I am currently trying to read Quillen's Cobordism paper - On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298. Still on the first page ...
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204 views

Properties of Surgery on Manifolds?

I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; ...
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211 views

Smith normal form of graded modules. (Major edit)

Ok, this is a major rewriting of my previous entry which no one answered. Let us have two graded $F[t]$-modules M and N with bases $m_1, \ldots, m_m$ and $n_1, \ldots, n_n$, respectively, and $F$ is ...
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161 views

Invariance of Wall's self-intersection under the regular homotopy

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
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356 views

Chain map inducing isomorphism in homology

If $X$ is a CW complex, show that there is a chain map $W_*(X) \to S_*(X)$ inducing isomorphisms in homology. Here $W_p(X) = H_p(X^p,X^{p-1})$ Let $E$ be the CW decomposition of $X$ and let $M$ ...
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434 views

Is every CW complex homotopic to a Delta-Complex?

Both answers to this question seem equally reasonable to me. If the answer is positive, I have no idea what the construction of such a space would look like.... If the answer is negative, I assume ...
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47 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
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43 views

Minimum number of sets required for a good open cover

A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible. For example, $S^2$ has an ...
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49 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
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59 views

Properties of $\mathbb{C}P^n$

I'm currently working on a somewhat deformed version of $\mathbb{C}P^2$ and want to check some properties from a geometrical and/or topological point of view. Of course, $\mathbb{C}P^2$ is Kähler ...
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72 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
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54 views

3-manifolds with certain fundamental group

Suppose $M$ is a 3-manifold with $\pi_1(M)=\mathbb{Z}/p\mathbb{Z}$. Is it true that $M$ is diffeomorphic to a lens space $L(p,q)$?
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90 views

Mapping $f: (S^n, pt) \to (S^n, pt)$ induces multiplication by $deg(f)$ on $H_*(S^n \wedge X, pt)$.

I'm studying for my final exam in Algebraic Topology and got stuck with somewhat straightforward looking preparation problem. Let $\tilde H$ be any reduced homology theory and $f: (S^n, pt) \to ...
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40 views

Definitions of the ring $R/(x_0^\infty,\ldots,x_{n-1}^\infty)$

Let $R$ be a ring, and let $I = (x_0,\ldots,x_{n-1})$ be a finitely-generated ideal inside of $R$, generated by a regular sequence. In algebraic topology one often encounters a ring, usually denoted ...
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68 views

Does naturality for characteristic classes imply the classifying space is universal for them?

Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup. To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, ...
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86 views

Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact ...
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63 views

Proof of the isomorphism $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$

I want to prove the following: $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$ via the map $[f] \mapsto f^* w_1(\gamma^1)$ where $\gamma^1$ denotes the tautological line bundle over ...
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121 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
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26 views

Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
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44 views

Inductively Constructing Chain Homotopies

Let $f,g:C_\bullet \rightarrow D_\bullet$ be chain maps. Let $T$ be a generator (possibly not the only one) of $C_n$ and assume $H_n(C)=0$. I'm trying to prove that for every $T$ there's a solution ...
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69 views

a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even ...
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55 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
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150 views

On the path-connectedness of $X_{i} \setminus X_{i-2}$

Suppose $X$ is an $n$-dimensional regular CW-space (a space with a regular CW decomposition). What are the weakest sufficient conditions required on $X$ to ensure that all regular CW decompositions ...
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40 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
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61 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
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64 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
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216 views

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
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55 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
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126 views

De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
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156 views

Standard spaces with non-standard topology and around

The following series of questions gives me no rest. Let $\mathbb{\widetilde{R}}^n$ be $\mathbb{R}^n$ with Zariski topology, i.e. we say that $A\subset \mathbb{R}^n$ is closed if it is given by the ...
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170 views

Higher homotopy groups: Basepoint independence.

Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ ...
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95 views

Motivation behind definition of homologous cycles

Two cycles are said to be homologous if their difference is a boundary.(usual meanings implied) What is the motivation behind this definition or the intuitive meaning it carries. I am looking of ...
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63 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
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250 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 ...
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102 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 ...
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95 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 } ...
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102 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
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86 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
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62 views

Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
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108 views

what is the homology groups some quotient space of torus

what is the homology group for The quotient space of $S^1 \times S^1$ obtained by identifying points in the circle $S1 \times\{x_0\} $ that differ by $\frac{2 \pi}{m}$ rotation and identifying points ...
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57 views

Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then ...
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34 views

Isomorphism of the etale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$ of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think ...
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72 views

Fiber Bundle of Manifolds

I want to show the following: Let $E \rightarrow B$ be a fiber bundle with fiber $F$. Show that if $B$ and $F$ are manifolds, then so is $E$. Solving this problem seems easy enough simply by ...