2
votes
1answer
59 views

About existence of Morse functions

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = ...
1
vote
1answer
47 views

Identity in Thom spaces.

Let $T$ be the one-point compattification, $E$ a real vector bundle, $\epsilon$ the trivial line bundle and $\Sigma$ the suspension operation. How can I prove that $$ T(\epsilon \oplus E) \simeq ...
7
votes
0answers
128 views

Lemma 6.3 of Milnor's Lectures on the h-cobordism theorem

Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$, ...
2
votes
0answers
68 views

How to do this surgery?

Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points. Take a neighborhood $U$ of ...
1
vote
0answers
57 views

Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...
0
votes
1answer
121 views

Definition of a 4-cobordism with boundary

Definition of a 3-cobordism (in my context) is a pair $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is a closed orientable topological 3-manifold and $\partial M$ is a disjoint union of ...
1
vote
0answers
32 views

Cylindrical structure and homology

Let us consider a cobordism $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is homeomorphic to $T \times I$, here $T$ is a torus $S^1 \times S^1$ and $I=[0, 1]$. I encountered the statement ...
1
vote
2answers
64 views

Neutral elements in the unoriented cobordism.

Let $M$ be an $n$-dimensional manifold and let $[M]$ denote the unoriented bordism class of $M$. Forming the usual commutative graded ring $\text{MO}_n$ we know that $$\text{MO}_* \simeq ...
2
votes
1answer
105 views

Opposite Orientation of Boundary in Bordisms

In Lurie's "On the Classification of Topological Field Theories" (and certainly other places) he defines the category $\mathbf{Cob}(n)$ who objects are oriented $(n-1)$ manifolds. Given ...
2
votes
1answer
127 views

Embedding a manifold in the disk

I don't understand a sentence made by Hirsch in his Differential Topology at page 175: If $k > n+1$ and $M^n = \partial W^{n+1}$, then an embedding $M^n \hookrightarrow S^{n+k}$ extends to a neat ...
4
votes
0answers
260 views

Definition of Cobordism

I am currently trying to read Quillen's Cobordism paper. Still on the first page though so the emphasis is on the trying :P. A map of manifolds $f: Z \to X$ can be give an complex orientation in the ...
2
votes
1answer
125 views

Bounding projective spaces

For which $n$ does there exist a (topological, smooth, PL, complex) manifold $M^n$ such that $\partial M = \mathbb{R}\mathbb{P}^m$. Obvously, $m = n -1 $ (at least an in the real case). There are a ...