Yeah, sorry to say, it certainly does look like rotten ol' crabgrass to me too.

And, unfortunately, once it's started to emerge (grow), normally from around mid to late June here in zone 7A (CG's emergence varies depending upon the region/climate it is growing in). CG can and will quickly take over and can ruin and even kill an entire lawn.

If the CG is spread all around in many different large areas of your property than it probably did NOT just come from only your neighbors lawn or soil, it was probably already in your soil and also under your neighbors soil as well and was just dormant and playing coy until the right combination of nutrients, temp, humidity and time of year caused it to emerge!

I feel for you, CG also emerged in my yard too, three summers ago, after never seeing it for well over 20 years, the outbreak of it here was real bad, the CG quickly took over and killed my entire KBG/Rye/Fescue mixed lawn that I has just spent the previous two years growing and working on!

I ended up learning a LOT about CG from reading many blogs, forums, lawn care pages and library books on the stuff, and lots on how to fight/kill/prevent it, I can gladly pass you some info and tips I've learned, if you don't mind reading a lot of info and tips that is, lol..

Once CG has emerged, like what we see in your photo above, I believe there are only liquid CG killers on the market out there to kill it with, stuff like Spectracide brand's ready to use and their concentrated Lawn Weed Killer with CG killer, they make two types, one does NOT contain the CG killer, but both are safe to use on lawns. That's if you wanted to kill the CG now.

None of the many powdered or slow-release pellet products out there meant for spreading over a lawn for feeding/weed killing/weed preventing/insect killing can kill CG! At least none that I have ever seen, read about, come across or tried to use.

Keep CG cut very short so that it can't take over your lawn, and also so that It never gets a chance to grow it's seed pods and start dropping it's seeds everywhere.

CG typically dies out by the end of Summer (late August to mid September here) and it's seeds go dormant again down in the soil until next Spring/Summer when it will re-emerge if a preventive product is not used.

When it does finally die off you will most likely have a few, or many bare spots where it was to deal with!

You will have to wait for the cooler temps to rake up and reseed those bare spots, since seed doesn't grow when temps are too warm and waiting for your existing grass to fill in the spots can take way too long, if it even happens before any new weeds start popping up in them bare spots or Winter hits.

You'll want to deeply water ONLY over the newly seeded spots each day for at least 2 to 3 weeks as the new seeds begin to grow out, than cut the watering back to every other day for a week or so once it's grown to about two inches high, than to watering three days apart for about a week or so, and so on, cutting the watering day back till you are only watering at 1 inch, and only once per week. The rest of an existing lawn should only get about 1 inch of water only once per week. That will help your roots develop nice and strong which will help crowd out any weeds, and it will also train your lawn to become more drought tolerant and will keep moss, mold, fungus and mildew away.

Try to NEVER water or mow a lawn during the hottest hours of the day, and never mow a wet or very damp lawn, as it can cause severe burns as it dries out, plus the wet clumps can choke out a mower, the lawn and cause too thick of a thatch.

Wait till the new seeds have grown to about 3 inches tall before giving it it's first mowing, and than wait till after it has been mowed at least four to five times before feeding your lawn.

Scotts brand, believe it or not, actually makes a pretty decent CG pre-emergent weed and feed product, the one in the blue bag with their "Haltz CG preventer" in it, which claims it can prevent CG for about 3 to 4 months and kills many other weeds too. I have been putting that down every early spring (Late March to mid April here) for the last three years now and I can say that it really DOES work. I never see any CG pop up in June or July anymore, even though every (lazy/cheap) neighbor around me has it in large abundances each Summer, (and I seriously hope it never comes back), it also helps feed and green my lawn back up and gets it kick started again after the Winter.

I also apply a Grub killer/preventer a few weeks later in mid to late spring, that keeps my lawn safe from them for the entire year. And I also feed my lawn with another Scott's weed and feed product right before the hotter Summer temps come rolling in, I use the one in the yellow bag with the Dandelion preventer/killer in it, I get that down around the mid to end of June to help feed the lawn for the upcoming summers and I can tell you that my grass stays nice and green and weed/Dandelion-free all summer long now.

Once the Summer is over, I top dress with "Builders Sand" (play sand and river sand are too smooth and not that good for use in top dressing) mixed with "Composted Top Soil" to fill in any low spots and level the yard off, and I'll also over seed in the early fall (mid to late Sept.) and than I wait about three to four weeks and get my lawn fed one last time to ready it for the coming Winter.

I have read countless times over the last couple of years about how much better seeding/reseeding/over seeding/feeding in the Fall is for your lawn (so is planting trees/shrubs/bulbs). Ever since I read all about that stuff I've only been over seeding and planting in Autumns now and my lawn and Gardens seems to look much better each spring. For an example, last Fall (2014) ended very early here, about 4 or 5 weeks early, snow and winter hit us earlier than normal, my seeds were barely starting to emerge in the few bare spots that I had when it all got frozen over, I thought that was it, it's all going to die off now, it's ruined and I'll have to start all over again next year, how wrong I was! lol, When the Spring thaw came this year it all started to grow right back up again, nice and thick and filled in all the spots that I had last Fall.

I've also read that you should only aerate a lawn once every 4 or 5 years, and only if you plan to re-seed or heavily over-seed a lawn, otherwise don't ever aerate.

You should only de-thatch a lawn if and when your thatch is an inch thick or more, or if it is showing signs of mold, fungus or mildew, otherwise it is fine to leave it and that it is also beneficial to leave a thatch which is less than one inch thick as it's decay helps feed the soil which feeds the lawn and also that the thatch helps keep some moisture on/in the soil and also protects the grass roots from heat and Sun.

Anyway, I hope some of this helps you out (or at least somebody out there) and that it wasn't too much to have to read.

Good Luck!