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Tuesday, October 25, 2016

Basic topological constructions

 Preliminary exam prep

Let X,Y be topological spaces based at x0,y0, respectively, and I=[0,1] the unit interval.
coneCX=X×I/X×{0}suspensionΣX=X×I/X×{0},X×{1}reduced suspension˜ΣX=X×I/X×{0},X×{0},{x0}×IwedgeXY=XY/{x0}{y0}smashXY=X×Y/X×{y0},{x0}×YjoinXY=X×Y×I/X×{y}×{0} yY{x}×Y×{1} xXconnected sumX#Y=(XDnX)(YDnY)/DnXDnY
In the last description, X and Y are assumed to be n-manifolds, with DnX a closed n-dimensional disk in X (similarly for Y). The quotient identification may also be made via some non-trivial map. In fact, only the interior of each n-disk is removed from X and Y, so that the quotient makes sense.

Remark: Some of the above constructions may be expressed in terms of others, for example
XY=X×Y/XY,XY=Σ(XY).
The first is clear by viewing X=X×{y0} and Y={x0}×Y as sitting inside X×Y. The second is clear by letting X×{y}×{0} be identified to {x0}×{y}×{0} for every yY, and analogously with Y.

Example:
Here are some of the constructions above applied to some common spaces.
CXptΣSn=Sn+1SnSm=Sn+mΣX=S1XSnSm=Sn+m+1
Remark: We may also calculate the homology of the new spaces in terms of the old ones.
˜Hk(CX)=0via homotopy˜Hk(ΣX)=˜Hk1(X)via Mayer--Vietoris˜Hk(XY)=˜Hk(X)˜Hk(Y)via Mayer--Vietoris˜Hk(XS)=˜Hk(X)via Kunneth˜Hk(X#Y)=˜Hk(X)˜Hk(Y)via Mayer--Vietoris and relative homology
The last equality holds for k<n1, for M and N both n-manifolds, and for k=n1 when at least one of them is orientable.

References: Hatcher (Algebraic Topology, Chapters 0, 2)

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