# Tag Info

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### Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?

An explicit cobordism is given by Stong: R. E. Stong, A Cobordism, Proceedings of the American Mathematical Society Vol. 35, No. 2 (Oct. 1972), pp. 584-586 I do like the short title "A Cobordism". ...
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### Nilpotence of the stable Hopf map via framed cobordism

Answer Summary Let $\eta$ be the framed 1-manifold which is the Lie group framing on the circle and let $\nu$ be the Lie group framing on $S^3 = Spin(3)$. I am probably going to conflate these framed ...
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### Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?

Thom wrote two notes in the proceedings of the "Colloque de Topologie de Strasbourg", which was a topology seminar organized by Ehresmann at that time: "Quelques propriétés des ...
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### Homology theory represented by Madsen-Tillmann spectra

This is an exercise in understanding the Pontrjagin--Thom correspondence. The group $\pi_k(MTO(n) \wedge X_+)$ is in bijection with tuples of a $(n+k)$-manifold $M$, an $n$-dimensional vector ...
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### Critical dimensions D for "smooth manifolds iff triangulable manifolds"

All smooth manifolds are triangulable, as you say. This follows from Morse theory, which dictates that you only need to know how to triangulate (PL) handle-attachments, which one can do by hand. The ...
• 7,891
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### What is known about exotic spheres up to stable diffeomorphism?

The inertia group $I_M$ of a closed oriented $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$. Wall ...
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### Does Spin cobordism vanish in dimension $4k-1$?

I believe the bordism groups are nonzero in every dimension after some relatively small finite dimension, just by looking at the Poincaré polynomial in Anderson-Brown-Peterson's earlier paper "Spin ...
• 12.7k
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### Super-cobordisms

There are a number of technical issues with making what you describe precise, for example: what precisely is a supermanifold with boundary? how can you glue/compose bordisms? etc. I am going to ignore ...
• 25.8k
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### Sphere spectrum, Character dual and Anderson dual

The Anderson dualizing spectrum $I_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded ...
• 4,773
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### The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum

One can prove that $\mathrm{Map}(H\mathbf{F}_p,MU)$ is contractible. We know that $H\mathbf{F}_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology ...
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### Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?

Here is an argument that "most" points in the $SL(2, {\mathbb C})$-character variety $X(F)$ of the surface $F$ do not correspond to representations extendible to 3-manifold groups (as in the question)....
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### Book recommendation for cobordism theory

Perhaps the Notes on cobordism by Haynes Miller could be of some help too. Another possibility (but geared primarily towards applications in symplectic geometry) is the book V. Guillemin, V. ...
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### Cobordisms and Euler characteristics

René Thom proved that two closed $n$-manifolds $M$ and $N$ are (unoriented) cobordant iff their Stiefel-Whitney numbers agree: for any partition $i_1 + \dotsb + i_k = n$, [M]\frown w_{i-1}(M)w_{i_2}...
• 6,496
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### Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)

Most of the main results needed for this calculation can be found in Wall's paper "Determination of the oriented cobordism ring", but this note by Gwynne might be helpful to express this in ...
• 48k
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### Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney ...
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### Cobordism and Kirby calculus

As Golla pointed out that since every smooth $4$-manifold has a handle decomposition, you can draw a Kirby diagram. See the following pretty nice picture from Akbulut's lecture notes (now it is a ...
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### Künneth formulas/theorem for bordism groups and cobordisms?

The Künneth formula for ordinary homology as you present it works only when $R$ is a PID (or more generally of cohomological dimension 1). For a general well-behaved homology theory[1] (this includes ...
• 15.6k