Pluag topology yog hom topology uas kawm txog qhov txawv ntawm cov manifolds thiab cov duab kos muaj ntau yam. Nrog kev nce qib ntawm algebraic topology thiab qhov txawv ntawm geometry, nws rov tshwm sim rau xyoo 1930. H. Whitney tau muab cov ntsiab lus dav dav ntawm kev sib txawv ntawm qhov txawv txav thaum xyoo 1935 thiab ua pov thawj tias nws tuaj yeem ib txwm nyob rau qhov chaw siab Euclidean. Txhawm rau kawm txog vector teb ntawm qhov sib txawv ntawm manifold, nws kuj tau npaj lub tswvyim ntawm cov khoom siv fiber ntau, yog li muaj ntau qhov teeb meem geometric cuam tshuam nrog homology (chav kawm cim) thiab teeb meem homotopy
Xyoo 1953, Rene Thom' s lub tswv yim kev hloov chaw tsim ib qho xwm txheej qhov twg cov topology sab saum toj thiab algebraic topology tau nce mus rau ib sab. Ntau yam teeb meem nyuaj ntawm qhov teeb meem topology tau hloov mus ua algebraic topology teeb meem thiab daws, uas kuj tseem txhawb algebraic topology. Kev tsim kho txuas ntxiv. Xyoo 1956, Milno nrhiav pom tias ntxiv rau cov qauv kev sib txawv ntawm lub ntsej muag uas muaj xya qhov ntev, kuj tseem muaj qhov tsis xws txawv qhov tsis xws luag. Tom qab, cov manifolds uas tsis tuaj yeem tsim ib qho qauv txawv txawv tau tsim los ntawm tib neeg. Tag nrho cov no qhia tau hais tias peb pawg ntawm topological manifolds, txawv plawg manifolds, thiab piecewise linear manifolds nyob rau hauv nruab nrab muaj qhov sib txawv loj, cov ntsiab lus topology tau suav txij li tau lees paub tias yog ib qho kev ywj pheej ntawm topology. Xyoo 1960, Smail muaj pov thawj raws li Poincaré conjecture rau cov khoom sib txawv ntawm cov neeg muaj ntau dua tsib qhov ntev. JW Milno li al. tsim ib txoj kev yooj yim los daws qhov txawv ntawm qhov sib txawv if ─ 剜 讓 擜 so, kom qhov kev faib tawm ntawm manifolds nrog ntau dua tsib qhov ntev tau maj mam ua algebraic.
Cov chaw muaj txiaj ntsig zoo yog kev sib raug zoo ntawm cov saum toj no peb qeb ntawm manifolds thiab kev faib tawm ntawm peb-seem thiab plaub-sab. Cov kev ua tiav loj hauv thaum xyoo 1980s muaj cov ntawv pov thawj ntawm plaub feem ntawm Poincaré conjecture thiab kev tshawb pom ntawm cov txheej txheem txawv txawv hauv plaub-seem Euclidean chaw. Qhov kev tshawb nrhiav no feem ntau hu ua geology topology txhawm rau hais txog nws cov xim geometric, uas yog qhov txawv ntawm cov lej algebraic homotopy theory.